Tonehole Filter Design

The tone-hole reflectance and transmittance must be converted to discrete-time form for implementation in a digital waveguide model. Figure 6.12 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 [409,412]. 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 [437, 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 [294] due to its ability to design optimal IIR filters via quadratic minimization. In the spirit of the well-known Steiglitz-McBride algorithm [293], 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 minimiz