This section presents a series of examples, working up from very
simple to more complicated situations.
Suppose we wish to model a situation in which a mass
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.
shows the ``wave digital mass
'' derived previously.
The derivation consisted of inserting an infinitesimal
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
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
, respectively (i.e.
, we are dropping
the subscript `d' in Fig.F.2
The physical force associated with the mass is
The zero-force case is therefore obtained when
. This is illustrated in Fig.F.8
Wave digital mass in flight at a constant velocity.
a (left portion) illustrates what we derived
by physical reasoning, and as such, it is most appropriate as a
of the constant-velocity mass. However, for actual
b would be more typical in
practice. This is because we can always negate the state variable
if needed to convert it from
. It is
very common to see final simplifications like this to maximize
Note that Fig.F.8
b can be interpreted physically as a wave
digital spring displaced
by a constant force
Since we are using a force-wave
simulation, the state variable
(delay element output) is in units of physical force
. (The physical force is, of
course, 0, while its traveling-wave
components are not 0 unless the
is at rest.) Using the fundamental relations relating traveling
force and velocity
here, it is easy to convert the state variable
other physical units, as we now demonstrate.
The velocity of the mass, for example, is given by
Thus, the state variable
can be scaled by
to ``read out''
the mass velocity.
The kinetic energy
of the mass is given by
, the square of the state variable
can be scaled by
to produce the physical kinetic energy associated with the mass.
Suppose now that we wish to drive
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:
must be computed as
. This is shown in Fig.F.9
The simplified form in Fig.F.9
b can be interpreted as a wave
with applied force
delivered from an infinite
. That is, when the applied force goes to zero, the
termination remains rigid at the current displacement
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
gives the physical picture of a free mass
an external force in one dimension. Figure F.11
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
Physical diagram of an external force driving a mass
sliding on a frictionless surface.
Electrical equivalent circuit of the force-driven mass in Fig.F.10.
The next step is to convert the voltages and currents in the
electrical equivalent circuit to wave variables
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
, as we did in §F.1.1
to derive Eq.
Intermediate equivalent circuit for the
force-driven mass in which an infinitesimal transmission line section
has been inserted to facilitate conversion of the mass impedance
into a wave-variable reflectance.
Intermediate wave-variable model of the
force-driven mass of Fig.F.11.
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
). 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
interface). After a short period
determined by the reflectance of the mass,F.4
``return waves'' from the mass result in an ultimately
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
(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.
Interconnection of the wave digital mass
with an ideal force source by means of a two-port parallel adaptor.
The symbol ``'' is used in the WDF literature to signify a
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
shown in Fig.F.14
) we have that the common junction force is equal to
from which we conclude that
The outgoing waves are, by Eq.
for this model, the reflection
seen on port 1 is
from port 1 is
. In the opposite
direction, the reflection coefficient
on port 2 is
the transmission coefficient from port 2 is
. The final
result, drawn in Kelly-Lochbaum
form (see §F.2.1
diagrammed in Fig.F.15
, as well as the result of some
elementary simplifications. The final model is the same as in
, as it should be.
Wave digital mass driven by external force .
For this example, we have an external force
driving a spring
which is terminated on the other end at a rigid wall.
shows the physical diagram
and the electrical equivalent circuit
is given in
External force driving a spring terminated
by a rigid wall.
Electrical equivalent circuit of the compressed
spring of Fig.F.16.
depicts the insertion of
an infinitesimal transmission line, and
shows the result of converting the spring
to wave variable
Intermediate equivalent circuit for the
force-driven spring in which an infinitesimal transmission line
section has been inserted to facilitate conversion of the spring
impedance into a wave-variable reflectance.
Intermediate wave-variable model of Fig.F.17.
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
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.8
b 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
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
This problem is easier than it may first appear since an ideal ``force
, 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
model shown in Fig.F.25
. However, we will go
ahead and analyze this case more formally since it has some
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
External force driving a spring which in turn drives a free mass sliding on a frictionless surface.
Electrical equivalent circuit of the
spring/mass system of Fig.F.21.
For this example we need a three-port parallel adaptor
, as shown in
(along with its attached mass and spring).
The port impedances are 0,
, yielding alpha parameters
. 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
Wave digital filter model of an external force
driving a mass through a spring. The mass force-wave components are
denoted , while those for the spring are denoted .
From this and the preceding example, we can see a general pattern:
Whenever there is an ideal force source driving a parallel
and all other port admittances
finite. In this case, we always obtain
This is our first example illustrating a series
elements. Figure F.26
gives the physical scenario of
a simple mass
system, and Fig.F.27
. Replacing element voltages and currents in the
equivalent circuit by wave variables
in an infinitesimal waveguides
External force driving a mass which in turn
drives a dashpot terminated on the other end by a rigid wall.
