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 [406,409]. 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 [432, 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 [290]
due to its ability to design optimal IIR filters via quadratic
minimization. In the spirit of the well-known Steiglitz-McBride algorithm
[289], 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 [299], [432, p. 46] in conjunction with weighted equation-error minimization. Kopec's method is based on linear prediction:

- Given a desired complex frequency response , compute an allpole model using linear prediction
- Compute the error spectrum .
- Compute an allpole model
for
by
minimizing

This optimization criterion causes the filter to fit the

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 [242].

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University