Data Processing for Filter Design

• If measured data is to be used within a synthesis environment that includes a feedback loop (such as to characterize the body response of a violin), it is imperative that the resulting digital filter structure be designed in such a way as to guarantee passivity.

• Measured responses tend to have some noise, especially at low and high frequencies. The filter design should thus allow smoothing or the ability to exclude certain parts of the response.

• One typical approach is to assume a system can be well modeled as a sum of modes or complex exponential functions, each having a center frequency, damping factor (or bandwidth) and gain. If these parameters can be correctly estimated, the system response can subsequently be implemented using a bank of parallel second-order (or mode) filters.

• Various approaches have been used for the estimation of modal parameters, such as frequency-domain peak picking or time-domain Prony-based methods.

  Figure 15: A measured violin admittance and a discrete-time fit using a parallel bank of second-order filters.

  

• A popular iterative peak-picking approach to mode-fitting of wind instrument impedances was proposed in Ablitzer (2021) (website demonstration, GitHub code repository).

  Figure 16: Peak-picking of a theoretical cylinder input impedance using the Ablitzer (2021) Matlab interface.

  

• The Ablitzer (2021) method determines modal parameters $A_n$, $\omega_n$ and $\xi_n$, corresponding to real amplitudes, radian frequencies and damping ratios, in a continuous-frequency representation of the input impedance of the form
    $$Z(\omega) = \sum_{n=1}^N \frac{j \omega A_n}{\omega_n^2 - \omega^2 + j 2 \xi_n \omega_n \omega}.$$ (1)

• One approach to using those parameters to derive a parallel bank of second-order discrete-time mode filters involves the bilinear transform s- to z-plane mapping, which is discussed in a subsequent week. An example of using this approach to is shown in Fig. 17.

  Figure 17: Continuous- and discrete frequency bilinear transform fit of three impedance modes determined using Ablitzer (2021) software: All three modes (top); closeup of peak around 707 Hz.