## 3 Architecture of the Proposed HDL Detector
For instance, if \(\mathcal{T}_{\text{train}}\) is set too low, the model may fail to generalize adequately, as it will not account for the noise impact  
\[\mathbf{\hat{y}}=\frac{y\cdot h}{|h|^{2}+\sigma^{2}} \tag{5}\]2 Training loss of the proposed HDL model for different equalizers with data setup \((N,A,M)=(4,1,4)\): (a) training loss for the ZF equalizer, (b) training loss for the MMSE equalizer, and (c) training loss for the DFE equalizer. \[\mathbf{\hat{y}}=\frac{y\cdot h}{|h|^{2}+\sigma^{2}} \tag{5}\]3 Training loss of the proposed HDL model for different modulation orders and data combinations with the ZF equalizer: (a) training loss for the \((N,A,M)=(4,1,4)\) setup, (b) training loss for the \((N,A,M)=(8,2,8)\) setup, and (c) training loss for the \((N,A,M)=(8,4,16)\) setup. during training. The simulation results presented in the next section detail the process of selecting an appropriate training SNR for each experimental setup. We trained our proposed HDL model over 1000 epochs across all experimental settings, with each epoch comprising 20 batches and each batch containing 1000 data samples (often referred to as the batch size). This setup resulted in a total of \(2\times 10^{4}\) batches, yielding approximately \(2\times 10^{7}\) unique training samples. The samples for each batch were generated randomly during the training process. The HDL model was trained for various configurations of \((N,A,M)\). Specifically, we trained the proposed model with modulation orders of QPSK, 8 QAM, and 16 QAM. All training configurations for the proposed model are detailed in \[\mathbf{\hat{y}}=\frac{y\cdot h}{|h|^{2}+\sigma^{2}} \tag{5}\]4 Our proposed model was trained using a batch size of 1000 over 1000 epochs. The proposed system can operate in real time to estimate data bits across different channel fading scenarios without the need for further modifications to \(\theta\) after it has been trained offline. Upon receiving the signal and channel characteristics, the HDL model efficiently generates an estimated bit and performs comparably to the ML detector in the presence of channel estimation errors. We applied 100,000 samples to test the proposed model. Figure 6 presents the confusion matrix of the proposed model under the \((4,1,4)\) data setup and with a training SNR of 15 dB, providing a bit-wise performance evaluation on the test data. The confusion matrix shows the model's accuracy in predicting \"0\" and \"1\" bits at the specified SNR level. Test samples with true bit labels were passed through the trained model to obtain predicted values, which were thresholded at 0.5 to produce binary predictions. The confusion matrix indicates that the model correctly classified 197,769 \"0\" bits and 197,822 \"1\" bits, with 2214 \"0\" bits misclassified as \"1\" bits and 2195 \"1\" bits misclassified as \"0\" bits. These results highlight the model's strong bit-wise accuracy with a low misclassification rate. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Training samples & 10,000,000 \\ \hline Testing samples & 100,000 \\ \hline Modulation scheme & QPSK, 8 QAM, 16 QAM \\ \hline Noise type & AWGN \\ \hline Fading channel & Rayleigh fading channel \\ \hline Number of hidden layers & 128 \\ \hline Learning rate & 0.01, 0.005, 0.002 \\ \hline Number of epoch & 1000 \\ \hline Batch size & 1000 \\ \hline Training SNR (dB) & 0, 5, 10, 15, 20, 25 \\ \hline N & 4, 8 \\ \hline A & 1, 2, 3, 4 \\ \hline Optimizer & Adam, RMSProp, Adagrad, Gradient Descent \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters for the proposed system.