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Page 00000001 From Piano String Vibrations to the Acoustic Field Radiated by the Soundboard Mitsuko ARAMAKI, Philippe GUILLEMAIN, Nathalie HOEB, Alain ROURE C.N.R.S. - Laboratoire de Mecanique et d'Acoustique 31, chemin Joseph Aiguier 13402 Marseille Cedex 20, France email: [aramaki, guillem, hoeb, roure]@lma.cnrs-mrs.fr Abstract We propose to characterize the relationships between the vibrations and the acoustic radiation of the piano soundboard. For that purpose, we use a signal processing approach. In this context, the purpose of this study is not to describe the mechanical and acoustical phenomena that occur during the sound production. Conversely, we model the influence of the soundboard in the piano sound production by transfer functions between the vibrations of strings at the bridge level and the acoustic field radiated at the level of the ears of the pianist. These transfer functions are estimated from the analysis of experimental signals collected on a grand piano placed in an anechoic room. The accuracy of the analysis method and the obtained results are then validated by the resynthesis of sounds using binaural and transaural techniques. 1 Introduction The physical mechanisms that are responsible of the piano sound production are mainly: the interaction between the strings and the hammer, the vibrations of strings coupled by the bridge, the soundboard vibrations and its acoustic radiation. This work focuses on this last point and is included in a larger project related to the synthesis of piano tones. Assuming that the most important listener is the pianist itself since there is a strong feedback mechanism between the pianist play and the acoustic field he is immersed into, we restrain the soundboard vibration and the radiation modeling problems at the level of the ears of the pianist. Indeed, though the soundboard plays a role in the vibrational behaviour of the string by coupling them through the bridge, the field it radiates can be deduced from the string vibrations at the bridge level up to a linear filtering. Such an approach is relevant, considering that there exist efficient techniques for the resynthesis of coupled string vibrations at the bridge level using for example coupled digital waveguides. The estimated transfer functions would partly give some informations on the con tribution of the soundboard in piano tones from a perceptive point of view. In this article, we shall present the experimental setup, which has been used to build a relevant database of real signals. The second step will consist in the design of a deconvolution technique based on digital waveguide models to estimate the two transfer functions between the string vibrations and the acoustic pressures perceived by the pianist. The results we obtained, will be commented and will let us discuss relationships between the estimated transfer functions of different notes. The relevancy of the method we used, is validated by a resynthesis of sounds using various restitution techniques. 2 Experimentation The first step of this study has consisted in collecting experimental signals from a calibrated setup. We have placed a Disklavier grand piano in an anechoic room. We simultaneously measured the string vibrations at the bridge level with an accelerometer placed close to the played note and the radiated acoustic pressures at the location of the ears of the pianist with a dummy head. We piloted the piano (equipped with MIDI input and output) with a computer placed outside the anechoic room, to minimize the perturbations created by human presence during the acquisitions. This system of piloting has also the advantage of being reproducible since the velocities applied on the keys are controlled. We have played the eigthy-eight notes of the keyboard. In particular, we have played about ten notes with several velocities of the hammer. 3 Analysis of experimental signals For each note we have measured three signals (strings velocity at bridge level, acoustic pressures at the pianist ears level). The aim is to calculate two transfer functions which describe the wave propagation from the bridge, through the soundboard and air, to the right and left ears of the pianist.
