## 2 Materials and Methods
### Glottal Flow Model for the Kalman Filter
A diagram of this augmented system is shown in Figure 3. The white noise \(w_{2}(n)\) is the input to the shaping filter, the latter being the Rosenberg model convolved with the periodic input (Figure 2). The output of this filter is the colored noise \(w_{1}(n)\) modeling the GVV signal, which is the state noise to the canonical MA system (physical system in Figure 3), whose output \(z(n)\) is the observed signal, i.e., the neck-skin acceleration. The new state-space equations in discrete-time are:  
\[\mathbf{X}_{T}(n+1) =\mathbf{A}_{T}\mathbf{X}_{T}(n)+\mathbf{B}_{T}w_{2}, \tag{19}\] \[z(n) =\mathbf{C}_{T}\mathbf{X}_{T}(n)+v(n), \tag{20}\]  
where  
\[\mathbf{A}_{T}= \begin{bmatrix}\mathbf{A}&\mathbf{C}_{SF}\\ \mathbf{0}&\mathbf{A}_{SF}\end{bmatrix}  \tag{21}\]  
\[\mathbf{B}_{T}= \begin{bmatrix}\mathbf{0}\\ \mathbf{B}_{SF}\end{bmatrix}  \tag{22}\]  
\[\mathbf{C}_{T}= \begin{bmatrix}\mathbf{C}&\mathbf{0}\end{bmatrix}  \tag{23}\]  
\[\mathbf{X}_{T}= \begin{bmatrix}\mathbf{x}\\ \mathbf{x}_{SF}\end{bmatrix} . \tag{24}\]An example of the estimated GVV using matrix \(\mathbf{A_{T}}\) is shown in Figure 4 and compared to the estimated GVV using the original matrix \(\mathbf{A}\). The upper plot shows the tracking of the first time step state \(\tilde{x}(n-N+1|n)\), which corresponds to the smoothed (time delayed estimate) GVV. Can be noticed that there are no differences between the original MA Kalman filter and the one incorporating a colored state noise. There is, however, a noticeable difference in the tracking of the last time step state \(\tilde{x}(n|n)\) of the GVV, which corresponds to the the filtered GVV estimate considering all the observation information up to the current sample \(n\). The original Kalman filter produces a zero-mean signal, while the modified Kalman filter with colored state noise modeling the glottal spectrum tracks an expected GVV signal. Figure 3: Diagram of modified Kalman Filter with colored state noise process. The physical system corresponds to the standard MA Kalman Filter, with a shaping Kalman filter based on a autoregressive noise process from the spectrum of a periodic Rosenberg glottal flow excitation. Figure 2: Spectrum of periodic input \(P(z)\) multiplied in the frequency domain by Rosenberg model \(G(z)\), which corresponds to an ARMA model of the glottal source (\(f_{0}=210\) Hz). The proposed implementation of the IBIF method in a Kalman filter framework has two important additions: the adaptive tracking of the GVV signal using the accelerometer and the modeling of state and observation noise. In the first case, the adaptive tracking is performed through the sample by sample correction of the predicted accelerometer signal by the Kalman gain \(\mathbf{K}(n)\). In our hypothesis, the correction term helps to improve the estimation of the GVV signal by minimizing the deviations from the GVV signal obtained with IBIF. The process noise variance \(\sigma_{w}^{2}\) (mL\({}^{2}\)/s\({}^{2}\)) and the observation noise variance \(\mathbf{R}_{v}=\sigma_{v}^{2}\) (cm\({}^{2}\)/s\({}^{4}\)) were selected using a grid-search process to compare the root-mean-square error (RMSE, mL/s) between the Kalman state \(x(n-N+1)\) and a reference GVV signal obtained by inverse filtering of the OVV signal [9]. Figure 5 shows different values of \(\sigma_{w}^{2}\) and \(\sigma_{v}^{2}\) where multiple minima (\(RMSE=17.268\)) are found within a range for one subject producing the vowel /a/. Most blue RMSE values in Figure 5 correspond to \(RMSE=17.273\) which are very close to the minimum. Similar trends were found for other subjects and vowels. We selected \(\sigma_{w}^{2}=100\) and \(\sigma_{v}^{2}=1\) in this work, which are plausible values for the state and measurement noises due to the assumption of higher process noise due to glottal flow variance with low observation noise, while they produce the minimum RMSE value. Figure 4: **Top** panel: GVV estimates (\(\hat{x}(n-N+1|n)\)) using **A** (blue) and \(\mathbf{A_{lp}}\) (red).