Modeling Approaches

• 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 $M_{o}(s) = (\rho x_{o}/A_{o}) \, s$, where $\rho$ 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:
  $$\mathcal{R}_{o}(s) \stackrel{\Delta}{=} \frac{P^{-}_{o}(s)}{P^{+}... ...= \frac{M_{o}(s) - Z_{o}}{M_{o}(s) + Z_{o}} = \frac{x_{o} s - c}{x_{o} s + c},$$
  where $Z_{o} = \rho c / A_{o}$ is a real, locally defined characteristic impedance parameter.

• The reflectance filter, discretized with the bilinear transform, is
  $$\mathcal{R}_{o}(z) = \frac{-a_{1} - z^{-1}}{1 + a_{1} z^{-1}}, \h... ...\mbox{where} \hspace{0.1in} a_{1} = \frac{c - \alpha x_{o}}{c + \alpha x_{o}},$$
  and $\alpha$ 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.

  Figure 18: A closed-open conic structure (top) and its digital waveguide block diagram (bottom).

  

• 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.