## 2 Perceptually Informed Friction Sound Synthesis
### Source Modeling
#### 2.1.3 Signal Model  
From the empirical signal observations presented in previous sections, the following general synthesis model of the source \(e(t)\) is proposed:  
\[e(t)=\sum_{k=1}^{N_{f_{0}}}A_{k}(t)sin(\Omega_{k}(t)) \tag{2}\]  
where \(N_{f_{0}}=\left\lfloor\frac{f_{s}}{2f_{0}}\right\rfloor\) and \(f_{s}\) the sample frequency. The instantaneous frequency is given by \(\frac{d\Omega_{k}}{dt}(t)=2\pi kf_{0}(t)\), and \(A_{k}\) the amplitude modulation law for the \(k\)-th harmonic. From a signal point of view, the sound morphologies for squeaking, creaking and singing objects differ by the temporal evolution of the fundamental frequency \(f_{0}(t)\) and the amplitudes \(A_{k}(t)\).  
* _Fundamental frequency behavior._ The following general expression for \(f_{0}\) is proposed: \[f_{0}(t)=(1+\epsilon(t))\Phi(\Gamma(t))\] (3) \(\Phi\) represents the fundamental frequency defined from a given mapping \(\Gamma(t)\). \(\epsilon(t)\) is a low pass filtered noise with cutoff frequency defined below 20 Hz. This stochastic part models the chaotic variations of the fundamental frequency observed in the case of squeaking or creaking sounds. In the case of self-sustained oscillations, \(\epsilon\) is null. For a more physically informed mapping, the fundamental frequency can be defined from the dynamical parameters of the two bodies in contact, in particular \(\Gamma(t)=(p(t),v(t))\), by assuming that the friction sounds mainly depend on the dynamic pressure \(p(t)\) and the relative velocity \(v(t)\) between objects. The mapping can be defined with other descriptors according to the desired applications as mentioned in the introduction and developed below in the applications section.
* _Amplitude Modulation Behavior._ In the specific case of the singing wineglass low frequency beats are observed in the generated sounds. The frequency of the amplitude modulation is directly linked to the velocity \(v(t)\) of the wet finger which is rubbing the glass rim and the diameter of the glass \(D\). An explicit expression of \(A_{k}(t)\) as been proposed [24; 25]: \[A_{k}(t)=\frac{1}{k}sin\left(2\pi\int_{0}^{t}\frac{v(t)}{\pi D}dt\right)\] (4) In other words, the smaller the diameter, the higher the velocity and the amplitude modulation, and vice versa. For other situation than the singing wineglass we proposed an amplitude modulation law defined by \(A_{k}=\frac{1}{k}\). The parameters of the model related to different friction phenomena are summarized in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sound** & **Frequency Modulation** & **Amplitude Modulation** \\  & \(f_{0}(t)\) & \(AM_{k}(t)\) \\ \hline Creaky Door & & \\ Squeaky Vessel & & \\ Squeaky Wineglass & & \\ \hline Singing Wineglass & \(\tilde{f}_{(n,0)}\) & \(\frac{1}{k}sin\left(2\pi\int_{0}^{t}\frac{v(t)}{\pi D}dt\right)\) \\ \hline Bowed String & \(\tilde{f}_{(n,0)}\) & \(\frac{1}{k}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mapping between descriptors and signal parameters – \(\Gamma(t)\) represents the mapping between the descriptors and the fundamental frequency, the velocity and the pressure – \(\tilde{f}_{(n,0)}\) is the \(n\)-th mode of the vibrating object (wineglass or string). Note that in each case the resonant filter bank that models the material and the shape of the object has to be calibrated accordingly [15; 26].