# Highly Sensitive SPAD-Based Receiver for Dimming Control in LiFi Networks +
## 2 System Model
The received photon rate that corresponds to the highest detected photon count is \(N_{\text{a}}/\tau_{\text{d}}\), and the corresponding detected photon count is \(N_{\text{a}}T_{\text{s}}/(e\tau_{\text{d}})\), where \(N_{\text{a}}\) is the number of SPAD array pixels, \(T_{\text{s}}\) is the symbol time, and \(e\) is Euler's number [37]. The SPAD array receiver's relatively restricted dynamic range (defined by the number of micro-cells and the dead time) causes saturation when the received optical power exceeds a certain threshold. As the optical power is increased further, the PQ SPAD detector's performance degrades significantly. To address this problem, the proposed receiver employs a VOA to attenuate incident light on the SPAD array as necessary. To ensure that the SPAD array receiver performs optimally, the proposed adaptive receiver first determines the optimal transmittance, \(\xi\), from a lookup table and/or the provided equations based on the estimated received signal and ambient power values, \(P_{\text{R}}\) and \(P_{\text{b}}\), and then controls the VOA adaptively by the receiver controller that is employed at the signal demodulator unit. The receiver controller monitors both the received signal and background optical power. When the channel condition changes (e.g., a shift in receiver location or orientation or a change in brightness), the controller needs to readjust the optimal transmittance of the VOA to achieve reliable performance. The incident photon rate can be presented as [38]  
\[\lambda_{\text{i}}(\xi)=I_{\text{s}}(\xi)\gamma x_{t}+I_{\text{n}}(\xi), \tag{4}\]where \(I_{\rm s}(\xi)\gamma x_{t}\) and \(I_{\rm n}(\xi)\) denote the signal photon rate and photon rate induced by DCR and background light, respectively. They can be written as  
\[ \begin{split} I_{\rm s}(\xi)&=\frac{\xi Y_{\rm PDE} \zeta(1+\varrho_{\rm AP}+\varrho_{\rm CT})}{E_{\rm ph}},\\ I_{\rm n}(\xi)&=(\theta_{\rm DCR}+\xi\theta_{\rm B })(1+\varrho_{\rm AP}+\varrho_{\rm CT}),\end{split}  \tag{5}\]  
knowing that \(Y_{\rm PDE}\) is the PDE of the SPAD, \(E_{\rm ph}\) is the photon energy, \(\theta_{\rm B}\) represents the background photon rate, and \(\theta_{\rm DCR}\), \(\varrho_{\rm AP}\)m and \(\varrho_{\rm CT}\) denote the DCR of the array, the afterpulsing and crosstalk probabilities, respectively. It is noted that when \(\xi=1\), the proposed adaptive SPAD receiver converges to a traditional one in the absence of VOA. The background photon rate \(\theta_{\rm B}\) is equal to \(Y_{\rm PDE}P_{\rm b}/E_{\rm ph}\) where \(P_{\rm b}\) denotes the received background light power. For an array of PQ SPADs with \(N_{\rm a}\) SPAD pixels and each with a \(\tau_{\rm d}\) dead time, the average detected photon count during the time-domain OFDM sample duration, \(T_{\rm s}\), can be expressed as [32; 37]  
\[\mu_{\rm a}(x)=\lambda_{\rm i}(\xi)T_{\rm s}{\rm exp}\left(-\frac{\lambda_{\rm i }(\xi)\tau_{\rm d}}{N_{\rm a}}\right). \tag{6}\]  
Assuming the sample duration is larger than the dead time, i.e., \(T_{\rm s}\geq\tau_{\rm d}\), the variance of the detected photon count of the array is given by [39; 40]  
\[\sigma_{\rm a}^{2}(x)=\lambda_{\rm i}(\xi)T_{\rm s}{\rm exp}\left(-\frac{ \lambda_{\rm i}(\xi)\tau_{\rm d}}{N_{\rm a}}\right)-\frac{\lambda_{\rm i}^{2} (\xi)T_{\rm s}\tau_{\rm d}}{N_{\rm a}}{\rm exp}\left(-\frac{2\lambda_{\rm i}( \xi)\tau_{\rm d}}{N_{\rm a}}\right)\left(2-\frac{\tau_{\rm d}}{T_{\rm s}} \right). \tag{7}\]  
VLCs employ intensity modulation and direct detection (IM/DD), wherein a direct-detection receiver generates a photocurrent proportionate to the incident optical power. In contrast to standard VLC systems employing linear PDs, which assume a simple additive white Gaussian noise (AWGN) channel, the dead time of SPAD introduces non-linear signal distortion and complicated signal-dependent shot noise, which must be accounted for in the performance analysis [11].