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.