# Deep Learning-Enhanced Autoencoder for Multi-Carrier Wireless Systems
## 3 Proposed GRU-MC-AE Network Architecture
#### Training and Testing Procedure  
A collection of randomly generated incoming messages \(s\) or their corresponding one-hot vectors is used to train the MC-AE model offline. During training, randomly generated noise \(\mathbf{w}\) and channel \(\mathbf{H}\) are added to the encoder's output. The model is trained with 200,000 date samples. The training and testing parameters used in this experiment are shown in Table 1. In DL, the selection of an appropriate loss function is crucial as it directly influences model performance and guides the optimization process. By quantifying errors, the loss function ensures that the model learns relevant patterns for specific tasks, such as regression or classification. For GRU-MC-AE training, we use the mean squared error (MSE) loss function [42], which can be expressed as follows:  
\[\mathcal{L}(\mathbf{s}_{i},\mathbf{\hat{s}}_{i};\theta)=\frac{1}{n}\sum_{i=1} ^{n}(s_{i}-\hat{s}_{i})^{2}, \tag{10}\]  
where \(\theta\) denotes the weight and bias of the model. \(n\) is the training batch size and \(\mathbf{\hat{s}_{i}}\) is the prediction of \(\mathbf{s_{i}}\). For randomly selected batches from the data sample, the stochastic gradient descent (SGD) algorithm can update the model parameter \(\theta\) in the following manner:  
\[\theta^{+}\coloneqq\theta-\eta\
abla\mathcal{L}(\mathbf{s}_{i},\mathbf{\hat{ s}}_{i};\theta), \tag{11}\]  
where \(\eta\) is the learning rate and step size of SGD. Optimizers play a crucial role in DL by enabling efficient weight updates, leading to quicker convergence and improved model performance [43]. The adaptive moment estimation (Adam) optimizer is widely used in DL due to its adaptive learning rates, which combine the strengths of momentum and RMSProp for faster and more stable convergence. It is efficient, robust to noisy gradients, and performs effectively across various tasks, especially with large models and datasets. Leveraging these advantages, we train our proposed model using the Adam optimizer [44] which is extensively accessible in several commercial DL libraries, including TensorFlow [45]. During the training, data labeling is conducted using one-hot encoding. Each data sample represents one of the possible \(M=16\) modulation symbols. The label for each sample is a one-hot encoded vector of length \(M\), where the position corresponding to the true modulation symbol is set to 1, and all other positions are set to 0. This allows the neural network to learn the mapping between the transmitted symbol and its corresponding one-hot label during training. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Training samples & 200,000 \\ \hline Testing samples & 30,000 \\ \hline Fading channel & Rayleigh fading channel \\ \hline Noise type & AWGN \\ \hline Number of hidden layer & 256, 128 \\ \hline Learning rate & 0.01, 0.005, 0.001, 0.0005 \\ \hline Batch size & 64, 128, 256, 512 \\ \hline Number of epoch & 250, 500, 1000 \\ \hline Training SNR (dB) & 5, 7, 10, 15 \\ \hline (N,M) & (4, 16), (4, 32), (4, 64) \\ \hline Optimizer & Adam \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The simulation parameters for the proposed system. Figure 4 illustrates the classification performance metrics of the proposed GRU-based MC-AE model for the (4, 16) data combination trained at 7 dB SNR. In Figure 4a, the precision-recall curve for each modulation class shows a gradual precision decrease as recall increases, reflecting the model's accuracy across different modulation classes. The inset zooms in on variations in the middle precision-recall range, revealing slight class differences in performance. Figure 4b depicts the training loss over epochs, with an initial sharp decline indicating rapid adaptation to the data. The loss stabilizes around the 200 th epoch, suggesting the model has reached a stable, low-error state and effectively minimized prediction errors.