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 as in §C.8.1, 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.

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