# Enhancing Target Speech Based on Nonlinear Soft Masking Using a Single Acoustic Vector Sensor
## 2 Problem Formulation
### AVS Data Model  
An AVS unit generally consists of \(f\) co-located constituent sensors, including one omnidirectional sensor (denoted as _o_-sensor) and _J_-1 orthogonally oriented directional sensors. Figure 1 shows the data capture setup with a single AVS. It is noted that the left bottom plot in Figure 1 shows a 3D-AVS unit implemented by our team, which consists of one o-sensor with three orthogonally oriented directional sensors depicted as the _u_-sensor, _v_-sensor, and _w_-sensor, respectively. In theory, the directional response of the oriented directional sensor has dipole characteristics, as shown in Figure 2a, while the omnidirectional sensor has the same response in all of the directions, as shown in Figure 2b. In this study, one target speaker is considered. As shown in Figure 1, the target speech \(S(t)\) is impinging from (\(\theta_{s}\),\(\phi_{s}\)) meanwhile, interference \(S_{i}(t)\) are impinging from (\(\theta_{j}\),\(\phi_{j}\)), where \(\phi_{s}\), \(\phi_{i}\)\(\in\) (0\({}^{\circ}\),360\({}^{\circ}\)) are the azimuth angles, and \(\theta_{s}\), \(\theta_{i}\)\(\in\) (0\({}^{\circ}\),180\({}^{\circ}\)) are the elevation angles.  
Figure 1: Illustration of a single acoustic vector sensor (AVS) for data capturing.  
Figure 2: (**a**) The directional response of oriented directional sensor; (**b**) The directional response of omnidirectional sensor.  
For simplifying the derivation, without considering room reverberation, the received data of the AVS can be modeled as [13]:  
\[\mathbf{x}_{\textit{arcs}}(t)=\mathbf{a}(\theta_{s},\phi_{s})s(t)+\sum_{i=1}^{M_{i}}\bm {a}(\theta_{i},\phi_{i})s_{i}(t)+\mathbf{n}_{\textit{arcs}}(t) \tag{2}\]  
where \(\mathbf{x}_{\textit{arcs}}(t)\), \(\mathbf{n}_{\textit{arcs}}(t)\) and \(\mathbf{a}(\theta_{s}\phi_{s})\) are defined respectively as:  
\[\mathbf{x}_{\textit{arcs}}(t)=[x_{u}(t),x_{v}(t),x_{w}(t),x_{o}(t)] \tag{3}\]  
\[\mathbf{n}_{\textit{arcs}}(t)=[u_{u}(t),u_{v}(t),n_{w}(t),n_{o}(t)] \tag{4}\]  
\[\mathbf{a}(\theta_{s},\phi_{s})=[u_{s},v_{s},w_{s},1]^{T}=[\sin\theta_{s}\cos \phi_{s},\sin\theta_{s}\sin\phi_{s},\cos\theta_{s},1]^{T} \tag{5}\]  
\[\mathbf{a}(\theta_{i},\phi_{i})=[u_{i},v_{i},w_{i},1]^{T}=[\sin\theta_{i}\cos \phi_{i},\sin\theta_{i}\sin\phi_{i},\cos\theta_{i},1]^{T} \tag{6}\]  
In Equation (3), \(x_{u}(t)\), \(x_{v}(t)\), \(x_{w}(t)\), \(x_{o}(t)\) are the received data of the \(u\)-sensor, \(v\)-sensor, \(w\)-sensor, and \(o\)-sensor, respectively; \(n_{u}(t)\), \(n_{v}(t)\), \(n_{w}(t)\), \(n_{o}(t)\) are assumed as the additive zero-mean white Gaussian noise captured at the \(u\)-sensor, \(v\)-sensor, \(w\)-sensor, and \(o\)-sensor, respectively; \(s(t)\) is the target speech; \(s_{i}(t)\) are the \(i\)th interfering speech; the number of interferences is \(M_{i}\); \(\mathbf{a}(\theta_{s}\phi_{s})\) and \(\mathbf{a}(\theta_{j}\phi_{j})\) are the steering vectors of \(s(t)\) and \(s_{i}(t)\), respectively. \([.]^{T}\) denotes the vector/matrix transposition.  
From the AVS data model given in Equation (2), taking the short-time Fourier transform (STFT), for a specific TF point (\(\tau\),\(\omega\)), we have:  
\[\mathbf{X}_{\textit{arcs}}(\tau,\omega)=\mathbf{a}(\theta_{s},\phi_{s})S(\tau,\omega )+\sum_{i=1}^{M_{i}}\mathbf{a}(\theta_{i},\phi_{i})S_{i}(\tau,\omega)+\mathbf{N}_{ \textit{arcs}}(\tau,\omega) \tag{7}\]  
where \(\mathbf{X}_{\textit{arcs}}(\tau\),\(\omega)\) = [\(X_{u}(\tau\),\(\omega)\), \(X_{v}(\tau\),\(\omega)\), \(X_{w}(\tau\),\(\omega)\)]\({}^{T}\); \(X_{u}(\tau\),\(\omega)\), \(X_{v}(\tau\),\(\omega