## 4 Results
The BER performance of the proposed model with different optimizers is illustrated in Figure 10. Specifically, we examined the performance of the HDL model using the Adam, RMSProp, Adagrad, and Gradient Descent optimizers. The figure clearly indicates that the proposed model demonstrates acceptable performance across all optimizers. However, its performance with the Gradient Descent optimizer is somewhat less favorable compared to the others. Notably, while the HDL detector exhibits nearly comparable performance with both Adam and RMSProp, the Adam optimizer achieves a slightly superior performance in comparison to RMSProp. Figure 11 illustrates the BER performance of the HDL detector in the OFDM-IM system with different modulation orders under imperfect channel conditions. We evaluated the performance with the QPSK, 8 QAM, and 16 QAM modulation schemes, where the configurations were set as follows: \((N,A,M)=(4,1,4)\) for QPSK, \((8,2,8)\) for 8 QAM, and \((8,4,16)\) for 16 QAM. We observed that increasing the modulation order and data setup leads to a reduction in performance; specifically, the model achieves 5 dB and 9 dB superior performance for the \((4,1,4)\) configuration compared to the \((8,2,8)\) and \((8,4,16)\) configurations, respectively. Figure 12 compares the BER performance of the proposed HDL-based detector against ML, GD, DeepIM [27], and LSTM-IM [34] detectors. This evaluation used a 15 dB training  
Figure 11: BER performance of the proposed HDL-based detector for various modulation orders and data setup. SNR and a 0.005 learning rate along with the ZF equalizer and Adam optimizer. The results clearly demonstrate that the HDL detector consistently outperforms GD, ML, DeepIM, and LSTM-IM models with the \((4,1,4)\) data setup. Specifically, the proposed detector shows approximately 2.5 dB improved performance over GD and achieves nearly 0.9 dB better BER performance compared to both the ML and DeepIM detectors under imperfect CSI conditions. Additionally, we compared the performance of the proposed model with the \((4,3,4)\) data setting against a CNN-based model [31]. Table 2 presents the BER performance of all comparable models at an SNR of 20 dB, with the results illustrated in Figure 12. The simulation results confirm that the HDL model outperforms the CNN model at higher data rates. To further demonstrate the robustness of the proposed HDL-based OFDM-IM system, we calculated the throughput and spectral efficiency (SE), as shown in Figure 6:0 These metrics were evaluated for the (4,1,4), (4,2,4), and (4,3,4) data configuration, with each setup representing a distinct combination of active indices. The results indicate that both throughput and SE increase with higher active indices. The model demonstrated the highest performance with (4,3,4) setup. These results confirm that our proposed HDL model effectively achieves high throughput and SE. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**SNR** & **Model Name** & **BER** & **Model Name** & **BER** \\ \hline \multirow{2}{*}{20 dB} & HDL & 0.0033074783 & DeeIM & 0.00387089132682052 \\  & LSTM-IM & 0.00355340813397947 & ML & 0.0039256792218776 \\  & HDL & 0.0074599785 (A = 3) & CNN & 0.00854427859235968 (A = 3) \\  & GD & 0.00585573211245060 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparative BER performance at 20 dB SNR for all models. Figure 12: BER performance comparison of the proposed HDL-based detector with other detectors under imperfect CSI conditions for the \((4,1,4)\) data combinations.