## 3 Proposed SE Framework with DWT Features
### Fusion of Two-Level DWT Features and Time-Domain Features  
As introduced in the previous section, the \(L\)-level DWT splits the lower sub-band from the \((L-1)\)-level DWT into two sub-bands. A two-level DWT can extract low-frequency information from short-time speech signals for further analysis with a higher resolution than a one-level DWT. In addition, a higher-level DWT somehow agrees more with the human perception of hearing. Therefore, we propose using a two-level DWT to build features for the frame signals and then integrate them with the time-domain features for the encoder of an SE network such as Conv-TasNet. Figure 9: (**a**) The BPF module of DWT features (**b**) the encoder part of CD-TCN with time-domain and one-level DWT features by fusion and concatenation. The flowchart of the Conv-TasNet with the two-level DWT outputs as one source of encoder features is depicted in Figure 10. Referring to the previous section, each frame signal \(\mathbf{x}_{k}\) passes through a two-level DWT to produce its three sub-band signals, denoted by \(\mathbf{c}_{k}^{D_{1}}\), \(\mathbf{c}_{k}^{D_{2}}\) and \(\mathbf{c}_{k}^{A_{2}}\), respectively. Then, we organize these sub-band signals to produce three feature matrices, \(C_{D_{1}}\), \(C_{D_{2}}\) and \(C_{A_{2}}\), consisting of \(\mathbf{c}_{k}^{D_{1}}\), \(\mathbf{c}_{k}^{D_{2}}\) and \(\mathbf{c}_{k}^{A_{2}}\) as columns, respectively. Next, for the one-level DWT outputs in the previous section, we process the three DWT outputs, \(C_{D_{1}}\), \(C_{D_{2}}\) and \(C_{A_{2}}\), individually with a 1-D trainable convolution (together with the nonlinear function \(H\)) to produce the other three matrices \(W_{D_{1}}\), \(W_{D_{2}}\) and \(W_{A_{2}}\) with the same size as the time-domain feature matrix \(W_{T}\), which comes from Equation (1). The operations are formulated as  
\[W_{D_{1}}=H(U_{D_{1}}C_{D_{1}}),\ \ W_{D_{2}}=H(U_{D_{2}}C_{D_{2}}),\ \ W_{A_{2}}=H(U_{A_{2}}C_{A_{2}}), \tag{12}\]  
where \(U_{D_{1}}\), \(U_{D_{2}}\) and \(U_{A_{2}}\) denote the 1-D convolution matrices for \(C_{D_{1}}\), \(C_{D_{2}}\), and \(C_{A_{2}}\), respectively. As an extension of the BPF-wise used for the one-level DWT introduced in the previous section, we choose to directly concatenate the time-domain features \(W_{T}\) and a fusion of two-level DWT features as the ultimate encoding features. However, we currently have three DWT feature matrices, and we have put forward three methods to create their fusion with three multiplicative masks. These three methods are as follows:  
* **Use two BPF modules for two pairs of DWT feature matrices:** This method is illustrated in Figure 11. We choose the DWT feature matrices with closer frequencies as a pair and apply a BPF module to each pair. Then, the resulting two BPF outputs are added to form the final DWT matrix. The whole process is formulated by: \[M_{1} =\sigma(\psi_{M_{1}}([W_{D_{1}};W_{D_{2}}],\theta_{M_{1}})),\] (13) \[M_{2} =\sigma(\psi_{M_{2}}([W_{D_{2}};W_{A_{2}}],\theta_{M_{2}})),\] (14) and \[W_{DWT} =\ M_{1}\odot W_{D_{1}}+(\  
where \(M_{1}\) and \(M_{2}\) are the applied two BPF matrices. * **Use an MPF module with intra-channel softmax for three DWT feature matrices:** This method is illustrated in Figure 12. It extends the idea of BPF in order to linearly combine more than two sources, which is termed multiple projection fusion (MPF). Rather than the sigmoid function used in BPF, the softmax function is used to obtain three multiplicative masks for three DWT feature matrices, respectively. More precisely, we apply the softmax function to three DWT features individually _for each channel_ (here, a channel indicates an entry of the frame-wise feature vector), as we carried out in the BPF process for two DWT feature matrices. The whole process is formulated by:  
\[ \begin{split}\left[M_{1}^{(i,j)},M_{2}^{(i,j)},M_{3}^{(i,j)} \right]&=\texttt{softmax}(\psi_{M_{s}}([W_{D_{1}};W_{D_{2}};W_{A_{ 2}}],\theta_{M_{s}})),\\ i&=1,2,...,N,\;\;j=1,2,...,K,\end{split}