# A Unified Speech Enhancement System Based on Neural Beamforming With Parabolic Reflector
## 2 Materials and Methods
### Two Approaches to Utilize Masks
#### 2.3.2 Neural Beamformer  
The frequency-domain beamformer is used to reduce computational complexity. \(\mathbf{w}(n,k)\) denotes the corresponding beamforming filter coefficients. Using a spatial filtering approach, an estimation of the desired signal is given as a linear combination of the microphone signals \(\mathbf{y}(n,k)\), at each \(\mathbf{T}-\mathbf{F}\) bin, as illustrated in Equation (6):  
\[\hat{X}_{d}(n,k,ref_{m})=\mathbf{w}^{\mathrm{H}}(n,k)\mathbf{y}(n,k), \tag{6}\]  
where H represents conjugate transpose. In this work, the MVDR criterion was used to compute filter coefficients, and in this criterion, coefficients were found by minimizing the power of noise components at output, given by Equations (7) and (8). \[\mathbf{w}(n,k)=\underset{\mathbf{w}}{\mathrm{argmin}}\mathbf{w}^{\mathrm{H}} \mathbf{\Phi}_{\mathrm{n}}\mathbf{w}, \tag{7}\]  
subject to:  
\[\mathbf{w}^{\mathrm{H}}(n,k)\mathbf{a}(n,k)=1, \tag{8}\]  
where \(\mathbf{a}(n,k)\) denotes the Relative Transfer Function (RTF) vector. Considering the individual signal components in Equation (2) to be uncorrelated, the Power Spectral Density (PSD) matrix of microphone signals can be expressed as Equation (9):  
\[\mathbf{\Phi}_{\mathbf{y}}(n,k)=\mathrm{E}\!\left[\mathbf{y}(n,k)\mathbf{y}^{ \mathrm{H}}(n,k)\right]=\mathbf{\Phi}_{\mathrm{x}_{\mathrm{d}}}(n,k)+\mathbf{ \Phi}_{\mathrm{n}}(n,k), \tag{9}\]  
where \(\mathrm{E}\!\left\{\cdot\right\}\) represents the expectation operator, \(\mathbf{\[PERSON]}_{\mathrm{x}_{\mathrm{d}}}(n,k)\) denotes the rank-one PSD matrix of the desired signal, and \(\mathbf{\Phi}_{\mathrm{n}}(n,k)\), denotes the PSD matrix of noise components. PSD matrices are robustly estimated using the expectation with respect to time-frequency masks as Equations (10) and (11):  
\[\mathbf{\Phi}_{\mathrm{x}_{\mathrm{d}}}(k)=\frac{1}{\sum_{n}^{N}I_{(\cdot)}(n, k)}\!\sum_{n=1}^{N}I_{(\cdot)}(n,k)\mathbf{y}(n,k)\mathbf{y}^{\mathrm{H}}(n,k), \tag{10}\]  
\[\mathbf{\Phi}_{\mathrm{n}}(k)=\frac{1}{\sum_{n}^{N}1-I_{(\cdot)}(n,k)}\!\sum_{ n=1}^{N}\left(1-I_{(\cdot)}(n,k)\right)\mathbf{y}(n,k)\mathbf{y}^{\mathrm{H}}(n,k), \tag{11}\]The closed-form solution to the optimization problem is given by Equation (12) in [21]:  
\[\mathbf{w}(n,k)=\frac{\mathbf{\Phi}_{\mathrm{n}}^{-1}\mathbf{a}(n,k)}{\mathbf{a}^ {\mathrm{H}}(n,k)\mathbf{\Phi}_{\mathrm{n}}^{-1}\mathbf{a}(n,k)}. \tag{12}\]  
Adopting the optimization formalization [21], the explicit dependence of the above filter on the relative transfer functions can be eliminated, and the following form can be obtained, as illustrated in Equation (13):  
\[\mathbf{w}=\frac{\left(\mathbf{\Phi}_{\mathrm{n}}(k)\right)^{-1}\mathbf{\Phi }_{\mathrm{x}_{\mathrm{d}}}(k)}{\mathrm{Tr}\!\left(\left(\mathbf{\Phi}_{ \mathrm{n}}(k)\right)^{-1}\mathbf{\Phi}_{\mathrm{x}_{\mathrm{d}}}(k)\right)} \mathbf{u}, \tag{13}\]  
where \(\mathbf{u}\) is a one-hot vector representing a reference microphone, and \(\mathrm{Tr}(\cdot)\) represents matrix trace operation. The overall flowchart of the proposed speech enhancement framework is illustrated in Figure 4. The circular microphone array captures the noisy speech. The speech is processed by a neural beamformer, which removes most of the interference. So much high frequency information is lost. In this approach, any microphone in the circular array can be defined as the reference microphone to estimate the mask. The physical amplification characteristic of the designed PR model has significant effects in speech enhancement, and some speech distortions are also introduced into the target speech. The signals captured by the microphone at the focus and the output of the neural beamformer1 have complementary information of the desired speech in the frequency domain. By processing the signals from two approaches mentioned earlier, the fusion operation aims to make a trade-off between speech distortion and speech intelligibility. The reflector microphone signal is utilized to estimate the mask, and the direct mask application is used to enhance the desired speech.