Page 00000002 We note the two functions T, (for the right ear) and T, (for the left ear). 3.1 Estimation of transfer functions The transfer functions Tr(w) and T (w) satisfy the relation: R(w) = Tr(w)V(w) L(w) = Ti(w)V(w), with V the Fourier transform of strings velocity signal at bridge level, R the Fourier transform of the right ear pressure signal and L the Fourier transform of the left ear pressure signal. The transfer functions can be directly calculated from these expressions but the ratios between the Fourier transforms are unstable because of many anti-resonances. Coupled E digital waveguide l ^r - ET F shall see that such a modeling let us reduce the deconvolution problem to the ratio of the excitation spectra between the signals from the ears and the velocity signal at the bridge level. Moreover, it let us get rid of parasitic components appearing in the ear signals such as mechanical noises due to the hammer movement, which pose problems when using the deconvolution techniques mentioned above. 3.2 Coupled digital waveguide model The coupled digital waveguide chosen for the modeling of the measured signals is shown in Figure 2. ________ G I ________ Figure 2: Coupled digital waveguide: E input signal, S output signal, Ci et C2 coupling filters, Gi et G2 elementary digital waveguides which consists of a loop that contains two filters: D represents the propagation time and F represents the dissipation and dispersion phenomena. This model has been used to model coupled piano string vibrations (Aramaki, Bensa, Daudet, Kronland-Martinet, and Guillemain 1999). The transfer function of the model is given by: Figure 1: Analysis process for the transfer functions estimation: V is the Fourier transform of strings velocity signal at bridge level, R the Fourier transform of the right ear pressure signal and L the Fourier transform of the left ear pressure signal. We have tested various methods of transfer function estimation, such as FIR or IIR estimation, Least Mean Square minimization or spectral methods. The results obtained were unconvincing. Indeed, the estimated transfer functions must satisfy two requirements: preserve the attack transient, which requires a brief impulse response and modify accurately the amplitude and the phase of each partial. Using the methods mentioned above, it turned out that these two requirements were incompatible: accurate amplitude and phase modifications were done to the detriment of the length of the impulse response. These difficulties forced us to reconsider this deconvolution problem. For that purpose, each experimental signal is modeled as the output of a coupled digital waveguide model which will be described in the next section. The model having a linear behaviour, we can establish an equivalence relation between the output ratios and the input ratios of the model. Indeed, we choose to use the excitation of the coupled waveguide model known to be more smooth and more regular since it corresponds to the force applied by the hammer on the strings for the vibration signal model. The estimation process is described in Figure 1 in solid line. We S(w)_ Gi(w) E(w) 1- Ci(w)C2(w)Gi(w)G2(w) (1) SFi(w)e-iWD with Gi (w) = - w)e-iwD 1 - Fi(w)e-iwD i [1, 2]. All the filter expressions are estimated by identifying the impulse response of the system to a double sum of exponentially damped sine waves. In the frequency domain, it corresponds to identify the transfer function to a double sum of Lorentz functions: T(wZ) = + bk (2) k + (W - Wk) k+ (W - W2k) where ak and bk are the amplitudes, ak and A3k the damping coefficients, clk and W2k the eigen frequencies of the component k. By a local identification around each resonance k, we can deduce the filter expressions Ci(w), C2(w), F, (w), F2(w) and D from the physical parameters ak, bk,ak, Pak,41k and W2k of each component k which calculation steps are detailed in (Aramaki, Bensa, Daudet, Kronland-Martinet, and Guillemain 2001).
Page 00000003 The input E(w) of the model is deduced and can be expressed by: E(wk) E( D(akX(wlk) + bkY(W2k))2 ( E(w k) E(W k) = (3) akX(wlk)2 + bkY(w2k)2 WlkD-2krr w2kD-22krr with X(wlk) = Y(W2k) = W2 akD + 1 AkD + 1 We can notice that all the expressions above are only defined at the eigen frequencies. Though this model is obviously not representative of the vibrations of more than two coupled strings, it let us reproduce the beats phenomena, which is a perceptually important feature of piano tones (Weinreich 1977). 3.3 Results For each note, we have to estimate two functions describing the transfer from the string vibrations signal at the bridge level to the two pressure signals at the pianist ears. Thus, they are defined as the ratio of the corresponding excitation of the coupled waveguide model: MIDI velocities. The assumption of a linear behaviour of the synthesis model is overall justified. Nevertheless, we observe differences at the high frequencies, which can be explain by error estimation due to a weak excitation (especially for the smallest MIDI velocity 65). We can also compare transfer functions estimated for close notes. Assuming that they are defined on discrete values of frequencies, we can consider a global curve that merge all these points. Such a comparison is relevant for close notes since the bridge spreads on the soundboard the excitation provided by the strings. Thus, we obtain the results represented in Figure 5 for the two ears estimated for five notes in the medium scale located side by side (B2, C3, C3d, D3, D3d). Ph... Tg ý itess. M I D 1 65(-) 76(-.) 82(..) 4 3 2 0 -1 --2 -3 100 200 300 000 500 600 7001000 900 Tr(Wk) = E El (wk) Tl(wk) = El(wk) E, (wk) (4) Figure 4: Phase of Tr(bottom) and Ti(top) for the note D4 for three different MIDI velocities: 65(-), 76(-.), 82(..) as function of frequency. where E,, Er, El are the respective inputs of the models corresponding to the strings vibration signal, the right ear signal and the left ear signal. As the excitations are estimated on eigen frequencies, it let us get rid of the unstability due to the minima of the spectra of the signals. In order to obtain Module Tg - vitesse MIDI:65(-) 76(-.) 82(..) 0.12 0.1 0.08 -0.06 1 - 0.04 -0.02 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Module Td - vitesse MIDI 65(-) 76(-.) 82(..) 0.2 0.15 0.1 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 ( Figure 3: Modulus of Tr(bottom) and Ti(top)for the note D4 for three different MIDI velocities: 65(-), 76(-.), 82(..) as function of frequency. excitations, we first have to extract the physical parameters (ak, bk, ak, k, Wlk and W2k given in 3.2 of each component k) from the experimental signals using a band pass filtering to isolate each doublet and the parametric SteiglitzMac Bride method (Steiglitz and McBride 1965) to extract each component within a doublet. We then deduce the excitation expressions from (3) and finally the two transfer functions from (4). Figures 3 and 4 show the modulus and phases of Tr and T, estimated for the note D4, for three different Figure 5: Modulus of the global transfer function estimated for five notes located side by side (B2, C3, C3d, D3, D3d), for the right ear (right) and left ear (left). The horizontal axis represents the frequencies on the logarithmic scale. On each plot, is superimposed a continuous curve obtained by fitting the merged transfer functions by a eighth order polynom. The global transfer functions for the two ears are similar. Thus, the corresponding wave propagation pathes are very close and the small differences that are observed are due to the diffraction by pianist head. We have also estimated this global transfer function for other groups of close notes in the low and high scale. In all cases, we can recognize the global vibrational behaviour of the soundboard which has been shown in previous works (Giordano 1998).
