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.

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General Parallel Adaptor for Force Waves
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Unit Elements