Adaptors for Wave Digital ElementsAn 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 WavesFigure F.5a illustrates a generic parallel two-port connection in terms of forces and velocities.
wave digital element ports being connected in parallel, we have the physical constraints
The derivation for the two-port case extends to the -port case without modification:
The outgoing wave variables are given by
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:
We see that multiplies and additions are required. However, by observing from Eq.(F.17) that
reflection coefficient seen at port is defined as
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:
where denotes the transmission coefficient from port to port . Starting with Eq.(F.19) and substituting Eq.(F.18) gives
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.
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
denote this parallel combination, in admittance form. Then we must have
Let's check this ``physical'' derivation against the formal definition Eq.(F.20) leading to in Eq.(F.22). Toward this goal, let
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),
F.6a illustrates a generic two-port description of the series adaptor.
where are the port impedances (attached element reference impedances). In terms of the beta parameters, the force-wave series adaptor performs the following computations:
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:
We see that we have multiplies and additions as in the parallel-adaptor case. However, we again have from Eq.(F.26) that
velocity reflection coefficient seen at port is defined as
Representing the outgoing velocity wave as the superposition of the reflected wave plus the transmitted waves from the other ports, we have
where denotes the velocity transmission coefficientvelocity!transmission coefficient from port to port . Substituting Eq.(F.29) into Eq.(F.30) yields
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.
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
denote this series combination. Then we must have, as in Eq.(F.25),
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
series adaptor, we require
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.
Wave Digital Modeling Examples
Wave Digital Elements