# Mixed-Carrier Communication for Technology Division Multiplexing
## 4 Technology Division Multiplexing Trade-Off
Subsequently, BPM BER in Equation (16) is adjusted to include the impact of embedded OFDM as shown in Equation (17), where \(\text{SINR}_{\text{OFDM}}\) represents the OFDM signal to interference and noise ratio including ACO-OFDM waveform clipping beyond MCC envelope as explained in [7]. Figure 5 illustrates that BPM BER deteriorates as OFDM SINR increases. As BPM modulation order \(L_{1}\) increases, BPM BER behavior is enhanced for the same OFDM SINR and same BPM \(E_{b}/N_{o}\). \[\text{BER}_{\text{BPM}}\approx Q(\sqrt{\frac{E_{b-\text{BPM}}/N_{o}\;log_{2}(L _{1})}{1+2\;\text{SINR}_{\text{OFDM}}}}) \tag{17}\]  
In [7], a detailed analysis of the optimum ACO-OFDM SINR within an MCC envelope; including the clipping impact, is discussed. \(\lambda_{top}\), defined as the ratio between the top optical clipping level of the ACO-OFDM waveform and its standard deviation, \(\sigma_{\text{OFDM}}\), is investigated to decide the optimum OFDM modulation based on SINR. Using the analysis in [7] to target an FEC BER threshold of \(3.8\times 10^{-3}\) as a metric of maintaining minimum performance per embedded mixed carrier modulations, Figure 6 illustrates the relation between \(\lambda_{top}\) and OFDM SINR. As shown in the figure, increasing the OFDM M-QAM modulation order, requires more SINR for the same \(\lambda_{top}\) to target the FEC BER threshold. The required SINR starts to flatten beyond \(\lambda_{top}\) equals 1.5. While MCC provides different modulation techniques to serve simultaneous technologies, it is essential to investigate the spectral efficiency gain, \(\eta_{S}\), MCC offers. According to [26; 27], \(L_{1}\)-PPM bandwidth \(\text{BW}_{\text{PPM}}\) can be approximated as follows:  
\[\text{BW}_{\text{PPM}}\approx 1/T_{b}\approx\frac{R_{b-\text{PPM}}\;L_{1}}{ log_{2}(L_{1})} \tag{18}\]  
Thus, PPM is attractive for a range of applications where bandwidth is not of major concern as power efficiency. However, BPM bandwidth requirements are even more where BPM spectral efficiency \(\eta_{\text{BPM}}\) is constrained by the minimum PWM pulse duration \(T_{H}(n)_{\text{min}}\) sampling the beacon as shown in Equation (19):  
\[\eta_{\text{BPM}}=R_{b-\text{BPM}}\;T_{H}(n)_{\text{min}} \tag{19}\]  
Figure 6: Impact of MCC-OFDM clipping on required SINR for target BER. To investigate the spectral efficiency gain by the low-speed modulations embedded within MCC, i.e., MCC-LS, \(L_{\rm 2-BmPSK}\) is considered as in Equation (20). Without loss of generality, different types of VLC compatible OFDM modulation techniques can be incorporated; however, this analysis considers ACO-OFDM and more optical OFDM types will be considered for benchmarking in our future work. Thus, MCC overall spectral efficiency \(\eta_{\rm MCC-(ACO-OFDM)}\) considering M-QAM for ACO-OFDM modulation is explained by Equation (21), assuming the number of cyclic prefix sub-carriers to be 1/4 of the total sub-carriers:  
\[\eta_{\rm MCC-LS}=\frac{log_{2}(L_{1}L_{2})\ T_{\rm OFDM}}{L_{1}L_{2}T_{\rm PWM}} \tag{20}\]  
\[\text{BER}_{\text{OOK}}\approx Q(\sqrt{\frac{E_{b}}{N_{o}}}) \tag{14}\]0  
From Equations (20) and (21), the MCC spectral efficiency gain over conventional ACO-OFDM can be interpreted by Equation (22),  
\[\text{BER}_{\text{OOK}}\approx Q(\sqrt{\frac{E_{b}}{N_{o}}}) \tag{14}\]1  
where \(L=L_{1}\ L_{2}\) and \(\gamma_{f}\) is the ratio between \(T_{\rm PWM}\) and \(T_{\rm OFDM}\). The spectral efficiency gain is pictured in \[\text{BER}_{\text{OOK}}\approx Q(\sqrt{\frac{E_{b}}{N_{o}}}) \tag{14}\]2 The figure shows the inverse proportionality between \(\gamma_{f}\) and the spectral efficiency gain \(\eta g\). As the BPM and BnPSK modulation orders \(L_{1}\) and \(L_{2}\) increase, \(\eta g\) decreases due to the high bandwidth requirements resulting from reducing the minimum PWM pulse duration \(T_{H}(n)_{\rm min}\). In the following section, the optimum spectral efficiency is analyzed to maximize the MCC spectral gain. As noticed in Equation (16), the relation between \(\rm SINR_{OFDM}\) and \(BER_{\rm BPM}\) is inversely proportional.