Peak-picking identification technique for modal representation of input impedance of brass instruments

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Frédéric Ablitzer

Laboratoire d'Acoustique de l'Université du Mans, CNRS - UMR 6613, avenue Olivier Messiaen, 72085 Le Mans cedex 9, France

The video below illustrates the iterative approximation of the input impedance of a bass trombone, as described in Section 4.2. The green regions in the background indicate the current frequency range considered in the identification.

(use video controls to navigate through individual iterations)

Supplementary material for Section 3.3

Refining the identification by ascending or descending order of natural frequency

In Section 3.3, it is advised to update the poles in ascending order of natural frequency. The ascending order has been found to provide more robustness. To illustrate this, we propose the following procedure based on the result shown in Fig. 2:

starting from the solution at iteration 70, a random perturbation is applied to all poles (the natural frequencies are changed within a range of ±2%, the damping ratios are changed within a range of ±50%);
then, 20 refinement iterations are performed.

The process is repeated for 15 different sets of random perturbations.

When choosing the ascending order in each refinement step, the 20 iterations make all poles converge back towards their initial values.

On the contrary, choosing the ascending order in each refinement step leads to a significant dispersion of poles at the end of the process, especially in the upper frequency range.

Supplementary material for Section 4.2

Comparison between 4 different sequences

Each sequence starts with the same step (adding a mode to capture the first resonance), then the 4 available actions are executed in different orders.

Blue: the sequence detailed in the paper (already shown in Fig. 2),
Magenta: a sequence where refinement actions (R) are performed more frequently,
Green: a sequence where high-frequency resonances are captured before low-frequency resonances,
Orange: a sequence where 16 modal components are introduced (A) before any other action is executed; the initial frequency estimates are defined by clicking at arbitrary locations (which do not necessarily coincide with a peak), so as to progressively cover the entire frequency range.

These 4 sequences lead to the same final result, except for the value of the 15th pole which differs slightly, due to the fact that the fitting frequency range (light green area) is not strictly identical between the 4 sequences.

Complementary results

Application of the method to a woodwind instrument (oboe)

Measured input impedance of a oboe for a fingering corresponding to a D4 note and its approximation using real modes (8 modes + 1 spurious). The spurious mode was arbitrarily set at 2500 Hz, with a 10% damping ratio.

(measured input impedance courtesy of J.P. Dalmont)

Measured input impedance of a oboe for a fingering corresponding to a B♭4 note and its approximation using real modes (6 modes + 1 spurious). The spurious mode was arbitrarily set at 2500 Hz, with a 10% damping ratio.

(measured input impedance courtesy of J.P. Dalmont)