## 3 Proposed SE Framework with DWT Features
### Fusion of One-Level DWT Features and Time-Domain Features  
Here, we employ one-level DWT features and integrate them with the time-domain features to serve as the encoder features for the SE network. Referring to the procedures of Conv-TasNet introduced in Section 2.1, the proposed novel SE framework consists of the following steps:  
1. **Create time-domain features and one-level DWT features** For the encoder end of the framework, one splits the input time-domain utterance into \(K\) overlapping frames of length \(L\), each frame represented by a vector \(x_{k}\in\mathbb{R}^{L\times 1}\). These \(x_{k}\) are concatenated to comprise a data matrix \(X\in\mathbb{R}^{L\times K}\). Then, the time-domain feature matrix \(W_{T}\in\mathbb{R}^{N\times K}\) is created as in Equation (1) mentioned in Section 2.1.1. To create another branch of features, we present to apply a one-level DWT to the data matrix \(X\) concerning each of its columns \(\mathbf{x}_{k}\). That is, each frame signal \(\mathbf{x}_{k}\) passes through a one-level DWT (as introduced in Section 2.3) to produce its approximation (low-pass) and detail (high-pass) sub-band signals, denoted by \(\mathbf{c}_{k}^{A}\) and \(\mathbf{c}_{k}^{D}\), respectively. Compared with \(\mathbf{x}_{k}\), \(\mathbf{c}_{k}^{A}\) and \(\mathbf{c}_{k}^{D}\) have half the length and bandwidth. As such, we organize these sub-band signals to produce two feature matrices, \(C_{A}\) and \(C_{D}\) with size \(\frac{L}{2}\times K\), consisting of \(\mathbf{c}_{k}^{A}\) and \(\mathbf{c}_{k}^{D}\) as columns, respectively. Furthermore, we process \(C_{A}\) and \(C_{D}\) individually with a 1-D trainable convolution (together with the nonlinear function \(H\)) to produce the other two matrices \(W_{A}\) and \(W_{D}\) with size \(N\times K\), the same size as the time-domain feature matrix \(W_{T}\). The operations are formulated as \[W_{A}=H(U_{A}C_{A}),\ \ W_{D}=H(U_{D}C_{D}),\] (6) where \(U_{A}\) and \(U_{D}\) denote the 1-D convolution matrices for \(C_{A}\) and \(C_{D}\), respectively. Figure 7: A schematic diagram of the cross-domain TCN (CD-TCN), which uses time-domain and frequency-domain features together with their fusion. 2. **Integrate time-domain features and DWT features** To date, we have three feature matrices: \(W_{T}\) (time-domain features), \(W_{A}\) (DWT-wise approximation features), and \(W_{D}\) (DWT-wise detail features). Notably, they have the same size, and we have come up with three ways to integrate them to constitute the ultimate encoder features \(W_{E}\): * **Addition:** Here, \(W_{E}\) is simply the weighted sum of three matrices: \[W_{E}=0.50W_{T}+0.25W_{A}+0.25W_{D},\] (7) which is illustrated by Figure 8a. Accordingly, the final feature matrix has the same size as each component matrix. * **Concatenation:** The other intuitive way for the integration is to concatenate the three matrices: \[W_{E}=[W_{T};W_{A};W_{D}],\] (8) which is illustrated by Figure 8b. Here, the final feature matrix is three times the size of each component matrix. * **Fusion and concatenation:** To extract the information across the two DWT features \(W_{A}\) and \(A_{D}\) more effectively, we exploit the bi-projection fusion (BPF) method [31]. BPF has been used to integrate time-domain and frequency-domain features and exhibits outperforming behavior in DPTNet [39]. In addition, the two DWT features, which reflect the low-pass and high-pass half-band short-time spectra in speech, are supposed to be of unequal importance. For example, the low-pass part, \(W_{A}\), contains more about vowels, and the high-pass part, \(W_{D}\), might correspond to consonants better. Furthermore, \(W_{A}\) and \(W_{D}\) usually have different signal-to-noise ratios (SNRs) due to the embedded speech components and the background noise. Therefore, the two complementary masks (being non-negative with a unity sum) of the BPF  
Figure 8: A schematic diagram of the encoder part of CD-TCN with time-domain and one-level DWT features fused by (**a**) addition (**b**) concatenation.