Wave Digital Modeling Examples
This section presents a series of examples, working up from very simple to more complicated situations.
``Piano hammer in flight''
Suppose we wish to model a situation in which a mass of size
kilograms is traveling with a constant velocity. This is an
appropriate model for a piano hammer after its key has been pressed
and before the hammer has reached the string.
Figure F.2 shows the ``wave digital mass'' derived previously.
The derivation consisted of inserting an infinitesimal
waveguideF.3 having (real) impedance
, solving for the force-wave reflectance of the mass as seen from
the waveguide, and then mapping it to the discrete time domain using
the bilinear transform.
We now need to attach the other end of the transmission line to a ``force source'' which applies a force of zero newtons to the mass. In other words, we need to terminate the line in a way that corresponds to zero force.
Let the force-wave components entering and leaving the mass
be denoted and
, respectively (i.e., we are dropping
the subscript `d' in Fig.F.2).
The physical force associated with the mass is


Figure F.8a (left portion) illustrates what we derived
by physical reasoning, and as such, it is most appropriate as a
physical model of the constant-velocity mass. However, for actual
implementation, Fig.F.8b would be more typical in
practice. This is because we can always negate the state variable
if needed to convert it from
to
. It is
very common to see final simplifications like this to maximize
efficiency.
Note that Fig.F.8b can be interpreted physically as a wave
digital spring displaced by a constant force
.
Extracting Physical Quantities
Since we are using a force-wave simulation, the state variable
(delay element output) is in units of physical force (newtons).
Specifically,
. (The physical force is, of
course, 0, while its traveling-wave components are not 0 unless the
mass is at rest.) Using the fundamental relations relating traveling
force and velocity waves

where here, it is easy to convert the state variable
to
other physical units, as we now demonstrate.
The velocity of the mass, for example, is given by



The kinetic energy of the mass is given by



Force Driving a Mass
Suppose now that we wish to drive the mass along a frictionless
surface using a variable force . This is similar to the
previous example, except that we now want the traveling-wave
components of the force on the mass to sum to
instead of 0:





The simplified form in Fig.F.9b can be interpreted as a wave
digital spring with applied force delivered from an infinite
source impedance. That is, when the applied force goes to zero, the
termination remains rigid at the current displacement.
A More Formal Derivation of the Wave Digital Force-Driven Mass
Above we derived how to handle the external force by direct physical reasoning. In this section, we'll derive it using a more general step-by-step procedure which can be applied systematically to more complicated situations.
Figure F.10 gives the physical picture of a free mass driven by
an external force in one dimension. Figure F.11 shows the
electrical equivalent circuit for this scenario in which the external
force is represented by a voltage source emitting volts,
and the mass is modeled by an inductor having the value
Henrys.
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The next step is to convert the voltages and currents in the
electrical equivalent circuit to wave variables.
Figure F.12 gives an intermediate equivalent circuit in which
an infinitesimal transmission line section with real impedance
has been inserted to facilitate the computation of the wave-variable
reflectance, as we did in §F.1.1 to derive Eq.
(F.1).
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Figure F.13 depicts a next intermediate equivalent circuit in
which the mass has been replaced by its reflectance (using ``''
to denote the continuous-time reflectance
, as derived in
§F.1.1). The infinitesimal transmission-line section is now represented
by a ``resistor'' since, when the voltage source is initially
``switched on'', it only ``sees'' a real resistance having the value
Ohms (the waveguide interface). After a short period of time
determined by the reflectance of the mass,F.4 ``return waves'' from the mass result in an ultimately
reactive impedance. This of course must be the case because the
mass does not dissipate energy. Therefore, the ``resistor'' of
Ohms is not a resistor in the usual sense since it does not convert
potential energy (the voltage drop across it) into heat. Instead, it
converts potential energy into propagating waves with 100%
efficiency. Since all of this wave energy is ultimately reflected by
the terminating element (mass, spring, or any combination of masses
and springs), the net effect is a purely reactive impedance, as we
know it must be.
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To complete the wave digital model, we need to connect our wave
digital mass to an ideal force source which asserts the value
each sample time. Since an ideal force source has a zero internal
impedance, we desire a parallel two-port junction which connects the
impedances
(
) and
(
), as
shown in Fig.F.14. From
Eq.
(F.18) we have that the common junction force is equal to

