## 4 Proposed Modulation Recognition Scheme
### Combinatorial Residual Join  
As shown in Figure 3, we present combinatorial residual join (CRJ), which contains non-linear transformation as Equation (11), denoted by \(\Re^{1}(\cdot)\) and \(\Re^{2}(\cdot)\), respectively. The output of the \(l\)th CRJ is given by  
\[x_{l}=\Re^{1}_{l}(x_{l-1})+x_{l-1}+\Re^{2}_{l}\Big{(}\Re^{1}_{l}(x_{l-1})+x_{l -1}\Big{)} \tag{15}\]  
The CA mechanism is an advanced feature processing technique that incorporates the consideration of both channel information and location information related to direction [36; 37]. The CA mechanism can capture not only the channel information but also the location information related to the direction. With the assistance of global average pooling, the CA mechanism captures the channel statistical properties of the feature map. These properties are then normalized by a Softmax function to generate a set of channel weights that are applied to the original feature map. Integrating the CA mechanism into the residual block enables adaptive adjustment and optimization of the feature map. This allows the model to identify and locate time-frequency features more accurately. It achieves better performance with little additional computational burden, as shown in Figure 3.  
Let the input feature map be \(x\in\mathbb{R}^{C\times H\times W}\), where \(C\), \(H\), \(W\) are the number of channels, the height, and the width. An average pooling operation is conducted on the input feature maps, respectively, along the height and width directions, to obtain two feature maps as follows. \[x^{h}_{avg}(c,h,1)=\frac{1}{W}\sum_{i=1}^{W}x(c,h,i) \tag{16}\]  
Figure 3: Schematic of combinatorial residual join. Feature sharing and enhancement are accomplished by means of a \(1\times 1\) convolution kernel and residual structure. Meanwhile, a \(3\times 3\) convolution kernel is used for feature extraction. The CA mechanism is integrated into the residual block to achieve adaptive adjustment and optimization of the feature map. \[x^{w}_{avg}(c,1,w)=\frac{1}{H}\sum_{j=1}^{H}x(c,j,w) \tag{17}\]  
These two feature maps are fed into two convolutional layers to obtain two intermediate feature maps. \[x^{h}_{mid}=\delta(\text{Conv2d}(x^{h}_{avg})) \tag{18}\]  
\[x^{w}_{mid}=\delta(\text{Conv2d}(x^{w}_{avg}))) \tag{19}\]  
where \(\delta\) is the activation function, which usually is the ReLU function. Conv2d represents the convolution operation. \(x^{h}_{mid}\) will be copied along the width direction to obtain Equation (20), and \(x^{w}_{mid}\) will be copied along the height direction to obtain Equation (21). \[\hat{x}^{h}_{mid}(c,h,w)=x^{h}_{mid}(c,h,1) \tag{20}\]  
\[\hat{x}^{w}_{mid}(c,h,w)=x^{w}_{mid}(c,1,w) \tag{21}\]  
Finally, these two feature maps are spliced to obtain the CA feature map, which is defined as follows. \[x_{CA}=\sigma(\text{Conv2d}([\hat{x}^{h}_{mid},\hat{x}^{w}_{mid}])) \tag{22}\]  
where \([\hat{X}^{h}_{mid},\hat{X}^{w}_{mid}]\) denotes that the sum is spliced over the channel dimension. \(\sigma\) is _Sigmoid_ activation function. The function's equation is as follows. Figure 3.0  
The above non-linear combination of the CA mechanism is represented by \(o(\cdot)\), and the dimensions of the parameters are shown in Figure 3.1 The output of the \(l\)th CRJ is rewritten as  
Figure 3.2  
To optimize the flow of feature information between combinations of residual connections, we introduce dense connection into the model. Specifically, the inputs of the \(l\)th layer are a concatenation of the outputs of all the preceding layers, that is,  
Figure 3.3  
This design enables any residual connection in the network to be directly connected to all previous residual connections. As shown in Figure 2b, this configuration of dense connection allows the later layers to receive and integrate feature information from all the previous dense block layers \(x_{0},x_{1},\cdots,x_{l-1}\), thereby enhancing the reusability of features and the expressive power of the network.