## 4 Results and Discussion
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Expression** & **Description** & **Multiplications** & **Summations** & **Total Flops** \\ \hline \(\alpha\mathbf{a}\) & Vector Scaling & \(N\) & & \(N\) \\ \(\alpha\mathbf{A}\) & Matrix Scaling & \(MN\) & & \(MN\) \\ Ab & Matrix-Vector Prod. & \(MN\) & \(M(N-1)\) & \(2 MN-M\) \\ AB & Matrix-Matrix Prod. & \(MNL\) & \(ML(N-1)\) & \(2 MNL-ML\) \\ AD & Matrix-Diagonal Prod. & \(MN\) & & \(MN\) \\ \(\mathbf{a}^{H}\mathbf{b}\) & Inner Prod. & \(N\) & \(N-1\) & \(2N-1\) \\ \(\mathbf{ac}^{H}\) & Outer Prod. & \(MN\) & & \(MN\) \\ \(\mathbf{A}^{H}\mathbf{A}\) & Gram & \(\frac{MN(N+1)}{2}\) & \(\frac{N(M-1)(N+1)}{2}\) & \(MN^{2}+N\left(M-\frac{N}{2}\right)-\frac{N}{2}\) \\ \(\|\mathbf{A}\|^{2}\) & Euclidean norm & \(MN\) & \(MN-1\) & \(2 MN-1\) \\ \(\mathbf{Q}^{-1}\) & Inverse of Pos. Definite & \(\frac{N^{2}}{2}+\frac{3N^{2}}{2}\) & \(\frac{N^{2}}{2}-\frac{N^{2}}{2}\) & \(N^{3}+N^{2}+N\) Including N roots \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complexity cost of matrix-vector operations [28]. can be achieved with the CNN-based estimator. On the other hand, with lower complexity cost, the proposed FCN can only provide a SNR gain of 3 dB at the BER value of \(10^{-6}\). The same observation is made in Case 2. In Figure 5, the CNN-based estimator achieves an SNR gain of approximately 3 dB, and SFC-DNN achieves an SNR gain of only 2 dB. More specifically, the proposed PCN provides a same performance with the conventional MAP detector in almost all SNR regions, except the SNR value between 0 and 5 dB. Additionally, in comparison with Case 1, due to a large number of employed subcarriers, an additional SNR increment of around 3 dB is required to maintain same value of \(10^{-6}\) BER. Therefore, the performance observed in all figures demonstrates the advantage of the proposed detector over the conventional MAP detector in terms of error rate performance, which is most important to the reliability of VLC systems. In Table 2, the complexity costs of the compared estimators are analyzed. To evaluate the run time of both IE-DNN estimators and the MAP estimator, we use MATLAB to convert the trained model from Keras and estimate the run time for each sample. It should be noted that only the active LED index estimation results are evaluated in this simulation because all other operations (e.g., the costly IFFT computation) are the whole part and identical for all estimators. For all cases, the run time increases with the number of subcarriers  
Figure 4: Comparison of BERs for Case 1. Figure 3: Comparison of IER: (**a**) Case 1 and (**b**) Case 2.  
hence the number of trainable parameters. The MAP estimator with its simple algorithm and poorer performance has the shortest estimation time; by contrast, the SFC-DNN with the highest number of trainable parameters has the longest estimation time. Interestingly, the CNN exhibits the best performance. At first glance, it seems that the SFC-DNN and CNN estimators are computational costly for GLIM-VLC systems. However, compared with other operations, such as IFFT and QAM demodulation (which significantly increase complexity), the active LED index estimation increase does not significantly impact the total complexity cost. Moreover, with the ever-growing computational power of mobile devices and required quality of services, the BER improvement is much more important than complexity costs.