from which we conclude that



Since
and
for this model, the reflection
coefficient seen on port 1 is
. The
transmission coefficient from port 1 is
. In the opposite
direction, the reflection coefficient on port 2 is
, and
the transmission coefficient from port 2 is
. The final
result, drawn in Kelly-Lochbaum form (see §F.2.1), is
diagrammed in Fig.F.15, as well as the result of some
elementary simplifications. The final model is the same as in
Fig.F.9, as it should be.
Force Driving a Spring against a Wall
For this example, we have an external force driving a spring
which is terminated on the other end at a rigid wall.
Figure F.16 shows the physical diagram
and the electrical equivalent circuit is given in
Fig.F.17.
Figure F.18 depicts the insertion of an infinitesimal transmission line, and Fig.F.19 shows the result of converting the spring impedance to wave variable form.
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The two-port adaptor needed for this problem is the same as that for the force-driven mass, and the final result is shown in Fig.F.20.
Note that the spring model is being driven by a force from a zero source impedance, in contrast with the infinite source impedance interpretation of Fig.F.8b as a compressed spring. In this case, if the driving force goes to zero, the spring force goes immediately to zero (``free termination'') rather than remaining fixed.
Spring and Free Mass
For this example, we have an external force driving a spring
which in turn drives a free mass
. Since the force on the
spring and the mass are always the same, they are formally
``parallel'' impedances.
This problem is easier than it may first appear since an ideal ``force source'' (i.e., one with a zero source impedance) driving impedances in parallel can be analyzed separately for each parallel branch. That is, the system is equivalent to one in which the mass and spring are not connected at all, and each has its own copy of the force source. With this insight in mind, one can immediately write down the final wave-digital model shown in Fig.F.25. However, we will go ahead and analyze this case more formally since it has some interesting features.
Figure F.21 shows the physical diagram of the spring-mass system driven by an external force. The electrical equivalent circuit appears in Fig.F.22, and the first stage of a wave-variable conversion is shown in Fig.F.23.
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For this example we need a three-port parallel adaptor, as shown in
Fig.F.24 (along with its attached mass and spring).
The port impedances are 0, , and
, yielding alpha parameters
and
. The final result, after the
same sorts of elementary simplifications as in the previous example,
is shown in Fig.F.25. As predicted, a force source driving
elements in parallel is equivalent to a set of independent
force-driven elements.
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From this and the preceding example, we can see a general pattern:
Whenever there is an ideal force source driving a parallel
junction, then
and all other port admittances are
finite. In this case, we always obtain
and
,
.
Mass and Dashpot in Series
This is our first example illustrating a series connection of wave digital elements. Figure F.26 gives the physical scenario of a simple mass-dashpot system, and Fig.F.27 shows the equivalent circuit. Replacing element voltages and currents in the equivalent circuit by wave variables in an infinitesimal waveguides produces Fig.F.28.
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The system can be described as an ideal force source connected
in parallel with the series connection of mass
and
dashpot
.
Figure F.29 illustrates the resulting wave digital filter.
Note that the ports are now numbered for reference. Two more symbols
are introduced in this figure: (1) the horizontal line with a dot in
the middle indicating a series adaptor, and (2) the indication of a
reflection-free port on input 1 of the series adaptor (signal
). Recall that a reflection-free port is always necessary
when connecting two adaptors together, to avoid creating a delay-free
loop.
Let's first calculate the impedance necessary to make input 1 of
the series adaptor reflection free. From Eq.
(F.37), we require

The parallel adaptor, viewed alone, is equivalent to a force source
driving impedance . It is therefore realizable as in
Fig.F.20 with the wave digital spring replaced by the
mass-dashpot assembly in
Fig.F.29. However, we can also carry out a quick analysis
to verify this: The alpha parameters are

Therefore, the reflection coefficient seen at port 1 of the parallel
adaptor is
, and the Kelly-Lochbaum scattering
junction depicted in Fig.F.20 is verified.
Let's now calculate the internals of the series adaptor in
Fig.F.29. From Eq.(F.26), the beta parameters are

