# Enhancing Target Speech Based on Nonlinear Soft Masking Using a Single Acoustic Vector Sensor
## 2 Problem Formulation
### Monotone Functions between ISDRs and the DOA
Furthermore, from Equation (5), we have \(u_{s}=\text{sin}\theta_{s}\cdot\text{cos}\phi_{s}\), \(v_{s}=\text{sin}\theta_{s}\cdot\text{sin}\phi_{s}\) and \(w_{s}=\text{cos}\theta_{s}\). Then, substituting Equation (5) into Equations (20)-(22), we obtain the following equations:  
\[I_{uv}(\tau,\omega)\approx\frac{\sin\theta_{s}\cos\phi_{s}}{\sin\theta_{s}\sin \phi_{s}}+\varepsilon_{1}(\tau,\omega)\ =\cot\phi_{s}+\varepsilon_{1}(\tau,\omega) \tag{23}\]  
\[I_{vu}(\tau,\omega)\approx\frac{\sin\theta_{s}\sin\phi_{s}}{\sin\theta_{s}\cos \phi_{s}}+\varepsilon_{2}(\tau,\omega)\ =\tan(\phi_{s})+\varepsilon_{2}(\tau,\omega) \tag{24}\]  
\[I_{vu}(\tau,\omega)\approx w_{s}+\varepsilon_{3}(\tau,\omega)\ =\cos(\theta_{s})+ \varepsilon_{3}(\tau,\omega) \tag{25}\]  
From Equations (23)-(25), it is desired to see that the approximate monotone functions between \(I_{uv}\), \(I_{vu}\), and \(I_{uv}\) and the DOA (\(\theta_{s}\) or \(\phi_{s}\)) of the target speaker have been obtained since arccot, arctan, and arccos functions are all monotone functions. However, except for \(I_{uv}\), \(I_{vu}\), and \(I_{uv}\), other ISDRs do not hold such a property. Let's take \(I_{uw}\) as an example. From the definition in Equation (13), we can get:  
\[I_{uw}(\tau,\omega)=\frac{u_{s}S(\tau,\omega)+N_{tu}(\tau,\omega)}{w_{s}S(\tau,\omega)+N_{tu}(\tau,\omega)}\ =\frac{u_{s}+N_{tu}(\tau,\omega)/S(\tau,\omega)}{w_{s}+N_{tu}(\tau,\omega)/S( \tau,\omega)}\ =\frac{u_{s}+\varepsilon_{tus}(\tau,\omega)}{w_{s}+\varepsilon_{ tus}(\tau,\omega)}=\frac{u_{s}}{w_{s}}+\varepsilon_{a}(\tau,\omega) \tag{26}\]  
where \(\varepsilon_{4}\) can be viewed as the ISDR modeling error with zero-mean introduced by unwanted noise. Obviously, Equation (26) is valid when \(w_{s}\) is not equal to zero. Substituting Equation (5) into Equation (26) yields:  
\[I_{uw}(\tau,\omega)\approx\frac{\sin\theta_{s}\cos\phi_{s}}{\cos\theta_{s}}+ \varepsilon_{4}(\tau,\omega)\ =\tan\theta_{s}\cos\phi_{s}+\varepsilon_{4}(\tau,\omega) \tag{27}\]  
From Equation (27), we can see that \(I_{uw}\) is a function of both \(\theta_{s}\) and \(\phi_{s}\). In summary, after analyzing all of the ISDRs, we find that the desired monotone functions between ISDRs and \(\theta_{s}\) or \(\phi_{s}\), which are given in Equations (23)-(25), respectively. It is noted that Equations (23)-(25) are derived conditioned by assuming \(v_{s}\), \(u_{s}\), and \(w_{s}\) are not equal to zero. Therefore, we need to find out where \(v_{s}\), \(u_{s}\), and \(w_{s}\) are equal to zero. For presentation clarity, let's define an ISDR vector \(\textbf{I}_{\text{isdr}}=[I_{uv},I_{vu},I_{uv}]\). From Equation (5), it is clear that when the target speaker is at angles of \(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\), and \(270^{\circ}\), one of \(v_{s}\), \(u_{s}\), and \(w_{s}\) becomes zero, and it means that \(\textbf{I}_{\text{isdr}}\) is not fully available. Specifically, we need to consider the following cases:  
**Case 1**: the elevation angle \(\theta_{s}\) is about \(0^{\circ}\) or \(180^{\circ}\). In this case, \(u_{s}=\text{sin}\theta_{s}\cdot\text{cos}\phi_{s}\) and \(v_{s}=\text{sin}\theta_{s}\cdot\text{sin}\phi_{s}\) are close to zero. Then, the denominator in Equations (20) and (21) is equal to zero, and we cannot obtain \(I_{uv}\) and \(I_{vu}\), but we can get \(I_{uv}\). **Case 2**: \(\theta_{s}\) is away from \(0^{\circ}\) or \(180^{\circ}\). In this condition, we need to look at \(\phi_{s}\) carefully. 1. \(\phi_{s}\) is about \(0^{\circ}\) or \(180^{\circ}\). Then, \(v_{s}=\text{sin}\theta_{s}\text{sin}\phi_{s}\) is close to zero, and the denominator in Equation (20) is equal to zero, which leads to \(I_{uv}\) being invalid. In this case, we can compute \(I_{vu}\) and \(I_{uv}\) properly. 2. \(\phi_{s}\) is about \(90^{\circ}\) or \(270^{\circ}\). Then, \(u_{s}=\text{sin}\theta_{s}\cdot\text{cos}\phi_{s}\) is close to zero, and the denominator in Equation (21) is equal to zero, which leads to \(I_{vu}\) being invalid. In this case, we can obtain \(I_{uv}\) and \(I_{uv}\) properly. 3.