## 2 Method
### Low Frequency Region f1
#### 2.2.1 Nearfield Recording  
The setup for the near field recordings is illustrated in Figure 3. A microphone array \(\mathbf{X}_{\text{near,m}}\) with equidistant microphone spacing of 40 mm is installed at a distance of 5 cm parallel to the harpsichord soundboard surface. The index \(m=1,\ldots,1500\) describes a microphone position above the harpsichord. An acoustic vibrator excites the instrument at the intersection point of string and bridge; the _string termination point_[32]\(\mathbf{\kappa}_{a}\). Here, the index \(a=1,\ldots,A\) describes the pressed key. For a harpsichord with 5 octaves, \(A=62\) keys exist. To obtain impulse responses from the recorded data the so-called exponential sine sweep (ESS) technique is utilized [37]. The method has originally been proposed for measurements of weakly non-linear systems in room acoustics (e.g., loudspeaker excitation in a concert hall) but can also be adapted to structure-borne sound [38]. For the excitation an exponential sine sweep  
\[s(t)=\sin(\omega(t)) \tag{1}\]  
Figure 5: Procedure to derive the impulse response \(R_{f1}(\mathbf{\Theta}_{l},\mathbf{\kappa}_{a},t)\) for each loudspeaker and pressed key in the radiation keyboard. The brown curve represents the bridge, the brown dots depict exemplary excitation points. The black dots represent microphones near the soundboard, the light gray dots represent equivalent sources on the soundboard. The gray dots represent a regular subset of equivalent sources, which are replaced by loudspeakers (circles) in the radiation keyboard. is used, where  
\[\omega(t)=\omega_{1}e^{\left(\frac{t\ln\left(\frac{\omega_{2}}{\omega_{1}}\right) }{t}\right)}. \tag{2}\]  
Here, \(\omega_{1}=2\pi\) rad s\({}^{-1}\) is the starting frequency, \(\omega_{2}=2\pi\times\) 24,000 rad s\({}^{-1}\) is the maximum frequency and \(T=25\) s is the signal duration. The vibrator excites the sound board with this signal. Figure 6 shows the spectrogram of an exemplary microphone recording \(p(\mathbf{X}_{1},t)\). Since the frequency axis has logarithmic scaling, the sweep appears as a straight line. Due to non-linearity in the shaker excitation the recording shows harmonic distortions parallel to the sinusoidal sweep. A deconvolution process eliminates these distortions. The deconvolution is realized by a linear convolution of the measured output \(p(\mathbf{X}_{m},t)\) with the function  
\[u(t)=s^{-1}(t)b(t). \tag{3}\]  
Here, \(s^{-1}(t)\) is the temporal reverse of the excitation sweep signal (2) and \(b(t)\) is an amplitude modulation that compensate the energy generated per frequency, reducing the level by 6 dB/octave, starting with 0 dB at \(t=0\) s and ending with \(-6\log_{2}(\omega_{2}/\omega_{1})\) dB at \(t=T\), expressed as  
\[b(t)=1/e^{\frac{t\ln\frac{\omega_{2}}{\omega_{1}}}{t}}. \tag{4}\]  
This linear deconvolution delays \(s(t)\) of an amount of time varying with frequency. The delay is proportional to the logarithm of frequency. Therefore, \(s^{-1}(t)\) stretches the signal with a constant slope, and compresses the linear part to a time delay corresponding to the filter length. The harmonic distortions have the same slope as the linear part and are, therefore, also packed to very precise times. If \(T\) is large enough, the linear part of an impulse response is temporally clearly separated from the non-linear pseudo IR. This deconvolution process yields one signal  
\[q^{{}^{\prime}}(\mathbf{X}_{m},t)=p(\mathbf{X}_{m},t)*u(t). \tag{5}\]  
This signal \(q^{{}^{\prime}}(\mathbf{X}_{m},t)\) is the linear impulse response \(q(\mathbf{X}_{m},t)\) preceded by the nonlinear distortion products, i.e., the pseudo-IRs. An example of \(q^{{}^{\prime}}(\mathbf{X}_{1},t)\) and \(q(\mathbf{X}_{1},t)\) is illustrated in Figure 7. The linear impulse response part can be obtained by a peak detection searching for the last peak in the time series. In the figure the final, linear impulse response \(q(\mathbf{X}_{1},t)\) is highlighted in red. The driving signal and the convolution are reproducible.