Following Eq.(F.30), the series adaptor computes

We do not need to explicitly compute
because it goes into a
purely resistive impedance
and produces no return wave. For the
same reason,
.
Figure F.30 shows a wave flow diagram of the computations derived,
together with the result of elementary simplifications.
![]() |
Because the difference of the two coefficients in Fig.F.30 is 1, we can easily derive the one-multiply form in Fig.F.31.
Checking the WDF against the Analog Equivalent Circuit
Let's check our result by comparing the transfer function from the input force to the force on the mass in both the discrete- and continuous-time cases.
For the discrete-time case, we have



We now need
.
To simplify notation, define the two coefficients as

From Figure F.30, we can write
![\begin{eqnarray*}
F^{-}_3(z) &=& -a\left[F(z)-z^{-1}F^{-}_3(z)\right] + b\left[-...
...\,\,\quad
F^{-}_3(z) &=& -a\frac{F(z)}{1-(a-b)z^{-1}F^{-}_3(z)}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4987.png)
Thus, the desired transfer function is

We now wish to compare this result to the bilinear transform of the corresponding analog transfer function. From Figure F.27, we can recognize the mass and dashpot as voltage divider:


Thus, we have verified that the force transfer-function from the driving force to the mass is identical in the discrete- and continuous-time models, except for the bilinear transform frequency warping in the discrete-time case.
Wave Digital Mass-Spring Oscillator
Let's look again at the mass-spring oscillator of §F.3.4, but this time without the driving force (which effectively decouples the mass and spring into separate first-order systems). The physical diagram and equivalent circuit are shown in Fig.F.32 and Fig.F.33, respectively.
Note that the mass and spring can be regarded as being in parallel or
in series. Under the parallel interpretation, we have the WDF shown
in Fig.F.34 and Fig.F.35.F.5
The reflection coefficient can be computed, as usual, from
the first alpha parameter:


Oscillation Frequency
From Fig.F.33, we can see that the impedance of the parallel combination of the mass and spring is given by
(using the product-over-sum rule for combining impedances in parallel). The poles of this impedance are given by the roots of the denominator polynomial in

The resonance frequency of the mass-spring oscillator is therefore
Since the poles


We can now write reflection coefficient (see Fig.F.35) as





DC Analysis of the WD Mass-Spring Oscillator
Considering the dc case first (), we see from Fig.F.35
that the state variable
will circulate unchanged in the
isolated loop on the left. Let's call this value
. Then the physical force on the spring is always equal to
The loop on the right in Fig.F.35 receives





![$ x_2=[0, 2x_0, 4x_0, 6x_0,\ldots]$](http://www.dsprelated.com/josimages_new/pasp/img5012.png)
At first, this result might appear to contradict conservation of energy, since the state amplitude seems to be growing without bound. However, the physical force is fortunately better behaved:
Since the spring and mass are connected in parallel, it must be the true that they are subjected to the same physical force at all times. Comparing Equations (F.41-F.43) verifies this to be the case.
WD Mass-Spring Oscillator at Half the Sampling Rate
Under the bilinear transform, the maps to
(half the
sampling rate). It is therefore no surprise that given
(
), inspection of Fig.F.35 reveals
that any alternating sequence (sinusoid sampled at half the sampling
rate) will circulate unchanged in the loop on the right, which is now
isolated. Let
denote this alternating sequence.
The loop on the left receives
and adds
to
it, i.e.,
.
If we start out with
and
, we obtain
, or



