## 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

*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*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:

#### 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*.

*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).

^{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.

### 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.### 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.

### 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.

*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

#### 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

*voltage divider*:

### 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 toThe 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

#### 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),

*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

#### 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

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Adaptors for Wave Digital Elements