# Wave Digital Filters

A *Wave Digital Filter* (WDF) [136] is a particular
kind of *digital filter* based on physical modeling principles.
Unlike most digital filter types, every delay element in a WDF can be
interpreted *physically* as holding the current state of a mass
or spring (or capacitor or inductor). WDFs can also be viewed as a
particular kind of *finite difference scheme* having unusually
good numerical properties [55]. (See Appendix D for an
introduction to finite difference schemes.) WDFs have been applied
often in music signal processing
[394,338,556,361,348,554,555,56,527,523,484].

Wave digital filters were developed initially by Alfred Fettweis
[135] in the late 1960s for digitizing lumped
*electrical* circuits composed of inductors, capacitors,
resistors, transformers, gyrators, circulators, and other elements of
classical network theory [136]. The WDF approach is
based on the traveling-wave formulation of lumped electrical elements
introduced by Belevitch [34].

A WDF is constructed by interconnecting simple discrete-time models of
individual masses, springs, and dashpots (or inductors, capacitors,
and resistors). The rules for interconnecting the elementary models
are based on *scattering theory* (discussed in §C.8).
This is a direct result of the fact that all signals explicitly
computed may be physically interpreted as *traveling wave
components* of physical variables.

## 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 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.)

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 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).

- 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

#### 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

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

Thus,

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:

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 Farads, the driving-point impedance is (see §7.1.3)

#### Inductor Reflectance

For an inductor of Henrys, we have

#### Resistor Reflectance

Finally, for a resistor of 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.

#### 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 gives

- For the inductor, setting gives

- And for the resistor, we set to obtain

#### Digitizing Elementary Reflectances by Bilinear Transform

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

(F.11) |

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.

### Summary of Wave Digital Elements

From Eq.(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.

### Wave Digital Mass

In the case of a mass , we have

*i.e.*, , the bilinear transform of is immediately seen to be

*difference equation*for the wave digital mass is

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 , we have the impedance

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 , we have

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 . 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.

### Limiting Cases

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

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

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).

## Adaptors for Wave Digital Elements

An *adaptor* is an -port memoryless interface which
interconnects wave digital elements. Since each element's ``port'' is
a connection to an infinitesimal waveguide section at some real wave
impedance , and since the input/output signals are wave
variables (traveling-waves within the waveguide), the adaptor must
implement *signal scattering* appropriate for the connection of
such waveguides. In other words, an -port adaptor in a wave
digital filter performs exactly the same computation as an -port
scattering junction in a digital waveguide network.^{F.2}

This section first addresses the simpler two-port case, followed by a derivation of the general -port adaptor, for both parallel and series connections of wave digital elements.

As discussed in §7.2, a physical connection of two or more
ports can either be in *parallel* (forces are equal and the
velocities sum to zero) or in *series* (velocities equal and
forces sum to zero). Combinations of parallel and series connections
are also of course possible.

### Two-Port Parallel Adaptor for Force Waves

Figure F.5a illustrates a generic parallel two-port connection in terms of forces and velocities.

As discussed in §7.2, a *parallel connection* is
characterized by a common force and velocities which sum to zero:

Following the same derivation leading to Eq.(F.2), and defining for notational convenience, we obtain

The outgoing wave variables are given by

Defining the reflection coefficient as

as diagrammed in Fig.F.5b. This can be called the
*Kelly-Lochbaum* implementation of the two-port force-wave
adaptor.

Now that we have a proper scattering interface between two reference impedances, we may connect two wave digital elements together, setting to the port impedance of element 1, and to the port impedance of element 2. An example is shown in Fig.F.35.

The Kelly-Lochbaum adaptor in Fig.F.5b evidently requires four multiplies and two additions. Note that we can factor out the reflection coefficient in each equation to obtain

which requires only one multiplication and three additions. This can
be called the *one-multiply* form. The one-multiply form is most
efficient in custom VLSI. The Kelly-Lochbaum form, on the other hand,
may be more efficient in software, and slightly faster (by one
addition) in parallel hardware.

#### Compatible Port Connections

*Note carefully* that to connect a wave digital element to port
of the adaptor, we route the signal
coming out of the
element to become
on the adaptor port, and the signal
coming out of port of the adaptor goes into the element
as
. Such a connection is said to be a
*compatible port connection*. In other words, the connections
must be made such that the arrows go in the same direction in the wave
flow diagram.

### General Parallel Adaptor for Force Waves

In the more general case of wave digital element ports being
connected in parallel, we have the physical constraints

(F.14) | |||

(F.15) |

The derivation for the two-port case extends to the -port case without modification:

The outgoing wave variables are given by

#### Alpha Parameters

It is customary in the wave digital filter literature to define the
*alpha parameters* as

where are the admittances of the wave digital element interfaces (or ``reference admittances,'' in WDF terminology). In terms of the alpha parameters, the force-wave parallel adaptor performs the following computations:

We see that multiplies and additions are required. However, by observing from Eq.(F.17) that

*dependent port*.

#### Reflection Coefficient, Parallel Case

The *reflection coefficient* seen at port is defined as

In other words, the reflection coefficient specifies what portion of the incoming wave is

*reflected*back to port as part of the outgoing wave . The total outgoing wave on port is the

*superposition*of the reflected wave and the transmitted waves from the other ports:

where denotes the

*transmission coefficient*from port to port . Starting with Eq.(F.19) and substituting Eq.(F.18) gives

Equating like terms with Eq.(F.21), we obtain

Thus, the th alpha parameter is the force transmission coefficient from th port to any other port (besides the th). To convert the transmission coefficient from the th port to the reflection coefficient for that port, we simply subtract 1. This general relationship is specific to force waves at a parallel junction, as we will soon see.