which agrees with the spring, as it must.
Linearly Growing State Variables in WD Mass-Spring Oscillator
It may seem disturbing that such a simple, passive, physically
rigorous simulation of a mass-spring oscillator should have to make
use of state variables which grow without bound for the limiting cases
of simple harmonic motion at frequencies zero and half the sampling
rate. This is obviously a valid concern in practice as well.
However, it is easy to show that this only happens at dc and ,
and that there is a true degeneracy at these frequencies, even in the
physics. For all frequencies in the audio range (e.g., for typical
sampling rates), such state variable growth cannot occur. Let's take
closer look at this phenomenon, first from a signal processing point
of view, and second from a physical point of view.
A Signal Processing Perspective on Repeated Mass-Spring Poles
Going back to the poles of the mass-spring system in Eq.(F.39),
we see that, as the imaginary part of the two poles,
, approach zero, they come together at
to create a
repeated pole. The same thing happens at
since
both poles go to ``the point at infinity''.
It is a well known fact from linear systems theory that two poles at
the same point
in the
plane can correspond to an
impulse-response component of the form
, in addition
to the component
produced by a single pole at
. In the discrete-time case, a double pole at
can
give rise to an impulse-response component of the form
.
This is the fundamental source of the linearly growing internal states
of the wave digital sine oscillator at dc and
. It is
interesting to note, however, that such modes are always
unobservable at any physical output such as the mass
force or spring force that is not actually linearly growing.
Physical Perspective on Repeated Poles in the Mass-Spring System
In the physical system, dc and infinite frequency are in fact strange
cases. In the case of dc, for example, a nonzero constant force
implies that the mass is under constant acceleration. It is
therefore the case that its velocity is linearly growing. Our
simulation predicts this, since, using
Eq.
(F.43) and Eq.
(F.42),
![\begin{eqnarray*}
v_m(n) &=& \frac{f^{{+}}_m(n)}{m} - \frac{f^{{-}}_m(n)}{m}
=...
...m} \left[2(n+1) + 2n\right]x_0
= \frac{1}{m} (4 n x_0 + 2 x_0).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img5032.png)
The dc term is therefore accompanied by a linearly growing
term
in the physical mass velocity. It is therefore
unavoidable that we have some means of producing an unbounded,
linearly growing output variable.
Mass-Spring Boundedness in Reality
To approach the limit of
, we must either
take the spring constant
to zero, or the mass
to infinity, or
both.
In the case of , the constant force must approach zero, and we
are left with at most a constant mass velocity in the limit (not a
linearly growing one, since there can be no dc force at the limit).
When the spring force reaches zero,
, so that only zeros
will feed into the loop on the right in Fig.F.35, thus avoiding
a linearly growing velocity, as demanded by the physics. (A constant
velocity is free to circulate in the loop on the right, but the loop
on the left must be zeroed out in the limit.)
In the case of
, the mass becomes unaffected by the spring
force, so its final velocity must be zero. Otherwise, the attached
spring would keep compressing or stretching forever, and this would
take infinite energy. (Another way to arrive at this conclusion is to
note that the final kinetic energy of the mass would be
.) Since the total energy in an undriven mass-spring
oscillator is always constant, the infinite-mass limit must be
accompanied by a zero-velocity limit.F.6 This means the mass's
state variable
in Fig.F.35 must be forced to zero in
the limit so that there will be no linearly growing solution at dc.
In summary, when two or more system poles approach each other to form
a repeated pole, care must be taken to ensure that the limit is
approached in a physically meaningful way. In the case of the
mass-spring oscillator, for example, any change in the spring constant
or mass
must be accompanied by the physically appropriate
change in the state variables
and/or
. It is
obviously incorrect, for example, to suddenly set
in the
simulation without simultaneously clearing the spring's state variable
, since the force across an infinitely compliant spring can
only be zero.
Similar remarks apply to repeated poles corresponding to
. In this case, the mass and spring basically change
places.
Energy-Preserving Parameter Changes (Mass-Spring Oscillator)
If the change in or
is deemed to be ``internal'', that is,
involving no external interactions, the appropriate accompanying
change in the internal state variables is that which conserves
energy. For the mass and its velocity, for example, we must have





If the spring constant is to change from
to
, the
instantaneous spring displacement
must satisfy

Exercises in Wave Digital Modeling
- Comparing digital and analog frequency formulas.
This first exercise verifies that the elementary ``tank circuit''
always resonates at exactly the frequency it should, according to the
bilinear transform frequency mapping
, where
denotes ``analog frequency'' and
denotes ``digital frequency''.
- Find the poles of Fig.F.35 in terms of
.
- Show that the resonance frequency is given by
denotes the sampling rate.
- Recall that the mass-spring oscillator resonates at
. Relate these two resonance frequency formulas via the analog-digital frequency map
.
- Show that the trig identity you discovered in this way is true.
I.e., show that
- Find the poles of Fig.F.35 in terms of
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Adaptors for Wave Digital Elements