- In order to implement the equivalent circuit of a conical waveguide using digital waveguide techniques, it is necessary to express the lumped impedance elements of Fig. 14 in terms of traveling-wave parameters and then convert these expressions to discrete-time filters.
- The impedance of the input inertance, given in terms of a Laplace transform, is
, where is the mass density of air,
*A*_{o}is the area of the spherical wavefront at the waveguide input, and*s*is Laplace transform frequency variable. - The effective impedance at the waveguide input is determined as the parallel combination of
*M*_{o}and an input load impedance. If the input is rigidly terminated, the input load is infinite and the pressure-wave reflectance is given by:

where is a real, locally defined characteristic impedance parameter. - The reflectance filter, discretized with the bilinear transform, is

and is the bilinear transform constant that controls frequency warping. - This first-order allpass filter accurately accounts for the phase delay experienced by pressure traveling-wave components reflecting from a rigid input termination in a conical waveguide.
- The output inertance,
*M*_{e}, tends to be less significant than that at the input, particularly in the presence of an open-end load impedance. In general, a single output reflectance filter can be designed based on the parallel combination of*M*_{e}and an appropriate open-end impedance characterization. - Figure 18 shows a truncated conic structure and the corresponding digital waveguide block diagram, using input and output reflectance filters as discussed above.
- The goal here is to model a conical bore instrument system by attaching a simple, memory-less, non-linear excitation mechanism to a conical air column representation.
- The traditional reed function/air column coupling, however, is derived for an input cylindrical section using a real wave impedance. It is not a simple process to re-derive the reed function using the complex wave impedance of a conic frustum. Even if we ignored this complication, direct coupling of the reed function to the allpass inertance element at the input to the conical ``circuit'' would produce a delay-free loop in the digital waveguide implementation. These constraints lead to the modeling approach discussed first in this section.

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