# Wiener Filter and Deep Neural Networks: A Well-Balanced Pair for Speech Enhancement
## 3 Proposal  
This paper takes advantage of the data-driven paradigm behind deep learning for modeling the relationship between noise and clean data. We propose to obtain an estimate of the full term \(G_{MMSE}(k,l)=\frac{\xi}{1+\xi}\), i.e., the Wiener filter directly from the noisy signal by means of a deep structured network. This term is subsequently used in the expression of the gain function of the optimally modified version of LSA according to Equations (4) and (6).  
When the observed signal is barely noise affected, i.e., it is mostly clean, the dynamic range of the SNR can rise to \(\infty\). In this case, the regression would be more sensitive to errors because there will be a huge amount of possible values to provide as result. However, as the \(G_{MMSE}\) depends on the SNR (Equation (9)), high SNR conditions provoke high gain values \(G_{MMSE}\to 1\), while low SNR conditions dump \(G_{MMSE}\to 0\). This way, the dynamic range for the regression to obtain \(G_{MMSE}\) would be bounded \([0,1]\), which is a more accurately achievable task for a deep structured network.  
\[G_{MMSE}=\frac{1}{1+\frac{1}{SNR}} \tag{9}\]  
Furthermore, the use of a deep structured network in this task was also motivated by the fact that we can implement a causal enhancement system. This means that as this is not necessarily dependent on future time frames, it can be employed in online applications. In addition, the network performs non-recursive estimations, which avoid the re-insertion of estimation errors from previous frames. The mentioned previous statistical SNR-estimators are usually based on recursive and causal schemes [9; 13].  
Figure 2 depicts the DL-based noise reduction method proposed. The deep structured network is trained in supervised mode with clean speech data and noise patterns through a standard data augmentation process. The goal of the network is to obtain an accurate estimate of the MMSE gain from the noisy signal. During training, in order to obtain this estimate, according to Equations (2) and (7), the network needs to know the PSD of the clean speech \(P_{x}(k,l)\) and the PSD of the noise \(P_{d}(k,l)\). We estimate those PSDs by means of the Welch method [29].  
During inference, the input to the network is only the noisy signal \(y(n)\). The \(G_{MMSE}(k,l)\) delivered by the network is directly applied along with the speech presence probability term \(p(k,l)\) to compute the final speech enhancement filter gain function \(G(k,l)\) according to Equations (4) and (6). There are many ways of estimating the speech presence probability. Since the Wiener filter provides a value between 0 (when noise energy is much higher than the speech energy) and 1 (when the noise energy is much lower than the speech energy) we have directly used this term as an accurate estimate of the speech presence probability for each time instant and for each frequency bin. This way of estimating the presence of speech in the noisy signal is also appropriate in the sense that it does not require the use of estimates of the clean signal or the noise signal, avoiding the appearance of feedback loops that involve the speech enhancement gain function estimated by the method.