## 4 Results  
The BER performance of the proposed HDL detector was evaluated with respect to SNR under imperfect CSI conditions. A model's learning rate balances speed and stability, determining how quickly it learns; an optimal rate accelerates convergence, prevents instability, and ensures efficient and accurate outputs [49]. Our proposed model demonstrates efficient performance across different learning rates, as shown in Figure 7a. The performance of the HDL-based detector at various learning rates was calculated with a training SNR of 15 dB and a \((4,1,4)\) data setup. At lower SNR levels, it performs similarly for learning rates of 0.01, 0.005, and 0.002; at higher SNR levels it shows a slight advantage, with a learning rate of 0.005 compared to the others. Consequently, all experimental setups were designed with a learning rate of 0.005. The BER performance of the proposed HDL-based model for different batch sizes is shown in Figure 7b. The results demonstrate that the model exhibits robust performance across all batch sizes. At lower SNR levels, the performance remains nearly consistent for batch sizes of 500, 1000, and 2000. However, at higher SNR levels, the model achieves slightly superior performance with a batch size of 1000. The training SNR is critical for DL models in communication systems, as it directly impacts the model's ability to generalize across various channel conditions. The performance of the proposed detector across various training SNR levels is illustrated in Figure 8. Notably, the model demonstrates the most stable performance compared to the other levels at a training SNR of 15 dB. Increasing or decreasing the training SNR away from this optimal point results in diminished performance. For training SNR levels of 0 dB and 10 dB, the HDL model performs well at lower SNRs but exhibits slightly weaker performance at higher SNRs. In contrast, at training SNRs of 20 dB and 25 dB the proposed model demonstrates good performance at higher SNR levels while showing comparatively weaker results at lower SNRs. Thus, the experimental results indicate that the proposed HDL-based detector achieves superior performance at a training SNR of 15 dB. Consequently, all experimental setups were designed with a training SNR of 15 dB. Figure 6: The confusion matrix of the proposed HDL-based model. In all experimental settings, the results were calculated using the ZF equalizer and Adam optimizer, except in Figures 9 and 10, where BER performance is compared across different equalizers and optimizers. Additionally, all experiments in this setup utilized \((N,A,M)=(4,1,4)\) except in Figure 11, which presents a comparison of model performance across different modulation orders and data setups. Figure 8: Performance of the HDL-based detector at various training SNRs for the \((4,1,4)\) data configuration. Figure 7: Performance of the HDL-based detector with (**a**) different learning rates and (**b**) different batch sizes in for the \((4,1,4)\) data combination. Figure 10: BER performance of the proposed HDL-based detector utilizing different optimizers for the \((4,1,4)\) data configuration. Figure 9: Performance of the proposed HDL-based detector with various equalizers for the \((4,1,4)\) data configuration. Equalizers play a crucial role in communication systems by mitigating channel distortions such as ISI and multipath fading. Equalizers ensure reliable data transmission in noisy environments by improving BER and overall system performance by providing improved signal quality and dependability [50]. The BER performance of the proposed model with various equalizers is depicted in Figure 9. The results reveal that the proposed detector achieves impressive BER performance with both the ZF and MMSE equalizers, while its performance with the DFE equalizer is notably less effective. Although the model delivers nearly equivalent performance with the ZF and MMSE equalizers, it demonstrates slightly enhanced performance when employing the ZF equalizer, highlighting its efficiency in this configuration.