## 3 Methodology
### Dual-Branch Low-Band Speech Enhancement Network LB-Net
#### 3.1.2 Complex Convolution-Augmented Gated Attention Units  
In the self-attention mechanism, query (\(Q\)), key (\(K\)), and value (\(V\)) can be obtained by mapping input \(D\):  
\[Q=DW_{Q} \tag{9}\]  
\[K=DW_{K} \tag{10}\]  
\[V=DW_{V} \tag{11}\]  
Figure 5: The architecture of MLB-Net. Figure 6: The architecture of CLB-Net. where \(W_{Q}\), \(W_{K}\), and \(W_{v}\) represent the learnable parameter matrixes of the linear layers. Then, the scaled dot-product attention is applied to the query, key, and value afterward as:  
\[Attention(Q,K,V)=\text{softmax}(\frac{QK^{\intercal}}{\sqrt{d}})V \tag{12}\]  
where \(d\) represents the projection dimension of \(W\). Softmax is an activation function. Given an input vector of size n, the softmax function computes a vector of n probabilities. The formula for softmax is:  
\[\text{softmax}(\mathbf{z})_{i}=\frac{e^{z_{i}}}{\sum_{j=1}^{n}e^{z_{j}}} \tag{13}\]  
where \(\mathbf{z}=(z_{1},z_{2},...,z_{n})\) is the input vector and \(\text{softmax}(\mathbf{z})_{i}\) is the probability assigned to the i-th class. The denominator ensures that the sum of all the probabilities is equal to one, making it a valid probability distribution. Accordingly, inspired by [PERSON] [26], when input \(D\) is a complex number, the calculation formula of \(Q\) is:  
\[Q_{r}=D_{r}W_{Q_{r}}-D_{i}W_{Q_{i}} \tag{14}\]  
\[Q_{i}=D_{r}W_{Q_{i}}+D_{i}W_{Q_{i}} \tag{15}\]  
where \(r\) and \(i\) represent the real part and imaginary part of \(D\). In addition, \(K\) and \(V\) are calculated in the same way. Therefore, the complex self-attention (CSA) is calculated as follows:  
\[CSA(Q,K,V)=CSA_{r}+i\times CSA_{i} \tag{16}\]  
\[K=DW_{K} \tag{10}\]0  
\[K=DW_{K} \tag{10}\]1  
The two complex CGAUs in the complex CGA block have the same structure. Taking the first complex CGAU as an example, \(D\in\mathbb{R}^{B\times T\times F\times C\times 2}\) is reshaped as \(D^{in}\in\mathbb{R}^{BF\times T\times C\times 2}\) and input into the complex CGAU. \(D^{in}\) is divided into two channels, one of which is sent to the complex convolution module. The output of the complex convolution module is \(D^{conv}\in\mathbb{R}^{BF\times T\times C\times 2}\). The real part's query \(Q_{r}\), key \(K_{r}\), value \(V_{r}\), and the virtual part's query \(Q_{i}\), key \(K_{i}\), and value \(V_{i}\) are all replicas of \(D^{conv}\). Then, we add the rotation position code \(b\) to Formula (12), and use Formula (16) to calculate the CSA. \[K=DW_{K} \tag{10}\]2  
\[K=DW_{K} \tag{10}\]3  
\[K=DW_{K} \tag{10}\]4  
\[K=DW_{K} \tag{10}\]5  
In the above formulas, \(\varnothing\) represents the swish activation function, while the linear layers' learnable parameter matrixes are represented by \(W\). The shared representation is represented by \(Z\). The query \(Q\) and key \(K\) are obtained by applying simple affine transformations to \(Z\), respectively. \(V\) represents values in the CSA mechanism. The swish activation function is a smooth, non-monotonic function that is continuous and differentiable everywhere. For input \(x\), the function is defined as:  
\[K=DW_{K} \tag{10}\]6Before calculating CSA, we need to add the rotation position code [27] in Formula (12), which is represented by variable \(b\):  
\[K=DW_{K} \tag{10}\]7  
where, \(Q^{\prime}\), \(K^{\prime}\), and \(V^{\prime}\) respectively represent the real or imaginary parts of the corresponding complex \(Q\), \(K\), and \(V\). After that, we use Formula (16) to calculate \(CSA\). The other feed of input \(D^{in}\) passes through the linear layer and is activated by swish to obtain \(U\). Finally, the Hadamard product of \(U\) and \(CSA\) is calculated and input into the linear layer to obtain output \(D^{out}\) so that the convolution-augmented attention information is introduced to the gated linear unit. The calculation formula is as follows, where \(\circ\) represents Hadamard product:  
\[K=DW_{K} \tag{10}\]8  
\[K=DW_{K} \tag{10}\]9  
\[V=DW_{V} \tag{11}\]0  
\[V=DW_{V} \tag{11}\]1  
\[V=DW_{V} \tag{11}\]2