# Highly Sensitive SPAD-Based Receiver for Dimming Control in LiFi Networks +
## 3 Single-Carrier Transmission Schemes with Dimming Control
### On-Off Keying (OOK)--Analogue Dimming
#### 3.1.2 Adaptive SPAD Receivers  
Differently from the PD receivers, SPAD receivers are shot noise limited and their performance depends on both signal and background optical power.The transmitted OOK signal is assumed to be with an ideal infinite extinction ratio so that the received power for bit \"0\" is simply \(P_{\text{B}}\). Therefore, the received photon rate for each micro-cell of the SPAD array when bit \"1\" and bit \"0\" are sent can be written as [32]  
\[\begin{split}\lambda_{1}^{\text{ook}}(\xi)&=\frac{ \xi\text{Y}_{\text{PDE}}(2(\zeta\gamma\overline{P}_{\text{TX}})+P_{\text{B}})} {N_{\text{a}}E_{\text{ph}}},\\ \lambda_{0}^{\text{ook}}(\xi)&=\frac{\xi\text{Y}_{ \text{PDE}}P_{\text{B}}}{N_{\text{a}}E_{\text{ph}}},\end{split} \tag{11}\]  
It should be noted that the photon rates in (11) are expressed as functions of the adjustable transmittance, \(\xi\). For the purpose of this study, the dead time is regarded as the primary non-ideal effect of the SPAD receiver, and other effects, such as crosstalk and dark count, are disregarded for simplicity. Utilizing the effective photon rate given in Equation (3), the average photon counts corresponding to bit \"1\" and bit \"0\" during the symbol duration \(T_{\text{s}}\) can be computed using Equation (6) as  
\[\begin{split}\mu_{1}^{\text{ook}}(\xi)&=N_{\text{a} }\lambda_{1}^{\text{ook}}(\xi)T_{\text{s}}e^{-\lambda_{1}^{\text{ook}}(\xi)T_{ \text{d}}},\\ \mu_{0}^{\text{ook}}(\xi)&=N_{\text{a}}\lambda_{0}^{ \text{ook}}(\xi)T_{\text{s}}e^{-\lambda_{0}^{\text{ook}}(\xi)T_{\text{d}}}. \end{split} \tag{12}\]  
The study in [32] assumes equal mean and variance for the detected photon counts, however, we are considering unequal moments which is a more accurate assumption. Therefore, the variance for bit \"1\" and bit \"0\" during the symbol duration \(T_{\text{s}}\) can be calculated from (7) as  
\[\begin{split}\sigma_{1,\text{ook}}^{2}(\xi)&=\mu_{1 }^{\text{ook}}(\xi)-\frac{\lambda_{1}^{\text{ook}}(\xi)^{2}T_{\text{s}}\tau_{ \text{d}}}{N_{\text{a}}}\text{exp}\left(-\frac{2\lambda_{1}^{\text{ook}}(\xi) \tau_{\text{d}}}{N_{\text{a}}}\right)\left(2-\frac{\tau_{\text{d}}}{T_{\text{ s}}}\right),\\ \sigma_{0,\text{ook}}^{2}(\xi)&=\mu_{0}^{\text{ook}} (\xi)-\frac{\lambda_{0}^{\text{ook}}(\xi)^{2}T_{\text{s}}\tau_{\text{d}}}{N_{ \text{a}}}\text{exp}\left(-\frac{2\lambda_{0}^{\text{ook}}(\xi)\tau_{\text{d}} }{N_{\text{a}}}\right)\left(2-\frac{\tau_{\text{d}}}{T_{\text{s}}}\right). \end{split} \tag{13}\]  
Assuming Gaussian distribution for detected photon counts of bit \"1\" (bit \"0\"), the received BER can be expressed as [42]  
\[\text{BER}_{\text{SPAD}}^{\text{ook}}=Q\left(\frac{\mu_{1}^{\text{ook}}(\xi)- \mu_{0}^{\text{ook}}(\xi)}{\sigma_{1,\text{ook}}(\xi)+\sigma_{0,\text{ook}}(\xi) }\right). \tag{14}\]  
The analytical expression of the optimal transmittance, \(\xi\), which provides the lowest BER, is mathematically intractable. However, the expression of the approximated optimal transmittance assuming equal mean and variance of the detected photon count is given by Proposition 1 in [32]. Hence, in the simulation, we will focus on using the approximatedoptimal VOA to calculate the BER given in (14). An approximated maximum achievable data rate of the SPAD array with OOK modulation at the target BER, \(P_{\text{e}}^{\text{th}}\), can be written as [43]  
\[R_{\text{SPAD}}^{\text{ook}}= \tag{15}\] \[\min\left[\frac{\left(\sqrt{N\lambda_{1}^{\text{ook}}(\xi)e^{- \lambda_{1}^{\text{ook}}(\xi)\tau_{d}}}-\sqrt{N\lambda_{0}^{\text{ook}}(\xi)e^{ -\lambda_{0}^{\text{ook}}(\xi)\tau_{d}}}\right)^{2}}{Q^{-1}(P_{\text{e}}^{\text {th}})^{2}},B_{\text{max}}\right].\]  
It is worth noting that when the VOA transmittance of the adaptive receiver is fixed at unity, its performance converges to that of a traditional SPAD receiver in the absence of VOA.