# Highly Sensitive SPAD-Based Receiver for Dimming Control in LiFi Networks +
## 3 Single-Carrier Transmission Schemes with Dimming Control
### Variable On-Off Keying (VOOK)--Digital Dimming
#### 3.2.2 Adaptive SPAD Receivers  
The received photon rate of each PQ SPAD micro-cell when bit \"0\" and bit \"1\" are sent for VOOK, can be expressed as [43]  
\[\begin{split}\lambda_{1}^{\text{vook}}(\xi)&= \frac{\xi\text{Y}_{\text{PDE}}(2\delta_{d}\zeta\gamma\overline{P}_{\text{TX} }+P_{b})}{N_{\text{a}}E_{\text{ph}}},\\ \lambda_{0}^{\text{vook}}(\xi)&=\frac{\xi\text{Y}_ {\text{PDE}}P_{b}}{N_{\text{a}}E_{\text{ph}}},\end{split} \tag{18}\]  
where \(\xi\) denotes the transmittance of the employed VOA for VOOK modulation. The average photon counts for bit \"0\" and bit \"1\" during the VOOK's symbol duration \(T_{s}\) are given by [43]  
\[\begin{split}\mu_{1}^{\text{vook}}(\xi)&=N_{\text{ a}}\lambda_{1}^{\text{vook}}(\xi)T_{s}e^{-\lambda_{1}^{\text{vook}}(\xi)\tau_{ d}},\\ \mu_{0}^{\text{vook}}(\xi)&=N_{\text{a}}\lambda_{0}^ {\text{vook}}(\xi)T_{s}e^{-\lambda_{0}^{\text{vook}}(\xi)\tau_{d}},\end{split} \tag{19}\]  
Similar to the OOK case, the variance of the photon counts can be expressed as  
\[\begin{split}\sigma_{1,\text{vook}}^{2}(\xi)&= \mu_{1}^{\text{vook}}(\xi)-\frac{\lambda_{1}^{\text{vook}}(\xi)^{2}T_{s}\tau_{ \text{d}}}{N_{\text{a}}}\text{exp}\Bigg{(}-\frac{2\lambda_{1}^{\text{vook}}( \xi)\tau_{\text{d}}}{N_{\text{a}}}\Bigg{)}\Bigg{(}2-\frac{\tau_{\text{d}}}{T_{ \text{s}}}\Bigg{)},\\ \sigma_{0,\text{vook}}^{2}(\xi)&=\mu_{0}^{\text{ vook}}(\xi)-\frac{\lambda_{0}^{\text{vook}}(\xi)^{2}T_{s}\tau_{\text{d}}}{N_{ \text{a}}}\text{exp}\Bigg{(}-\frac{2\lambda_{0}^{\text{vook}}(\xi)\tau_{\text{ d}}}{N_{\text{a}}}\Bigg{)}\Bigg{(}2-\frac{\tau_{\text{d}}}{T_{\text{s}}} \Bigg{)}.\end{split} \tag{20}\]  
Therefore, the BER expression for PQ-based SPAD array receiver when VOOK modulation is employed can be expressed as  
\[\text{BER}_{\text{SPAD}}^{\text{vook}}=Q[\Gamma(\xi)] \tag{21}\]  
where \(\Gamma(\xi)=\frac{\mu_{1}^{\text{vook}}(\xi)-\mu_{0}^{\text{vook}}(\xi)}{\sigma _{1,\text{vook}}(\xi)+\sigma_{\text{vook}}(\xi)}\).  
In the proposed receiver, the controller adjusts \(\xi\) so that the SPAD array can operate at its best. The optimal \(\xi\) minimizing \(\text{BER}_{\text{SPAD}}^{\text{vook}}\) is equivalent to the one maximizing the \(\Gamma(\xi)\). With the assumption of equal mean and variance of the detected photon count [32], the approximated optimal \(\xi\) for VOOK can be expressed as  
\[\xi=\text{min}\Bigg{[}\frac{2N\text{l}vy_{\text{root},1}}{\text{Y}_{\text{PDE }}\tau_{d}P_{R}},1\Bigg{]}, \tag{22}\]  
where \(y_{\text{root},1}\) denotes the single positive root of the non-linear equation  
\[\frac{2P_{b}}{P_{R}}y-1=\frac{2y}{\sqrt{\frac{P_{b}}{P_{R}+P_{b}}}e^{y}-1}, \tag{23}\]  
in the regime \(y\in(0,\ln\sqrt{\frac{P_{R}+P_{b}}{P_{b}}})\).  
The approximated maximum achievable data rate of the SPAD array for VOOK system at the target BER, \(P_{\rm e}^{\rm th}\), can be expressed as  
\[R_{\rm SPAD}^{\rm vook}=\min\left[\frac{\left(\sqrt{\mu_{1}(\xi)}-\sqrt{\mu_{0}( \xi)}\right)^{2}}{Q^{-1}(P_{\rm e}^{\rm th})^{2}},\delta_{d}^{\min}B_{\rm max} \right]. \tag{24}\]  
It should be indicated that (24) requires \(\xi<2 Nh\upsilon\ln\frac{P_{\rm e}+P_{\rm h}}{P_{\rm h}}/2\tau_{d}Y_{\rm PDE}P_{R}\) and the derived transmittance value (22) satisfies this condition.