## 3 Results  
Figure 5 shows waveforms involved in the modelling of an example of an amplitude modulated natural vocal fry GAW using the nonmodulating model (top plot) and the modulating model (bottom plot). The input GAW is an amplitude modulated signal with four pulses within a modulator cycle (CAMP = 4). The modulating model has a modelling error of \(-11.55\) dB for the example GAW, which is larger than the modelling error obtained using the nonmodulating model (\(-2.15\) dB). A smaller modelling error indicates a better fit to the input GAW. The modulating model is observed to fit the input GAW better than the nonmodulating model. In particular, the nonmodulating model fails to track the jitter and shimmer (instantaneous frequency and amplitude modulations) of the individual pulses of the input GAW. As a result, the individual extrema of the estimated unit pulse trains (not shown here) are equally high, which results in constant upper and lower envelopes. Hence, the modulator estimated using these constant envelopes has negligible fluctuation. On the other hand, in the modulating model, the quasi unit pulse trains track the instantaneous  
Figure 5: Modelling of an example of an amplitude modulated vocal fry GAW using the nonmodulating model (top plot) and the modulating model (bottom plot). frequency and amplitude modulation of the individual pulses of the input GAW. The modulator estimated using these quasi unit pulse trains fluctuates in accordance with the input GAW, which results in a modelling error smaller than the error achieved when using the nonmodulating model. Figure 6 shows waveforms involved in the modelling of an example of a euphonic GAW using the nonmodulating model (top plot) and the modulating model (bottom plot). The modulation is negligible as compared to what is observed in vocal fry GAW in Figure 5. Thus, the GAWs modelled by the two models are similar which makes the difference between the modelling errors smaller compared to the difference between the modelling errors obtained for the GAW shown in Figure 5.  
The subtypes of amplitude modulated vocal fry GAWs are distinguished using the entropy of the modulator's power spectral density (PSD) and the modulator-to-carrier frequency ratio \(r_{f}\). Figure 8 shows box plots of the entropy and \(r_{f}\) values of natural and synthetic vocal fry GAW subtypes. The entropy of the modulator's PSD is used to distinguish between cyclically and acyclically modulated vocal fry GAWs. The entropy of the modulators' PSD is larger for modulator with a homogeneous spectrum than for a modulator with inhomogeneous spectrum. This results in larger modulator spectrum entropies for AAMP than for CAMP. An Anova test is performed to check the variance of entropy between the subtypes of vocal fry GAWs. The \(p\)-values obtained between CAMP (CAMP = 2 and CAMP = 4) and AAMP are less than 0.01, which indicates significant differences between the group means of CAMP and AAMP. For the two cyclically modulated groups, the obtained p-value is 0.2526. Therefore, a distinction is possible between CAMP and AAMP type GAWs based on the entropies of the modulator spectrum. To make a distinction between the two subtypes of cyclically modulated vocal fry GAWs, the \(r_{f}\) ratio is used. The \(r_{f}\) ratio for double pulsing (CAMP = 2) is approximately 0.5 and for quadruple pulsing (CAMP = 4) is approximately 0.25. In double pulsing, a modulator cycle contains two pulses, whereas in quadruple pulsing, a modulator cycle contains four pulses. The Anova test for the \(r_{f}\) ratio resulted in a \(p\)-value of less than 0.05, which indicates a significant difference between the cyclically modulated vocal fry GAWs. Figure 8: Features used for distinguishing the subtypes of synthetic vocal fry GAWs: (**a**) entropy of the modulators’ power spectral density (PSD) for distinguishing cyclically vs. acyclically modulated vocal fry and (**b**) modulator-to-carrier frequency ratio \(r_{f}\) to distinguish CAMP = 2 and CAMP = 4 type vocal fry GAWs.