The tone-hole reflectance and transmittance must be converted to discrete-time form for implementation in a digital waveguide model. Figure 9.49 plots the responses of second-order discrete-time filters designed to approximate the continuous-time magnitude and phase characteristics of the reflectances for closed and open toneholes, as carried out in [#!ScavoneAndSmithICMC97!#,#!ScavoneT!#]. These filter designs assumed a tonehole of radius mm, minimum tonehole height mm, tonehole radius of curvature mm, and air column radius mm. Since the measurements of Keefe do not extend to 5 kHz, the continuous-time responses in the figures are extrapolated above this limit. Correspondingly, the filter designs were weighted to produce best results below 5 kHz.
The closed-hole filter design was carried out using weighted equation-error minimization [#!JOST!#, p. 47], i.e., by minimizing , where is the weighting function, is the desired frequency response, denotes discrete-time radian frequency, and the designed filter response is . Note that both phase and magnitude are matched by equation-error minimization, and this error criterion is used extensively in the field of system identification [#!LjungAndSoderstrom!#] due to its ability to design optimal IIR filters via quadratic minimization. In the spirit of the well-known Steiglitz-McBride algorithm [#!StoicaAndSoderstrom81!#], equation-error minimization can be iterated, setting the weighting function at iteration to the inverse of the inherent weighting of the previous iteration, i.e., . However, for this study, the weighting was used only to increase accuracy at low frequencies relative to high frequencies. Weighted equation-error minimization is implemented in the matlab function invfreqz() (§8.6.4).
The open-hole discrete-time filter was designed using Kopec's method [#!MG!#], [#!JOST!#, p. 46] in conjunction with weighted equation-error minimization. Kopec's method is based on linear prediction:
This optimization criterion causes the filter to fit the upper spectral envelope of the desired frequency-response. Since the first step of Kopec's method captures the upper spectral envelope, the ``nulls'' and ``valleys'' are largely ``saved'' for the next step which computes zeros. When computing the zeros, the spectral ``dips'' become ``peaks,'' thereby receiving more weight under the ratio-error norm. Thus, in Kopec's method, the poles model the upper spectral envelope, while the zeros model the lower spectral envelope. To apply Kopec's method to the design of an open-tonehole filter, a one-pole model was first fit to the continuous-time response, Subsequently, the inverse error spectrum, was modeled with a two-pole digital filter, The discrete-time approximation to was then given by
The reasonably close match in both phase and magnitude by second-order filters indicates that there is in fact only one important tonehole resonance and/or anti-resonance within the audio band, and that the measured frequency responses can be modeled with very high audio accuracy using only second-order filters.
Figure 9.50 plots the reflection function calculated for a six-hole flute bore, as described in [#!Keefe90!#].