## 2 Method
### Low Frequency Region f1
#### 2.2.2 Back Propagation  
The harpischord soundboard is a continuous radiator of sound, but can be simplified as a discrete distribution of \(N=1500\) radiating points \(\mathbf{Y}_{n}\), referred to as _equivalent sources_[23]. These equivalent sources sample the vibrating sound board. The validity of this simplification is restricted by the Nyquist-Shannon theorem, i.e., two equivalent sources per wave length are necessary. The following steps are frequency-dependent. Therefore, we transfer functions of time into frequency domain using the discrete Fourier transform. Terms in frequency domain are indicated by capital letters and the \(\omega\) in the argument. For example \(Q(\omega)\) represents the frequency spectrum of \(q(t)\). The relationship between the radiating soundboard \(Q(\mathbf{Y}_{n},\omega)\) and the spectra of the aligned, linearized microphone recordings \(Q(\mathbf{X}_{m},\omega)\) is described by a linear equation system  
\[Q(\mathbf{X}_{m},\omega)=\sum_{n=1}^{N}G(r,\omega)Q(\mathbf{Y}_{n},\omega), \tag{6}\]  
where  
\[G(r,\omega)=\frac{e^{i(kr)}}{r} \tag{7}\]  
is the Free field Greens' function. It is a _complex transfer function_ that describes the _sound propagation_ of the equivalent sources as monopole radiators. Here, the term \(r=||\mathbf{X}_{m}-\mathbf{Y}_{n}||_{2}\) is the Euclidean distance between equivalent sources and microphones, \(k\) is the wave number and \(i\), the imaginary unit. Equation (6) is closely related to the Rayleigh Integral which is applied in acoustical holography and sound field synthesis approaches, like wave field synthesis and ambisonics [39]. One problem with Equation (6) is that the linear equation system is ill-posed. The radiated sound \(Q(\mathbf{X}_{m},\omega)\) is recorded but the source sound \(Q(\mathbf{Y}_{n},\omega)\), which created the recorded sound pressure distribution, is sought. When solving the linear equation system, e.g., by means of Gaussian elimination or an inverse matrix of \(G(r,\omega)\)[40], the resulting sound pressure levels tend to be huge due to small numerical errors, measurement and equipment noise. This can be explained by the propagation matrix being ill-conditioned when microphone positions are close to one another compared to the considered wavelength. In this case the propagation matrix condition number is high. A regularization method relaxes the matrix and yields lower amplitudes. An overview about regularization methods can be found in [21; 23; 40]. For musical instruments, the Minimum Energy Method (MEM) [23; 41] is very powerful. The MEM is an iterative approach, gradually reshaping the radiation characteristic of \(G(r,\omega)\) from monopole at \(\Omega=0\) to a ray at \(\Omega=\infty\) using the formulation  
\[\Psi\left(\alpha,\omega,\Omega\right)=1+\Omega\times\left(1-\alpha\right), \tag{8}\]  
Figure 7: Obtained impulse response \(q^{{}^{\prime}}(\mathbf{X}_{m},t)\) of the signal in Figure 6 after deconvolution. The harmonic distortions are separated in time and precede the linear part \(q(\mathbf{X}_{m},t)\) (red), which starts at \(t\approx 3.5\,\mathrm{s}\). where \(\Psi\left(\alpha,\omega,\Omega\right)\) is multiplied by \(G(r,\omega)\) in Equation (6) to reshaped the complex transfer function, like  
\[Q(\mathbf{X}_{m},\omega)=G(r,\omega)\times\Psi\left(\alpha,\omega,\Omega \right)\times Q(\mathbf{Y}_{n},\omega). \tag{9}\]  
In Equations (8) and (9), \(\alpha\) describes the angle between equivalent sources \(\mathbf{Y}_{n}\) and loudspeakers \(\mathbf{X}_{m}\) as inner product of both position vectors  
\[\alpha_{m,n}=\left|\frac{\mathbf{Y}_{n}\mathbf{X}_{m}}{\left|\mathbf{X}_{m} \right|\left|\mathbf{Y}_{n}\right|}\mathbf{n}\right|. \tag{10}\]  
The angle \(\alpha\) is given by the constellation of source- and receiver positions and is 1 in normal direction \(\mathbf{n}\) of the considered equivalent source position and 0 in the orthogonal direction.