## 6 Discussion  
In the above analyses, we considered ten MSFs and found that there were high similarities between MSFs and the vocal-emotion-recognition results in the noisy reverberant environments. We also clarified that there are two common MSFs (MSKT\({}_{k}\) and MSTL\({}_{k}\)) in all daily environments. We focused on specific MSFs and discussed the contributions of two common MSFs to vocal emotion recognition from the following two perspectives: the emotion perception characteristics of different emotions, and the different properties of acoustic- and modulation-frequency channels. The distribution of each MSF in each emotion is a real-valued random variable, and it has a finite mean and variance. Therefore, this study assumed that the distribution of each MSF in each emotion conforms to a normal distribution. As shown in Figure 11, we estimated its distribution using the probability density functions (PDFs) of the normal distribution; \(\mu\) and \(\sigma^{2}\) are the mean and variance of each MSF taken across the 10 utterances of each emotion, which are the same as shown in Equation (15). Figure 11a-d represent the four environments, and the PDF results of MSKT\({}_{k}\) and MSTL\({}_{k}\) are shown in order from top to bottom in each environment; the five colors represent five different emotions, where \(K\) means the \(K\)th acoustic frequency channel. Due to the page limitation, only the highest similarity (from the first to fourth acoustic-frequency channel) of each MSF is shown in Figure 11.  
In Figure 11, MSTL\({}_{k}\)'s PDFs of sadness have less overlap with other emotions, as explained in Section 4, and the \(\bar{d}_{\text{MSF}}^{\prime}\) represents the physical distance of the MSFs between different emotions. Therefore, the large distance between sadness and other emotions indicates that using MSTL\({}_{k}\) makes it easier to distinguish sadness from other emotions. The reason can be explained by the emotion-perception characteristic of sadness. Sad speech is a type of breathy speech produced when the vocal fold motion is not broad enough to close the glottis completely during the vibration cycle. This phenomenon perceptually decreases the loudness of the speech unit produced, so that a breathy voice has a spectrum with a strong slope. Conversely, for loud speech, such as hot-anger speech and joyous speech, which see a rapid closure of the vocal folds and a short open phase of the glottis, the spectral envelope is flatter (the spectral tilt is less) [24; 25; 26]. Combining the analysis of sad speech, as the results show in Figure 8, we can also see that sadness has a different power  
Figure 11: PDFs of MSKT\({}_{k}\) and MSTL\({}_{k}\) for each emotion in noisy/reverberant/noisy reverberant environments. Only MSFs with highest similarity are shown: (**a**) clean environment [15], (**b**) noisy environment (SNR \(=5\) dB), (**c**) reverberant environment (\(T_{\text{R}}=1.0\) s), and (**d**) noisy reverberant environment (SNR \(=10\) dB, \(T_{\text{R}}=0.5\) s). pattern in time-average modulation representation. In all environments, only sadness had a lower RMS in the second and third acoustic-frequency channels than in the first and fourth. This might also be due to sad speech being more breathy than other emotional speech. It is known that breathy voices are characterized by a steep decrease in spectral level from the 0-2 to the 2-5 kHz band. Conversely, \"vocal fry/creaky\" voices, such as those for hot anger and joy, were correlated with a steep decrease in level from the 2-5 to 5-8 kHz band [24; 27]. As we can see in the time-average modulation representation in Figure 8, sadness has an RMS that steeply decreased in the lower acoustic-frequency range. Hot anger and joy had RMSs that steeply decreased in the higher acoustic-frequency range. Consequently, MSTL\({}_{k}\) is important in sadness recognition. For both MSKT\({}_{k}\) and MSTL\({}_{k}\), in most environments there was less overlap between sadness and other emotions.