# Enhancing Target Speech Based on Nonlinear Soft Masking Using a Single Acoustic Vector Sensor
## 2 Problem Formulation
### Monotone Functions between ISDRs and the DOA  
Definition and some discussions on the inter-sensor data ratio (ISDR) of the AVS are presented in our previous work [13]. In this subsection, we briefly introduce the definition of ISDR first, and then present the derivation of the monotone functions between the ISDRs and the DOA of the target speaker. The ISDRs between each channel of the AVS are defined as:  
\[I_{ij}(\tau,\omega)=X_{i}(\tau,\omega)/X_{j}(\tau,\omega)\;\;\text{where}\;(i \
eq j) \tag{13}\]  
where \(i\) and \(j\) are the channel index, which refers to \(u\), \(v\), \(w\), and \(o\), respectively. Obviously, there are 12 different computable ISDRs, which are shown in Table 1. In the following context, we carefully evaluate \(I_{ij}\), and it is clear that only three ISDRs (\(I_{uv}\), \(I_{vu}\) and \(I_{vu}\)) hold the approximate monotone function between ISDR and the DOA of the target speaker. According to the definition of ISDRs given in Equation (13), we look at \(I_{uv}\), \(I_{vu}\) and \(I_{uv}\) first. Specifically, we have:  
\[I_{uv}(\tau,\omega)=X_{u}(\tau,\omega)/X_{v}(\tau,\omega) \tag{14}\]  
\[I_{vu}(\tau,\omega)=X_{v}(\tau,\omega)/X_{u}(\tau,\omega) \tag{15}\]  
\[I_{uv}(\tau,\omega)=X_{w}(\tau,\omega)/X_{o}(\tau,\omega) \tag{16}\]  
Substituting Equations (9) and (10) into Equation (14) gives:  
\[I_{uv}(\tau,\omega)=\frac{u_{s}S(\tau,\omega)+N_{tu}(\tau,\omega)}{v_{s}S(\tau,\omega)+N_{tv}(\tau,\omega)}=\frac{u_{s}+N_{tu}(\tau,\omega)/S(\tau,\omega)}{ v_{s}+N_{tv}(\tau,\omega)/S(\tau,\omega)}=\frac{u_{s}+\varepsilon_{tus}(\tau, \omega)}{v_{s}+\varepsilon_{tus}(\tau,\omega)} \tag{17}\]  
where \(\varepsilon_{tus}(\tau,\omega)=N_{tu}(\tau,\omega)/S(\tau,\omega)\), and \(\varepsilon_{tus}(\tau,\omega)=N_{tv}(\tau,\omega)/S(\tau,\omega)\). Similarly, we get \(I_{uv}\) and \(I_{uv}\):  
\[I_{vu}(\tau,\omega)=\frac{v_{s}S(\tau,\omega)+N_{tv}(\tau,\omega)}{u_{s}S( \tau,\omega)+N_{tu}(\tau,\omega)}=\frac{v_{s}+N_{tv}(\tau,\omega)/S(\tau, \omega)}{u_{s}+N_{tu}(\tau,\omega)/S(\tau,\omega)}=\frac{v_{s}+\varepsilon_{ tus}(\tau,\omega)}{u_{s}+\varepsilon_{tus}(\tau,\omega)} \tag{18}\]  
\[I_{uv}(\tau,\omega)=\frac{w_{s}S(\tau,\omega)+N_{tw}(\tau,\omega)}{S(\tau, \omega)+N_{tu}(\tau,\omega)}=\frac{w_{s}+N_{tw}(\tau,\omega)/S(\tau,\omega)}{ \mathbf{1}+N_{tu}(\tau,\omega)/S(\tau,\omega)}=\frac{w_{s}+\varepsilon_{tus}( \tau,\omega)}{\mathbf{1}+\varepsilon_{tus}(\tau,\omega)} \tag{19}\]  
In Equation (19), \(\varepsilon_{tus}(\tau,\omega)=N_{tv}(\tau,\omega)/S(\tau,\omega)\) and \(\varepsilon_{tos}(\tau,\omega)=N_{to}(\tau,\omega)/S(\tau,\omega)\). Based on the assumption of TF sparsity of speech shown in Section 2.1, we can see that if the TF points (\(\tau,\omega\)) are dominated by the target speech from (\(\theta_{s}\),\(\phi_{s}\)), the energy of the target speech is high, and the value of \(\varepsilon_{tus}(\tau,\omega)\), \(\varepsilon_{tus}(\tau,\omega)\), \(\varepsilon_{tus}(\tau,\omega)\) and \(\varepsilon_{tos}(\tau,\omega)\) tends to be small. Then, Equations (17)-(19) can be accordingly approximated as:  
\[I_{uv}(\tau,\omega)\approx u_{s}/v_{s}+\varepsilon_{1}(\tau,\omega) \tag{20}\]  
\[I_{vu}(\tau,\omega)\approx v_{s}/u_{s}+\varepsilon_{2}(\tau,\omega) \tag{21}\]  
Table 1.0 \begin{table}
\begin{tabular}{c c c c c} \hline
**Sensor** & \(\mathbf{u}\) & \(\mathbf{v}\) & \(\mathbf{w}\) & \(\mathbf{o}\) \\ \hline \(u\) & NULL & \(I_{vu}\) & \(I_{vu}\) & \(I_{vu}\) \\ \(v\) & \(I_{uv}\) & NULL & \(I_{uv}\) & \(I_{vu}\) \\ \(w\) & \(I_{uv}\) & \(I_{vu}\) & NULL & \(I_{vu}\) \\ \(o\) & \(I_{uv}\) & \(I_{vu}\) & \(I_{uv}\) & NULL \\ \hline \end{tabular}
\end{table} 
Table 1: Twelve computable inter-sensor data ratios (ISDRs). where \(\varepsilon_{1}\), \(\varepsilon_{2}\), and \(\varepsilon_{3}\) can be viewed as the ISDR modeling error with zero-mean introduced by interferences and background noise. Moreover, \(\varepsilon_{i}(\tau\),\(\omega)\) (\(i=1\), \(2\), \(3\)) is inversely proportion to the local SNR at (\(\tau\),\(\omega\)).