Page 00000004 4 Synthesis and sound restitution In order to validate the analysis method and the results obtained, we have resynthesized sounds using the estimated transfer functions. For that purpose, knowing that all the functions are estimated only on eigen frequencies, we have to interpolate them. The synthesis process using the estimated transfer functions is shown in the Figure 1 in dotted line. We have excited the digital coupled waveguide model corresponding to the string vibrations signal with E,. Then, the output of the model, which is the resynthesis of the velocity signal has been filtered with the transfer functions Tr and T1. Finally, we got a stereo signal representing the resynthesized acoustic pressure at the pianist ears level. Figure 6 shows a comparison between the spectrum of the real signal and the one of the resynthesized signal. From a perceptual point of view, the simulated sounds are identical to the original ones. 1 10 5 -0 ---5 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 D~d oreille droite: tr-nformee du sig-al de syntht-e 10 5 0 --5 - - - - - - - - - - - - - - - - - - - - - Figure 6: Spectrum of the real signal (top) and spectrum of the resynthesized signal (bottom) for the note D3d, for the right ear of the pianist as function of the frequencies. Figure 7 represents a comparison between the real and resynthesized temporal signals. The phase relations are exactly respected and the attack transient is well restituted. Thus, the accuracy of the presented analysis method can be validated. The resynthesized binaural signals can be directly restituted through headphones. Moreover, we also used two loudspeakers to restitute the acoustic field at the pianist location. For that purpose, we have implemented classical transaural methods (Gardner 1995). Those transaural techniques minimize the diaphony in order to restitute the binaural signals with standard stereophonics techniques. Therefore, we have to compensate the right (left) loudspeaker contribution on the left (right) ear. These techniques are very sensitive and only valid for one listener which has to be motionless. But they are adapted to our problem as far as we restrained the restitution of the acoustic field at the pianist's ears. Moreover, the sound directivity has been taken into account so that the sensations felt by a pianist in front of a real piano were well reproduced. Indeed, we can feel that a low note seems to come from the left and a high note, from the right. Figure 7: Real time signal (top) and resynthesized temporal signal (bottom) for the note D3d, for the right ear of the pianist, for the first 3000 samples. 5 Conclusion In this paper, we have presented a method allowing the restitution of the acoustic field at the level of the ears of the pianist from the vibration signal at the level of the bridge. For that purpose, we have designed a specific experiment that let us collect simultaneously the vibratory signals on the bridge and the acoustic signals. Transfer functions between the signals have been estimated using the coupled digital waveguides model. These transfer functions let us reproduce the acoustic field with binaural or transaural techniques. Moreover, by grouping transfer functions of close notes, we obtain an estimation of the vibration response of the soundboard in accordance to experiments performed previously. References Aramaki, M., J. Bensa, L. Daudet, R. Kronland-Martinet, and P. Guillemain (1999). Resynthesis of coupled piano string vibrations based on physical modeling. Proceedings of the Digital Audio Effects Conference: DAFx99, Tronheim, Norway, 135-138. Aramaki, M., J. Bensa, L. Daudet, R. Kronland-Martinet, and P. Guillemain (2001). Resynthesis of coupled piano string vibrations based on physical modeling. Journal of New Music Research. to be published. Gardner, W. (1995). Transaural 3-d audio. M..T. Media Laboratory Perceptual Computing Section. Technical Report N 342. Giordano, N. (1998). Sound production by a vibrating piano soundboard: Experiment. Journal of Acoustical Society of America 104, 1648-1653. Steiglitz, K. and L. E. McBride (1965). A technique for the identification of linear systems. IEEE Trans. Automatic Control, 461-464. Weinreich, G. (1977). Coupled piano strings. Journal ofAcoustical Society of America 62, 1474-1484.