## 5 Dimensional Transformation Residual Network (DTRN)
However, this method typically maps channel feature vectors with fixed shapes into a format suitable for CNN inputs. It is evident that this approach has limitations when dealing with feature vectors with different dimensions extracted from MU-MIMO systems. Typical CNN architectures such as LeNet, AlexNet, VGG, ResNet, etc., usually expect inputs in the form of grayscale images (the shape of each image is \(w\times h\)) or RGB images (the shape of each image is \(3\times w\times h\)). In order to adapt to the network input, this method maps the feature vectors from the complex domain to the real domain. The real and imaginary parts are treated as two color channels, whereas the other two dimensions \((w,h)\) correspond to the number of resource blocks and antenna ports, respectively. As a result of this processing, the shape of the CSI image becomes \(2\times N_{rb}\times N_{antennas}\), which satisfies the input requirements of the CNN. Figure 3: Modulus graphs of \(\mathbf{V_{a}}\),\(\mathbf{V_{b}}\) and \(\mathbf{V_{c}}\). This figure shows the modulus graphs of the feature vectors extracted from three UEs, denoted as \(a\), \(b\), and \(c\),that transmit UL signals at a fixed location in the system. The antenna ports \(N_{p}\), \(N_{p}\), \(N_{p}\), for these devices are set to 1, 2, and 4, respectively. Alice extracts vectors, \(\mathbf{V_{a}}\), \(\mathbf{V_{b}}\), and \(\mathbf{V_{c}}\), from these devices. To compare the feature vectors, the dimensions corresponding to the number of antenna ports and the number of antennas, \(N_{p}\), and \(N_{a}\), are projected onto the same axis for each vector. However, this approach of handling feature vectors cannot adapt well to MU-MIMO systems due to the following reasons. According to Equation (8), the variation in the number of device antennas results in a significant change in \(E_{i}\), which in turn leads to a substantial change in the shape of CSI image. The output size of the convolutional layer may change with the input size. As a result, the output of the convolutional layer (or pooling layer) cannot be matched with the input of the fully connected layer, rendering the network unable to function properly. It can be seen that when the number of antennas in the system is indefinite, the CSI IMAGE method cannot flexibly handle such variations. Moreover, modifying the network parameters of the CNN itself to adapt to different inputs would significantly increase the system overhead, which goes against the original intention of physical layer authentication. The proposed DTRN network in this section aims to address the issue of abrupt changes in the shape of feature vectors. D-block is proposed to be incorporated into the DTRN network for handling shape-variable input vectors \(\mathbf{V}\) at the input. As shown in Figure 4, it initially processes the feature vector \(\mathbf{V}\) using a Transformation layer and a ports-mapping layer. The processed vector retains both structural characteristics and all data elements, while also meeting the parameter requirements of the subsequent network. The Transformation layer first records information from each dimension, expanding the complex dimension of the three-dimensional complex vector and the two antenna dimensions. It then determines the size of \(N_{p_{i}}\) to decide whether to augment the insufficient data. Data augmentation involves replicating the channel estimation of existing ports, adding noise, and filling in the gaps. Figure 4: A segment of dimensional transformation block (D-block). The figure shows the structure of the transformation layer, the ports mapping layer and the 5 \(\times\) 5 conv layer of the D-block as well as the transformation process of the input vector \(\mathbf{V}\). The transformation layer fixes \(\mathbf{V}\) to a real vector with a shape of 256 \(\times\) 106. The ports mapping layer then maps this real vector to the input channels of the first convolutional layer.