# Enhancing Target Speech Based on Nonlinear Soft Masking Using a Single Acoustic Vector Sensor
## 2 Problem Formulation
### Nonlinear Soft Time-Frequency (TF) Mask Estimation  
As discussed above, Equation (31) is valid when the (\(\tau\),\(\omega\)) points are dominated by target speech with high local SNR. Besides, we have three equations to solve two variables, \(\theta_{\text{s}}\) and \(\phi_{\text{s}}\). In this study, from Equation (31), we estimate \(\theta_{\text{s}}\) and \(\phi_{\text{s}}\) in the following way:  
\[\hat{\phi}_{\text{s1}}(\tau,\omega) =\text{arccot}I_{uv}(\tau,\omega)+\Delta\eta_{1} \tag{32}\] \[\hat{\phi}_{\text{s2}}(\tau,\omega) =\text{arctan}I_{vu}(\tau,\omega)+\Delta\eta_{2}\] (33) \[\hat{\phi}_{\text{s}}(\tau,\omega) =\text{mean}(\hat{\phi}_{\text{s1}},\hat{\phi}_{\text{s2}})\] (34) \[\hat{\theta}_{\text{s}}(\tau,\omega) =\text{arccos}I_{uv}(\tau,\omega)+\Delta\eta_{3} \tag{35}\]  
where \(\Delta\eta_{1}\) and \(\Delta\eta_{2}\) are estimation errors. Comparing Equation (31) and Equations (32)-(35), we can see that if the estimated DOA values (\(\hat{\phi}_{\text{s}}(\tau,\omega)\),\(\hat{\theta}_{\text{s}}(\tau,\omega)\)) approximate to the real DOA values (\(\theta_{\text{s}}\),\(\phi_{\text{s}}\)), then \(\Delta\eta_{1}\) and \(\Delta\eta_{2}\) should be small. Therefore, for the TF points (\(\tau\),\(\omega\)) dominated by the target speech, we can derive the following inequality:  
\[\big{|}\hat{\phi}_{\text{s}}(\tau,\omega)-\phi_{\text{s}}\big{|}<\delta_{1} \tag{36}\]  
\[\big{|}\hat{\theta}_{\text{s}}(\tau,\omega)-\theta_{\text{s}}\big{|}<\delta_{2} \tag{37}\]  
Figure 3: The decision tree of handling the special angles in computing \(\mathbf{I}_{\text{isdr}}\). where \(\hat{\phi}_{\rm s}(\tau,\omega)\) and \(\hat{\theta}_{\rm s}(\tau,\omega)\) are the target speaker's DOA estimated by Equations (34) and (35), respectively. \(\theta_{\rm s}\) and \(\phi_{\rm s}\) are given the DOA of the target speech for the SE task. The parameters \(\delta_{1}\) and \(\delta_{2}\) can be set as the predefined permissible parameters (referring to an angle value). Following the derivation up to now, if Equations (36) and (37) are met at (\(\tau\),\(\omega\)) points, we can infer that these (\(\tau\),\(\omega\)) points are dominated by the target speech with high probability. Therefore, using Equations (36) and (37), the TF points (\(\tau\),\(\omega\)) can be extracted, and a mask associated with these (\(\tau\),\(\omega\)) points dominated by the target speech can be designed accordingly. In addition, we need to take the following facts into account. (1) The value of \(\phi_{\rm s}\) belongs to (0,2\(\pi\)). (2) The principal value interval of the arccot function is (0,\(\pi\)), and the arctan function is (\(-\pi\)/2,\(\pi\)/2). (3) The value range of \(\theta_{\rm s}\) is (0,2\(\pi\)). (4) The principal value interval of the arccos function is [0,\(\pi\)]. (5) To make the principal value of the anti-trigonometric function match the value of \(\theta_{\rm s}\) and \(\phi_{\rm s}\), we need to add \(L\pi\) to avoid ambiguity. As a result, a binary TF mask for preserving the target speech is designed as follows:  
\[mask(\tau,\omega)=\left\{ \begin{array}{l}1,\ \mbox{if}\ \left\{\begin{array}{l} \Delta\phi(\tau,\omega)=\left|\hat{\phi}_{\rm s}(\tau,\omega)-\phi_{\rm s}+L \pi\right|<\delta_{1}\\ \Delta\theta(\tau,\omega)=\left|\hat{\theta}_{\rm s}(\tau,\omega)-\theta_{\rm s }+L\pi\right|<\delta_{2}\\ 0,\ \mbox{else}\end{array}\right.\end{array} \right. \tag{38}\]  
where \(L\) = 0, \(\pm\) 1. (\(\Delta\phi(\tau\),\(\omega\)), \(\Delta\theta(\tau\),\(\omega)\)) is the estimation difference between the estimated DOA and the real DOA of the target speaker at TF point (\(\tau\),\(\omega\)). Obviously, the smaller the value of (\(\Delta\phi(\tau\),\(\omega\)), \(\Delta\theta(\tau\),\(\omega)\)), the more probable it is that the TF point (\(\tau\),\(\omega\)) is dominated by the target speech. To further improve the estimation accuracy and suppress the impact of the outliers, we propose a nonlinear soft TF mask as:  
\[mask(\tau,\omega)=\left\{