## 3 Perceptual Distance
_Perceptual domain transformation:_ The target and converted loudness spectra \(\mathbf{s}_{t,n}=[S_{t,n}(0),...,S_{t,n}(Q-1)]^{T}\) and \(\mathbf{\hat{s}}_{t,n}=[\hat{S}_{t,n}(0),...,\hat{S}_{t,n}(Q-1)]^{T}\), which are perceptually closer to the actual human listening are obtained as follows, \[\mathbf{s}_{t,n}=T_{s}[\mathbf{H}\cdot\mathbf{x}_{t,n}],\ \mathbf{\hat{s}}_{t,n}=T_{s}[\mathbf{H}\cdot\mathbf{\hat{x}}_{t,n}]\] (10) where \(Q\) is the number of Bark bands, \(\mathbf{H}\) is a Bark transformation matrix that converts the power spectra \(\mathbf{x}_{t,n}=[\mathbf{X}_{t,n}(0),...,\mathbf{X}_{t,n}(M-1)]\), \(\mathbf{\hat{x}}_{t,n}=[\mathbf{\hat{X}}_{t,n}(0),...,\mathbf{\hat{X}}_{t,n} (M-1)]\) into the Bark spectra \(\mathbf{b}_{t,n}=[B_{t,n}(0),...,B_{t,n}(Q-1)]\), \(\mathbf{\hat{b}}_{t,n}=[\hat{B}_{t,n}(0),...,\hat{B}_{t,n}(Q-1)]\), respectively. \(T_{s}[\cdot]\) is the mapping function that converts each band of the Bark spectrum to a some loudness scale as follows, \[S_{t,n}(q)=s_{t}\Big{(}\frac{P_{0}(q)}{0.5}\Big{)}^{\gamma}\cdot\Big{[}\Big{(} 0.5+0.5\frac{B_{t,n}(q)}{P_{0}(q)}\Big{)}^{\gamma}-1\Big{]}\] (11) where \(s_{t}\) is a loudness scaling factor, \(P_{0}(q)\) is the absolute hearing threshold for the \(q\)-th Bark band and \(\gamma\) is set to 0.23 [25] (2) _Disturbances computation:_ A relative small difference between the target and converted loudness spectra can be negligible [25,30,41]. Accordingly, a center-clipping operator over the absolute difference between the loudness spectra was applied to compute the symmetrical disturbance vector as follows, \[\mathbf{d}_{n}^{(s)}=\max(|\mathbf{\hat{s}}_{t,n}-\mathbf{s}_{t,n}|-\mathbf{m }_{n},\mathbf{0})\] (12) where \(\mathbf{m}_{n}=0.25\cdot\min(\mathbf{\hat{s}}_{t,n},\mathbf{s}_{t,n})\) is a clipping factor and \(|\cdot|\), \(\min(\cdot)\), and \(\max(\cdot)\) are applied element-wise, while \(\mathbf{0}\) is a zero-filled vector of length \(Q\). The asymmetrical disturbance vector is obtained as \(\mathbf{d}_{n}^{(a)}=\mathbf{d}_{n}^{(s)}\odot\mathbf{r}_{n}\), where \(\odot\) denotes an element-wise multiplication and \(\mathbf{r}_{n}\) is a vector of asymmetry ratios with components computed from the Bark spectra, \[R_{n,q}=\Big{(}\frac{\hat{B}_{t,n}(q)+\epsilon}{B_{t,n}(q)+\epsilon}\Big{)}^{\lambda}\] (13) For the speech enhancement task, the constants \(\epsilon\) and \(\lambda\) were set to 50 and 1.2, respectively [25]. In this study, the experiments were carried out to optimally determine the two constants, \(\epsilon\) and \(\lambda\). The experimental results showed that the same values adopted in [25] also yielded the minimum \(D_{\text{PD}}\). The symmetrical and asymmetrical disturbance terms in (8) are given by the weighted sum of each disturbance vector,  
\[ \begin{array}{rcl}D_{n}^{(s)}&=&||\mathbf{w}_{b}||_{1}^{\frac{1}{ 4}}\cdot||\mathbf{w}_{b}\odot\mathbf{d}_{n}^{(s)}||_{2}\\ D_{n}^{(a)}&=&||\mathbf{w}_{b}\odot\mathbf{d}_{n}^{(a)}||_{1}=\mathbf{w}_{b}^{ \intercal}\cdot\mathbf{d}_{n}^{(a)}\end{array}  \tag{14}\]  
where the components of the weight vector \(\mathbf{w}_{b}\) are proportional to the width of the Bard bands, as explained in [30].