# Deep Learning-Enhanced Autoencoder for Multi-Carrier Wireless Systems
## 4 Simulation Results  
We utilize a variety of advanced MC schemes listed in Table 1 as baseline schemes for the GRU-MC-AE model. In this study, simulations are conducted using the Python environment, with an Intel Core i7-8700 CPU at 3.20 GHz and an NVIDIA GeForce GTX 1660 Ti GPU. The configuration of IM-based schemes, such as OFDM-IM [11], is denoted by \((N,K,M)\), where \(N\) represents the number of sub-carriers per block, \(K\) is the number of active sub-carriers, and \(M\) is the size of the conventional M-ary modulation. In particular, the configuration for OFDM [9] and S-OFDM [10] is represented by \((N,M)\). All of these systems, including OFDM, S-OFDM, and OFDM-IM utilized ML-based detectors, with the exception of the MMSE-based S-OFDM system in [13]. The configurations for both  
Figure 4: Classification performance metrics of the proposed GRU-based MC-AE model for the (4, 16) data combination trained at 7 dB SNR. **(a)** Precision-recall curve and **(b)** loss performance over epochs. the DNN-based MC-AE [32] and the proposed MC-AE are represented as \((N,M)\). It is essential to note that the \(M\) in our approach is distinct from that in the baseline schemes, as it signifies the size of the transmitted message for each block of \(N\) sub-carriers, where \(M\) is defined as \(2^{m}\), with \(m\) representing the number of data bits in each message. The speed at which a model updates its weights during training is determined by the learning rate, a crucial parameter in deep learning. Finding the ideal learning rate is crucial to striking a balance between convergence speed and stability. A low learning rate may result in sluggish or less-than-ideal convergence, whereas an excessively high learning rate may create instability [48]. The BLER of the proposed model under perfect CSI conditions, with a (4, 16) setup, is shown in Figure 5. From the results, we observe that the model demonstrates superior performance at a 0.001 learning rate. While the performance at lower SNR levels is nearly similar for all learning rates, the performance gap widens at higher SNR levels. Specifically, at 15 dB SNR, the 0.001 learning rate outperforms the 0.01, 0.005, and 0.0005 learning rates by approximately 3 dB, 1.2 dB, and 0.9 dB, respectively. Hence, the 0.001 learning rate is used for all setups in this experiment. Figure 6 shows the BLER performance of the proposed model across different batch sizes and epochs for the (4, 16) data setup. While the model demonstrates strong performance across all batch sizes and epoch configurations, it achieves the best results with a batch size of 512 and 1000 epochs. When the batch size decreases while keeping the epoch count at 1000, performance declines. Similarly, reducing the number of epochs while maintaining a batch size of 512 also results in decreased performance. Moreover, reducing both batch size and epochs further degrades performance. These results confirm that our proposed model performs optimally with a batch size of 512 and 1000 epochs. Figure 5: Comparative BLER performance of the proposed model for different learning rates under perfect CSI condition with (4, 16) setup. Training SNR significantly affects a DL model's performance. Low training SNR can hinder the model's ability to generalize in high-SNR conditions, while high SNR may reduce its robustness in noisy environments. A balanced training SNR ensures better overall performance across varying SNRs [46]. Figure 7 presents a comparison of the proposed model's BLER performance across different training SNRs. The BLER is measured with a batch size of 512 and a configuration of (4, 16). From the results, we observe that the proposed model performs well for all positive training SNR values. Although the GRU-MC-AE model shows slightly poorer performance at 15 dB SNR, it still demonstrates good and almost similar performance at 5 dB and 10 dB training SNRs. When it is trained at 7 dB SNR, the model performs better compared to other training SNRs.