#### Physical Derivation of Reflection Coefficient

Physically, the reflection coefficient seen at port is due to an
impedance step from , that of the port interface, to a new
impedance consisting of the parallel combination of all *other*
port impedances meeting at the junction. Let

denote this parallel combination, in admittance form. Then we must have

Let's check this ``physical'' derivation against the formal definition Eq.(F.20) leading to in Eq.(F.22). Toward this goal, let

*all*admittances connected to the junction. Then by Eq.(F.24), we have for all . Now, from Eq.(F.17),

and the result is verified.

#### Reflection Free Port

It is useful in practice, such as when connecting two adaptors
together, to make one port *reflection free*. A
*reflection-free port* is defined to have a zero reflection coefficient. For port
of a parallel adaptor to be reflection free, we must have, from
Eq.(F.25),

Connecting two adaptors at a reflection-free port prevents the
formation of a *delay-free loop* which would otherwise occur
[136]. As a result, multi-port junctions can be joined
without having to insert unit elements (see §F.1.7) to avoid
creating delay-free loops. Only one of the two ports participating in
the connection needs to be reflection free.

We can always make a reflection-free port at the connection of two adaptors because the ports used for this connection (one on each adaptor) were created only for purposes of this connection. They can be set to any impedance, and only one of them needs to be reflection free.

To interconnect three adaptors, labeled , , and , we may
proceed as follows: Let be augmented with *two* unconstrained
ports, having impedances and . Add a reflection-free
port to , and suppose its impedance has to be . Add a
reflection-free port to , and suppose its impedance has to be
. Now set and connect to via the
corresponding ports. Similarly, set and connect to
accordingly. This adaptor-connection protocol clearly extends to any
number of adaptors.

### Two-Port Series Adaptor for Force Waves

Figure F.6a illustrates a generic two-port
description of the *series* adaptor.

As discussed in §7.2, a *series connection* is
characterized by a common velocity and forces which sum to zero at the
junction:

The derivation can proceed exactly as for the parallel junction in
§F.2.1, but with force and velocity interchanged, *i.e.*,
, and with impedance and admittance interchanged,
*i.e.*,
. In this way, we may take the
*dual* of Eq.(F.14) to get

diagrammed in Fig.F.7. Converting back to force wave variables via and , and noting that , we obtain, finally,

as diagrammed in Fig.F.6b. The one-multiply form is now

### General Series Adaptor for Force Waves

In the more general case of ports being connected in
*series*, we have the physical constraints

The derivation is the dual of that in the parallel case (cf.
Eq.(F.16)), *i.e.*, force and velocity are interchanged, and impedance
and admittance are interchanged:

The outgoing wave variables are given by

#### Beta Parameters

It is customary in the wave digital filter literature to define the
*beta parameters*
as

where are the port impedances (attached element reference impedances). In terms of the beta parameters, the force-wave series adaptor performs the following computations:

However, we normally employ a mixture of parallel and series adaptors,
while keeping a force-wave simulation. Since
, we obtain, after a small amount of algebra, the following
recipe for the series force-wave adaptor:

We see that we have multiplies and additions as in the parallel-adaptor case. However, we again have from Eq.(F.26) that

*dependent port*.

#### Reflection Coefficient, Series Case

The *velocity reflection coefficient* seen at port
is defined as

Representing the outgoing velocity wave as the superposition of the reflected wave plus the transmitted waves from the other ports, we have

where denotes the

*velocity transmission coefficient*

*velocity!transmission coefficient*from port to port . Substituting Eq.(F.29) into Eq.(F.30) yields

Equating like terms with Eq.(F.32) gives

Thus, the th beta parameter is the velocity transmission coefficient from th port to any other port (besides the th). To convert the transmission coefficient from the th port to the reflection coefficient for that port, we simply subtract 1. These relationships are specific to velocity waves at a series junction (cf. Eq.(F.22)). They are exactly the dual of Equations (F.22-F.23) for force waves at a parallel junction.

#### Physical Derivation of Series Reflection Coefficient

Physically, the *force-wave* reflection coefficient seen at port
of a series adaptor is due to an impedance step from , that
of the port interface, to a new impedance consisting of the series
combination of all *other* port impedances meeting at the
junction. Let

denote this series combination. Then we must have, as in Eq.(F.25),

(F.36) |

Let's check this ``physical'' derivation against the formal definition Eq.(F.31) leading to in Eq.(F.33). Define the total junction impedance as

*all*impedances connected to the junction. Then by Eq.(F.35), for all . From Eq.(F.26), the

*velocity*reflection coefficient is given by

Since

#### Series Reflection Free Port

For port to be reflection free in a series adaptor, we require

That is, the port's impedance must equal the series combination of the other port impedances at the junction. This result can be compared with that for the parallel junction in §F.2.2.

The series adaptor has now been derived in a way which emphasizes its duality with respect to the parallel adaptor.

## 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
waveguide^{F.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

*I.e.*, the square of the state variable can be scaled by to produce the physical kinetic energy associated with the mass.

### 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*.

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).

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.

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.

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.

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.

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.

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

*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 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 are on the axis, there is no damping, as we expect.

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 and adds to that. Since , we see it is

*linearly growing*in amplitude. For example, if (with ), we obtain , or

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),

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

*momentum*of the mass

*is*changed, however, since

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
- 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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