# Highly Sensitive SPAD-Based Receiver for Dimming Control in LiFi Networks +
## 4 Multi-Carrier Transmission Schemes with Dimming Control
### Aco-Ofdm
#### 4.2.2 Adaptive SPAD Receivers  
For this SPAD OFDM system, the combined nonlinear distortion of the transmitted signal \(x[n]\) can be derived by substituting (46) and (4) into (6) to obtain  
\[\mu_{\text{a}}(x[n])= \begin{cases}\left(\beta_{1}\kappa+\lambda_{\text{n}} \right)T_{\text{s}}\exp\!\left[-\frac{\left(\beta_{1}\kappa+\lambda_{\text{n }}\right)T_{\text{d}}}{N_{\text{a}}}\right],&x[n]\geq\kappa,\\ \left(\beta_{1}x[n]+\lambda_{\text{n}}\right)T_{\text{s}}\exp\!\left[-\frac{ \left(\beta_{1}x[n]+\lambda_{\text{n}}\right)T_{\text{d}}}{N_{\text{a}}} \right],&0<x[n]<\kappa,\\ \lambda_{\text{n}}T_{\text{s}}\exp\!\left[-\frac{\lambda_{\text{n}}\tau_{ \text{d}}}{N_{\text{a}}}\right],&x[n]\leq 0.\end{cases}  \tag{51}\]  
where \(\beta_{1}=I_{\text{s}}(\xi)P_{\text{max}}/\kappa\). Similar to the DCO-OFDM case, the OFDM signal can be quantified by a gain factor (\(\eta_{\text{ACO}}\)) and an additional signal-independent noise distortion-induced noise (\(w_{d}[n]\)) according to the [PERSON] theorem. The gain factor \(\eta_{\text{ACO}}\) can be calculated as [48]  
\[\eta_{\text{ACO}}= \frac{\beta_{1}^{2}\tau_{\text{d}}T_{\text{s}}}{\sqrt{2\pi N_{ \text{a}}}}\left\{\exp\!\left[-\frac{\kappa}{2}-\frac{\tau_{\text{d}}}{N_{ \text{a}}}(\beta_{1}\kappa+\lambda_{\text{n}})\right]-\exp\!\left[-\frac{\tau_ {\text{d}}\lambda_{\text{n}}}{N_{\text{a}}}\right]\right\} \tag{52}\] \[+\beta_{1}T_{\text{s}}e^{-\frac{\lambda_{\text{n}}\tau_{\text{d} }}{N_{\text{a}}}+\frac{\beta_{1}^{2}\tau_{\text{d}}^{2}}{2N_{\text{d}}^{2}}} \Bigg{[}1+\frac{\beta_{1}^{2}\tau_{\text{d}}^{2}}{N_{\text{a}}^{2}}-\frac{ \lambda_{\text{n}}\tau_{\text{d}}}{N_{\text{a}}}\Bigg{]}\times\bigg{[}Q\! \left(\frac{\beta_{1}\tau_{\text{d}}}{N_{\text{a}}}\right)-Q\!\left(\kappa+ \frac{\beta_{1}\tau_{\text{d}}}{N_{\text{a}}}\right)\bigg{]}.\]  
The variance of the distortion-induced noise, \(\sigma_{w_{d}}^{2}\), can be calculated in a similar manner as that for the DCO-OFDM case from (42). By applying the same principle as in (44), the variance of the shot noise, \(W_{\text{a}}\), for the considered system with ACO-OFDM, denoted as \(\sigma_{W_{\text{a}}}^{2}\)is identical to that with DCO-OFDM which has been introduced above in reference [49], Equation (46). The SNR of the received signal for ACO-OFDM can be written as  
\[\Gamma_{\rm ACO}(\xi)=\frac{\eta_{\rm ACO}^{2}\sigma_{\rm X}^{2}}{\sigma_{\rm W_{ d}}^{2}+\sigma_{\rm W_{a}}^{2}}. \tag{53}\]  
It is noted that the closed-form expression for the BER performance of \(M\)-QAM in AWGN is presented as a sum of terms in [50]. Using only the first two terms yields an accurate approximation. Therefore, the BER performance of \(M\)-QAM O-OFDM for the \(k\)th subcarrier of the considered modulation techniques for both PD and SPAD systems can be written as  
\[P_{e}[k]=\frac{4(\sqrt{M}-1)}{\sqrt{M}\log_{2}(M)}Q\Bigg{(}\sqrt{\frac{3 \Theta}{M-1}}\Bigg{)}+\frac{4(\sqrt{M}-2)}{\sqrt{M}\log_{2}(M)}Q\Bigg{(}3 \sqrt{\frac{3\Theta}{M-1}}\Bigg{)}, \tag{54}\]  
where \(\Theta\) is the relevant SNR of the considered OFDM system as in (36), (45), (48) and (53). In the proposed receiver, the controller should adjust the VOA transmittance, \(\xi\), so that the SPAD array can operate at its best. The optimal \(\xi\) minimizing the BER is equivalent to the one maximizing its SNR. Hence, by taking the first derivative of \(\Gamma_{\rm DCO}(\xi)\) or \(\Gamma_{\rm ACO}\xi)\) with respect to \(\xi\) and letting it equal zero, it will lead to a non-linear equation which is mathematically intractable. Therefore, we will find the optimal VOA transmittance, \(\xi\), for the considered O-OFDM systems through an exhaustive search, and a look-up table of the optimal \(\xi\) under various \(P_{R}\) and \(P_{b}\) could be calculated and pre-saved at the receiver. Using the chart, the ideal transmittance of VOA may therefore be chosen for the communication. To explore the throughput of the SPAD-based OFDM system, the SNR terms of DCO- and ACO-OFDM should be substituted into (54) where we calculate the BER for each subcarrier.