Electrical equivalent circuit of the mass and dashpot system of Fig.F.26.
The system can be described as an ideal force source
with the series
connection of mass
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
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
Let's first calculate the impedance
necessary to make input 1 of
the series adaptor reflection free. From Eq.
), we require
That is, the impedance of the reflection-free port must equal the
series combination of all other port impedances
meeting at the
The parallel adaptor
, viewed alone, is equivalent to a force source
. It is therefore realizable as in
with the wave digital spring
replaced by the
mass-dashpot assembly in
. 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
, and the Kelly-Lochbaum scattering
depicted in Fig.F.20
Let's now calculate the internals of the series adaptor in
. From Eq.
), the beta parameters are
), 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
shows a wave flow diagram of the computations derived,
together with the result of elementary simplifications.
Wave flow diagram for the WDF modeling an ideal
force source in parallel with the series combination of a mass and
Because the difference of the two coefficients in Fig.F.30
we can easily derive the one-multiply form in Fig.F.31
One-multiply form of the
WDF in Fig.F.30.
Let's check our result by comparing the transfer function
to the force on the mass
in both the discrete- and
For the discrete-time case, we have
where the last simplification comes from the mass reflectance
. (Note that we are using the standard
notation for the adaptor
, so that the
signs are swapped relative to element-centric notation.)
We now need
To simplify notation, define the two coefficients as
From Figure F.30
, we can write
Thus, the desired transfer function is
We now wish to compare this result to the bilinear transform
corresponding analog transfer function. From Figure F.27
can recognize the mass and dashpot
Applying the bilinear transform yields
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
in the discrete-time case.
Let's look again at the mass-spring
, but this time without the driving force
effectively decouples the mass
and spring into separate first-order
systems). The physical diagram and equivalent circuit
are shown in
Elementary mass-spring oscillator.
Equivalent circuit for the mass-spring oscillator.
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
The reflection coefficient
can be computed, as usual, from
the first alpha parameter:
, is just the ``impedance
sum'', so no calculation was really necessary.
Detailed wave-flow diagram for the wave digital
, we can see that the impedance
of the parallel
of the mass
is given by
(using the product-over-sum rule for combining impedances
parallel). The poles
of this impedance are given by the roots of the
denominator polynomial in
The resonance frequency of the mass-spring oscillator
Since the poles
are on the
axis, there is
no damping, as we expect.
We can now write reflection coefficient
We see that dc
) corresponds to
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
to that. Since
, we see it is
in amplitude. For example, if
), we obtain
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
) verifies this to be the case.
Under the bilinear transform
). It is therefore no surprise that given
), inspection of Fig.F.35
that any alternating sequence (sinusoid
sampled at half the sampling
rate) will circulate unchanged in the loop on the right, which is now
denote this alternating sequence.
The loop on the left receives
If we start out with
, we obtain
However, the physical spring force
is well behaved, since
As a check, the mass
force is found to be
which agrees with the spring, as it must.
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
. This is obviously a valid concern in practice as well.
However, it is easy to show that this only happens at dc
and that there is a true degeneracy at these frequencies, even in the
. For all frequencies in the audio range (e.g.
, for typical
rates), such state variable growth cannot occur. Let's take
closer look at this phenomenon, first from a signal
of view, and second from a physical point of view.
Going back to the poles
of the mass-spring
system in Eq.
we see that, as the imaginary part of the two poles,
, approach zero, they come together at
to create a
. The same thing happens at
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
plane can correspond to an
component of the form
, in addition
to the component
produced by a single pole at
. In the discrete-time case, a double pole at
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
. It is
interesting to note, however, that such modes are always
at any physical
output such as the mass
force that is not actually linearly growing.
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
simulation predicts this, since, using
) and Eq.
The dc term
is therefore accompanied by a linearly growing
in the physical mass velocity
. It is therefore
unavoidable that we have some means of producing an unbounded,
linearly growing output variable.
To approach the limit of
, we must either
take the spring
to zero, or the mass
to infinity, or
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
is always constant, the infinite-mass limit must be
accompanied by a zero-velocity limit.F.6
This means the mass's
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
must be accompanied by the physically appropriate
change in the state variables
. It is
obviously incorrect, for example, to suddenly set
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
If the change in
is deemed to be ``internal'', that is,
involving no external interactions, the appropriate accompanying
change in the internal state variables
is that which conserves
. For the mass
and its velocity
, for example, we must have
denote the mass values before and after the change,
denote the corresponding velocities.
The velocity must therefore be scaled according to
since this holds the kinetic energy
of the mass constant. Note that
of the mass is
changed, however, since
If the spring
is to change from
instantaneous spring displacement
In a velocity-wave simulation, displacement is the integral of
velocity. Therefore, the energy-conserving velocity correction is
impulsive in this case.
- 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
where 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
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