# Software-Defined Visible Light Communication for Internet of Things: A Low-Complexity Approach
## 4 Modeling of CAP16 Modulator and SDLMS Equalizer for SD-VLC System  
We construct a simplified end-to-end simulation model to evaluate the proposed SD-VLC system. The Tx consists of a PRBS generator and a CAP16 modulator. The signal propagates through both AWGN and multipath channels, where the latter includes one dominant line-of-sight (LOS) path and several delayed non-LOS components, forming a Rician-like fading model. At the Rx, the CAP16 demodulator processes the signal, followed by an SDLMS equalizer to suppress ISI and nonlinear distortion. System performance is evaluated in terms of BER, which reflects transmission reliability, and mean square error, which characterizes the equalizer's convergence behavior. Simulations are carried out under varying SNR conditions, with different step sizes and tap lengths, to evaluate both transmission reliability and equalizer convergence behavior. Figure 7 illustrates the error estimation with the SDLMS algorithm. The SDLMS algorithm maintains stability even with slower convergence, ensuring effective adaptation to varying signal conditions. The step size \(\mu\) in SDLMS equalization plays a central role in determining both the convergence speed and steady-state performance. According to adaptive filter theory, a larger \(\mu\) accelerates convergence but may lead to greater residual error and even instability, whereas a smaller \(\mu\) improves stability and reduces the error floor at the cost of slower adaptation. The result illustrates the evolution of the error for different step sizes. As \(\mu\) increases from 0.01 to 0.05, the algorithm converges faster, but the steady-state error fluctuations also become more pronounced, consistent with the theoretical trade-off. For VLC systems, where channel conditions fluctuate, SDLMS's ability to maintain stability while adapting to the signal makes it ideal for resource-constrained applications. Therefore, selecting the right step size is essential for ensuring both efficient performance and robust equalization. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **LMS** & **SDLMS** \\ \hline Computational Complexity & \(O(N)\) (Multiplications + Additions) & \(O(1)\) (Only Additions) \\ Multiplications per Iteration & \(N+1\) & 0 \\ Additions per Iteration & \(N\) & \(N\) \\ Convergence Speed & Rapid & Slower \\ MSE Reduction & Rapid & Slower \\ Power Consumption & High & Low \\ Suitability for IoT VLC & High Processing Demand & Low-Power Applications \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quantitative comparison of LMS and SDLMS equalization in VLC systems. Figure 8 shows the BER performance of the VLC system employing a CAP16 modulator and an SDLMS equalizer. The results indicate that employing a 6-tap SDLMS equalizer significantly improves link performance compared to the case without equalization, where the multipath effects render the received signal almost entirely unrecognizable. The use of SDLMS decreases the BER by approximately two orders of magnitude in the multipath channel. In the inset, a scatter plot at an SNR of 10 dB illustrates the received constellation after equalization. With equalization, the constellation points are noticeably clustered closer to their ideal positions, demonstrating that the adaptive equalizer effectively mitigates ISI and recovers the transmitted symbols. These results underscore the importance of adaptive equalization in CAP-based VLC systems, especially under challenging channel conditions where multipath effects and channel distortions can severely degrade performance. In the multipath channel, the BER curve exhibits slight fluctuations beyond 8 dB. This is attributed to residual ISI becoming the dominant error source as noise diminishes, especially when the SDLMS equalizer has limited taps or fixed step size, preventing full compensation of the channel dispersion.