Tonehole Filter Design

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 $b = 4.765$ mm, minimum tonehole height $t_{w} = 3.4$ mm, tonehole radius of curvature $r_{c} = 0.5$ mm, and air column radius $a = 9.45$ 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 $\ensuremath{L_2}$ equation-error minimization [432, p. 47], i.e., by minimizing $\vert\vert\,W(\ejo)[{\hat A}(\ejo)H(\ejo) - {\hat B}(\ejo)]\,\vert\vert _2$ , where $W$ is the weighting function, $H(\ejo )$ is the desired frequency response, $\Omega$ denotes discrete-time radian frequency, and the designed filter response is ${\hat H}(\ejo) = {\hat B}(\ejo)/{\hat A}(\ejo)$ . 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 $i+1$ to the inverse of the inherent weighting ${\hat A}_i$ of the previous iteration, i.e., $W_{i+1}(\ejo) = 1/{\hat A}_i(\ejo)$ . 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 $H(\ejo )$ , compute an allpole model $1/{\hat A}(z)$ using linear prediction
• Compute the error spectrum $\hat E(\ejo)\isdef {\hat A}(\ejo)H(\ejo)$ .
• Compute an allpole model $1/{\hat B}(z)$ for $\hat E^{-1}(\ejo)$ by minimizing
  $$\left\Vert\,{\hat B}(\ejo)\hat E^{-1}(\ejo)\,\right\Vert _2 = \left\Vert\,\frac{{\hat B}(\ejo)}{{\hat A}(\ejo)}H^{-1}(\ejo)\,\right\Vert _2.$$

Use of linear prediction is equivalent to minimizing the $\ensuremath{L_2}$ ratio error

$$\left\Vert\,\hat E(\ejo)\,\right\Vert _2 = \left\Vert\,{\hat A}(\ejo)H(\ejo)\,\right\Vert _2.$$

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 $\ensuremath{L_2}$ 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 $\hat{H}_{1}(z)$ was first fit to the continuous-time response, $H(e^{j\Omega}).$ Subsequently, the inverse error spectrum, $\hat{H}_{1}(e^{j\Omega})/H(e^{j\Omega})$ was modeled with a two-pole digital filter, $\hat{H}_{2}(z).$ The discrete-time approximation to $H(e^{j\Omega})$ was then given by $\hat{H}_{1}(z)/\hat{H}_{2}(z).$

Figure 9.49: Two-port tonehole junction closed-hole and open-hole reflectances based on Keefe's acoustic measurements (dashed) versus second-order digital filter approximations (solid). Top: Reflectance magnitude; Bottom: Reflectance phase. The closed tonehole has one resonance in the audio band just above $16$ kHz. The open tonehole has one anti-resonance in the audio band near $10$ kHz. At dc, the open tonehole fully reflects, while the closed tonehole reflects close to nothing (from [406]).

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].

Figure 9.50: Reflection functions for note $G$ (three finger holes closed, three finger holes open) on a simple flute (from [406]). (top) Transmission-line calculation; (bottom) Digital waveguide two-port tonehole implementation.

The upper plot was calculated using Keefe's frequency-domain transmission matrices, such that the reflection function was determined as the inverse Fourier transform of the corresponding reflection coefficient. This response is equivalent to that provided by [242], though scale factor discrepancies exist due to differences in open-end reflection models and lowpass filter responses. The lower plot was calculated from a digital waveguide model using two-port tonehole scattering junctions. Differences between the continuous- and discrete-time results are most apparent in early, high-frequency, closed-hole reflections. The continuous-time reflection function was low-pass filtered to remove time-domain aliasing effects incurred by the inverse Fourier transform operation and to better correspond with the plots of [242]. By trial and error, a lowpass filter with a cutoff frequency around 4 kHz was found to produce the best match to Keefe's results. The digital waveguide result was obtained at a sampling rate of 44.1 kHz and then lowpass filtered to a 10 kHz bandwidth, corresponding to that of [242]. Further lowpass filtering is inherent from the first-order Lagrangian delay-line length interpolation technique used in this model [506]. Because such filtering is applied at different locations along the ``bore,'' a cumulative effect is difficult to accurately determine. The first tonehole reflection is affected by only two interpolation filters, while the second tonehole reflection is affected by four of these filtering operations. This effect is most responsible for the minor discrepancies apparent in the plots.