## 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

we have that the scattering relations for the two-port parallel adaptor are

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:
 0 (F.16)

The outgoing wave variables are given by

#### Alpha Parameters

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

 (F.17)

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:
 (F.18) (F.19)

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

we may implement , say, as in order to eliminate one multiply. In WDF terminology, port 1 is then a dependent port.

#### Reflection Coefficient, Parallel Case

The reflection coefficient seen at port is defined as

 (F.20)

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:

 (F.21)

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

 (F.22) (F.23)

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

 (F.24)

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

 (F.25)

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

denote the parallel combination of 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),

Thus, the port's impedance must equal the parallel combination of the other port impedances at the junction. In this case, the junction as a whole perfectly terminates'' the reflection free port, so no reflections come back from it.

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

 (F.26)

where are the port impedances (attached element reference impedances). In terms of the beta parameters, the force-wave series adaptor performs the following computations:
 (F.27) (F.28)

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:

 (F.29) (F.30)

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

so that we may implement one beta parameter as 2 minus the sum of the rest, thus eliminating a multiplication by creating a dependent port.

#### Reflection Coefficient, Series Case

The velocity reflection coefficient seen at port is defined as

 (F.31)

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

 (F.32)

where denotes the velocity transmission coefficientvelocity!transmission coefficient from port to port . Substituting Eq.(F.29) into Eq.(F.30) yields

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

 (F.33) (F.34)

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

 (F.35)

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

This is the series combination of 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

the result follows.

#### Series Reflection Free Port

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

 (F.37)

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.

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