## 3 Proposed UDS-Based SSI
### Perceptual Distance  
In the previous SSI methods, the speech parameters were obtained by minimizing the mean squared error (MSE). Although the resultant speech parameters approximated those of the original speech signals, there was no guarantee that the reproduced speech signals would be perceived as the original speech. In the present study, the conventional MSE-based objective function was modified by incorporating both a symmetrical disturbance \(D^{(s)}\) and an asymmetrical disturbance, \(D^{(a)}\)[34]  
\[D_{PD}=\frac{1}{N}\sum_{n}\left(w_{M}D_{MSE,n}+w_{s}D_{n}^{(s)}+w_{a}D_{n}^{(a )}\right) \tag{12}\]  
where \(D_{MSE}(n)\) is the MSE of the \(n\)-th spectrum  
\[D_{MSE,n}=\frac{1}{M}\sum_{m=0}^{M-1}\frac{1}{\sigma_{m}^{2}}\Bigg{(}\log\frac {|P_{m,n}|^{2}}{|\bar{P}_{m,n}|^{2}}\Bigg{)}^{2} \tag{13}\]  
where \(|P_{m,n}|^{2}\) and \(|\bar{P}_{m,n}|^{2}\) are, respectively, the original and predicted power spectra where the indexes, \(m\) and \(n\) denote frequency and frame, respectively. \(\sigma_{m}\) is the standard deviation of \(|P_{m}|^{2}\) and \(M\) is the number of frequency bins. In (12), \(w_{M}\), \(w_{s}\), and \(w_{a}\) are weighting factors for the MSE, and the symmetrical and asymmetrical disturbances, respectively. The symmetrical disturbance reflects the absolute difference between the converted and target loudness spectra when auditory masking effects are account for. When the symmetrical disturbance is applied to a SSI, it can be regarded as a distance function between the estimated and target speech represented in a domain that reflects the human auditory system. There are two types of difference patterns in a SSI, one where the target value is greater than the estimated value and vice-versa. Such difference patterns cannot be reflected in distance metrics such as the MSE or the the symmetrical disturbance. Whereas the signs  
Figure 3: Block diagram of the proposed ultrasonic SSI system. of the loudness differences are considered in the asymmetrical disturbance, the negative differences (under-estimation case) and positive differences (over-estimation case) are perceived differently due to the masking effects. By using the asymmetrical disturbance, the differences between the estimated and target speech signals can be described in more detail, which can lead to an improvement in the SSI performance. The calculation of symmetrical and asymmetrical disturbances reflects the human auditory system and is composed of several steps, as follows [34]:  
1. Perceptual domain transformation: The target and converted loudness spectra \(\mathbf{s}_{n}=[S_{0,n},\dots,S_{Q-1,n}]^{T}\) and \(\mathbf{\hat{s}}_{n}=[\hat{S}_{0,n},\dots,\hat{S}_{Q-1,n}]^{T}\), which are perceptually closer to human listening are obtained as follows, \[\mathbf{s}_{n}=T_{s}[\mathbf{H}^{(bt)}\cdot\mathbf{p}_{n}],\ \mathbf{\hat{s}}_{n}=T_{s}[\mathbf{H}^{(bt)}\cdot\mathbf{\hat{p}}_{n}]\] (14) where \(Q\) is the number of Bark bands, \(\mathbf{H}^{(bt)}\) is a Bark transformation matrix that converts the power spectra \(\mathbf{p}_{n}=[P_{0,n},\dots,P_{M-1,n}]\), \(\mathbf{\hat{p}}_{n}=[\hat{P}_{0,n},\dots,\hat{P}_{M-1,n}]\) into the Bark spectra \(\mathbf{b}_{n}=[B_{0,n},\dots,B_{Q-1,n}]\) and \(\mathbf{\hat{b}}_{n}=[\hat{B}_{0,n},\dots,\hat{B}_{Q-1,n}]\), respectively. \(T_{s}[\cdot]\) is a mapping function that converts each band of the Bark spectrum to a some loudness scale. A detailed description of this function is found in [34]. 2. Disturbances computation: A center-clipping operator over the absolute difference between the loudness spectra was applied to compute the symmetrical disturbance vector as follows, \[\mathbf{d}_{n}^{(s)}=\max(|\mathbf{\hat{s}}_{n}-\mathbf{s}_{n}|-\mathbf{m}_{n },\mathbf{0})\] (15) where \(\mathbf{m}_{n}=0.25\cdot\min(\mathbf{\hat{s}}_{n},\mathbf{s}_{n})\) is a clipping factor and \(|\cdot|\), \(\min(\cdot)\), and \(\max(\cdot)\) are applied element-wise. The designation \(\mathbf{0}\) is a zero-filled vector of length \(Q\).