BORE RECONSTRUCTION FROM MEASURED ACOUSTICAL INPUT IMPEDANCE; EQUIPMENT, SIGNAL PROCESSING, ALGORITHMS AND PREREQUISITES Wilfried Kausel University of Music and Performing Arts Vienna, Institut für Wiener Klangstil, Singerstrasse 26/a, A-1010 Vienna, Austria e-mail:
[email protected] Abstract Bore reconstruction of brass wind instruments has successfully been achieved in the past based on time domain measurements with a non reflecting sound source (pulse reflectometry). The reflecting sound source, which usually terminates the instrument's front end during a steady state input impedance measurement makes it difficult to numerically calculate the pressure pulse response, which is required as an input to the reconstruction algorithm. Although this forward algorithm is efficient and elegant, it tends to accumulate any errors instead of averaging it. On the other hand transmission line modelling using conical elements and taking visco-thermal losses into account, allows for accurate calculation of complex input impedance curves from any known duct geometry. Computer optimisation is proposed as a means to reconstruct a geometry by matching a measured input impedance curve with a calculated one. Reconstruction results of accurately known geometries are presented and discussed. The brass instrument analysis system BIAS [1] is used to measure acoustical input impedance curves accurately enough in order to serve as a starting point for a bore reconstruction. Measured and theoretical input impedance magnitudes and phases are compared for reference ducts. INTRODUCTION
There exists a unique relationship between the geometry of an acoustical duct and the pulse response or the acoustical impedance at its input cross-section. By physical modelling [2][3][4][5][6] it is possible to calculate the pulse response as well as the input impedance when the geometry is known. To reconstruct the geometry from a known pulse response or input impedance, is commonly referred to as the ‘inverse problem’ or as ‘acoustical bore reconstruction’. Basically all ‘inverse problems’ can be solved by optimisation if their associated ‘forward problem’ is numerically solvable. Especially the acoustical bore reconstruction problem would need an optimisation algorithm capable of dealing with a big number of parameters, because actually the complete geometry of the instrument is subject to optimisation. Therefore all attempts in the past were tackling the forward approach while optimisation has not yet been considered. The equation system for the geometry co-ordinates is probably over-determined because there usually will be much more frequency points available than variables to calculate. This fact is a strong indication, that optimisation of impedance matching might be a suitable reconstruction method, because it tends to average all measurement inaccuracies yielding the best possible compromise, the statistically best overall fit. The direct reconstruction algorithm, which is usually employed, is the so called ‘layer peeling algorithm’. It is a recursive algorithm, where numerical errors tend rather to accumulate instead of cancelling out. Anyhow, the inverse problem has been tackled and successfully solved only on the basis of a measured impulse response in the time domain up to now, and measurements of the steady state in the frequency domain have not yet been considered to be used as the basis of an acoustical bore reconstruction, although it is of course evident, that an impulse response in the time domain is mathematically equivalent to a complex spectrum in the frequency domain.
TIME VERSUS FREQUENCY DOMAIN
The system, which is investigated in the so called ‘pulse reflectometry’ [7][8] and the one employed in a typical frequency domain measurement are not completely identical. In the pulse reflectometry the instrument is coupled to a non reflecting sound source. The plane wave front created by a “Dirac impulse” is guided into the instrument passing an “infinitely” long tube with a characteristic impedance which is perfectly matched with the characteristic impedance of the circular cross-section at the entry point of the instrument. All reflections, which are observed at this entry point are therefore caused by impedance changes (= bore diameter changes) within the instrument and multiple reflections are minimised because of the non reflecting termination at the entry point. Practically this is achieved by making the tube of the sound source long enough that reflections from the end where the pulse is excited do not reach the instruments entry point during the time interval required to be considered. What is recorded, is the transient characteristic after a pulse excitation of a system, which has been in the perfect equilibrium state before. In the frequency domain a steady state is measured and a reflecting high impedance sound source is usually directly connected to the mouthpiece. To calculate the input impulse response IIR(e being the discretised frequency, required to do bore reconstruction analytically, the phases of the input impedance Zin(e ) must be known or properly reconstructed and the characteristic impedance Z0 of the effective coupling cross-section must be determined exactly. On top of that, the conversion M
M
( )
IIR e j θ
Z0 Z in e j θ = Z0 1+ Z in e j θ 1−
( )
(1)
( )
seems to present some numerical challenge, in order to get a result accurate enough for the numerically already rather sensitive incremental procedure for getting physical dimensions. Trumpet in Bb, Bore Reconstruction From Input Impedance, BW=8kHz From Pulse Reflectometry 18
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Figure 1: Layer Peeling Algorithm, Comparison
The figure above shows the accuracy difference between a bore reconstruction of a trumpet using time domain or frequency domain measurement data. The difference on the left side originates from different coupling adapters but on the right side accumulating numerical errors are apparent. The resolution of the frequency domain method is limited by the upper band edge, which directly determines the x-axes grid, but indirectly the lower band edge is an issue, too. Of course, it is possible to determine a minimum length of a pulse response which contains all first order reflections of the initial Dirac pulse, travelling along the whole length of the instrument and back to the microphone, but in any real duct, which is not a simple tube, there will be higher order
reflections travelling back and forth several times, until they are recorded. By applying the bilinear transform (1) the periodic time domain signal, the inverse Fourier transform of the input impedance is “unfolded” into a more or less quickly decaying non periodic signal, which is actually the signal obtained by pulse reflectometry. The lower band edge of the impedance limits the length of the pulse response and therefore truncates the decaying multiple reflections. The same applies to pulse reflectometry where the finite length of the source tube limits the measurement interval. The other way round – to use an optimiser [9][10] to match a simulated and a measured impedance curve, thus reconstructing the original geometry – solves many of these problems implicitly. It averages and cancels out measurement and numerical errors, it can work with much more general and accurate models – even including higher oscillation modes and other kinds of losses, it can match phases, if those are not redundant and if the model supports them and it can benefit from error propagation models by assigning confidence values to measured data. MODELING BRASS WIND INSTRUMENTS
Transmission line modelling [4][5] is a good compromise between simple but inaccurate lumpedparameter models and finite element modelling. Because of the one-dimensional assumptions higher oscillation modes are not taken into account. The frequency
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