An important class of DWFs can be constructed as a linear cascade chain of digital waveguide sections, as shown in Fig.C.24 (a slightly abstracted version of Fig.C.19). Each block labeled stands for a state-less Kelly-Lochbaum scattering junction (§C.8.4) interfacing between wave impedances and . We call this a ladder waveguide filter structure. It is an exact bandlimited discrete-time model of a sequence of wave impedances , as used in the Kelly-Lochbaum piecewise-cylindrical acoustic-tube model for the vocal tract in the context of voice synthesis (Fig.6.2). The delays between scattering junctions along both the top and bottom signal paths make it possible to compute all of the scattering junctions in parallel--a property that is becoming increasingly valuable in signal-processing architectures.
To transform the DWF of Fig.C.24 to a conventional ladder digital filter structure, as used in speech modeling [300,367], we need to (1) terminate on the right with a pure reflection and (2) eliminate the delays along the top signal path. We will do this in stages so as to point out a valuable intermediate case.
Terminated Waveguide Filters
A reflecting termination occurs when the wave impedance of the next section is either zero or infinity (§C.8.1). Since Fig.C.24 is labeled for force waves, an infinite terminating impedance on the right results in
Similarly, a zero terminating impedance on the right gives
For velocity waves, the signs are interchanged. Thus, a reflectively terminated ladder digital waveguide corresponds to a final feedback with gain at the end of the ladder. Such a termination will be used below to derive conventional ladder/lattice filters.