## 2 Materials and Methods
### Analyzer  
Figure 4 shows the block diagram of the analysis-by-synthesis approach used in the analyzer to model input GAWs. A similar approach was proposed in the past to model diplophonic GAWs [20] where two simultaneous fundamental frequency (\(f_{o}\)) tracks were used. In this study, instead of two \(f_{o}\) tracks, only one \(f_{o}\) track is estimated. First, the \(f_{o}\) track of the input GAW is extracted by a hidden Markov model (HMM) combined with repetitive execution of a Viterbi algorithm. An unmodulated quasi unit pulse train \(u_{t}\) is generated by an oscillator driven by the extracted \(f_{o}\) track. The pulse locations of this pulse train approximate the time instants of the maxima of the input GAW. An additional pulse train for indicating the locations of the minima of the input GAW is obtained by phase shifting the train's phase by \(180^{\circ}\). The instantaneous phase and amplitude of the maxima and minima are extracted from the quasi unit pulse train, which is explained in detail in Section 2.5.  
A Fourier synthesizer (FS) uses the extracted instantaneous phase \(\Theta\)(t) and the Fourier coefficients \(R_{k}\) obtained from pulse shapes \(r_{l}\) to model the input GAW. The pulse shapes  
Figure 4: The modified version of the analyzer used for modelling GAWs, modified from [20].  
Figure 3: Overview of the analyzer and the synthesizer used for modelling and synthesizing GAWs. The analyzer models and estimates parameters of input GAWs, which is described in Section 2.2. The synthesizer described in Section 2.4 generates synthetic GAWs \(y_{s}(t)\) is based on the distributions of the estimated parameters \(\psi_{i}\) that are fitted as described in Section 2.3.  
are obtained by cross-correlating the quasi unit pulse with each block of the input GAW of length 32 ms obtained using a Hanning window with a 50% overlap. These single-peak pulse shapes are then modelled using [PERSON]'s model [24]. [PERSON]'s model estimates the pulse parameters which are used for generating synthetic GAWs. The pulse shapes are transformed to the Fourier coefficient \(R_{k}\) using the discrete Fourier transform (DFT). The synthesized GAW \(y_{F}(t)\) is obtained by the Fourier synthesizer using the Fourier coefficients of the pulse shapes and the extracted instantaneous phase. The synthesized GAW \(y_{F}(t)\) is multiplied with an amplitude modulator \(m(t)\) to obtain a modelled GAW,  
\[\hat{y}(t)=y_{F}(t)\cdot m(t) \tag{1}\]  
The modelling error \(E\) is the level of the root mean squared difference between the input GAW \(y(t)\) and the modelled GAW \(\hat{y}(t)\) i.e.,  
\[E(t)=20\cdot log_{10}\left(\frac{\sqrt{\overline{\left(y(t)-\hat{y}(t)\right) ^{2}}}}{\sqrt{\overline{y^{2}(t)}}}\right)\ \ [dB] \tag{2}\]  
\(\hat{y}(t)\) obtained using the unmodulated quasi unit pulse train \(u_{t}\) is the output of a nonmodulating model where the frequency and amplitude modulation present in the original GAWs are not modelled. Random modulations of the individual pulses of the input GAW are modelled by minimizing the error \(E\) via modulating the quasi unit pulse trains, which approximates a nonlinear system in a linear way. Pulse heights and time instants of the unit pulse trains are modulated on a pulse-to-pulse time scale. Additionally, the modelled GAWs are clipped at the baseline to minimize the modelling error \(E\). The amplitude modulation vector \(h(\rho)\) and the pulse time modulation vector \(\Delta(\rho)\) of the quasi unit pulse train are estimated iteratively by minimizing the time domain modelling error \(E(t)\). \(\rho\) is the pulse index.