# Highly Sensitive SPAD-Based Receiver for Dimming Control in LiFi Networks +
## 4 Multi-Carrier Transmission Schemes with Dimming Control
### Dco-Ofdm
#### 4.1.2 Adaptive SPAD Receivers  
Theoretical performance analysis of SPAD-based OWC systems with DCO-OFDM and ACO-OFDM was conducted in [47; 48], and the derived equations will be used in this study to be suited for dimming scenarios. It is noted that (7) is a nonlinear function of the signal amplitude causing this shot noise to be signal-dependent and more complicated than the traditional APD shot noise, whose power is proportional to the signal amplitude [49]. Therefore, there are two nonlinear distortions in the SPAD OFDM system under consideration. Similarly to linear photodetectors, the clipping-induced distortion caused by the light source's low dynamic range as presented in (26) is one component. The second is the special distortion caused by SPAD as introduced in (7). The combined nonlinear distortion of the transmitted signal \(x[n]\) can be written as [47]  
\[\mu_{\text{a}}(x[n])= \begin{cases}(\beta_{1}\kappa_{t}+\beta_{2})\,T_{\text{s }}\exp\Bigl{[}-\frac{(\beta_{1}\kappa_{t}+\beta_{2})\tau_{\text{d}}}{N_{\text {a}}}\Bigr{]},&x[n]\geq\kappa_{t},\\ (\beta_{1}x[n]+\beta_{2})\,T_{\text{s}}\exp\Bigl{[}-\frac{(\beta_{1}x[n]+ \beta_{2})\tau_{\text{d}}}{N_{\text{a}}}\Bigr{]},&\kappa_{b}<x[n]<\kappa_{t}, \\ (\beta_{1}\kappa_{b}+\beta_{2})\,T_{\text{s}}\exp\Bigl{[}-\frac{(\beta_{1} \kappa_{b}+\beta_{2})\tau_{\text{d}}}{N_{\text{a}}}\Bigr{]},&x[n]\leq\kappa_ {b}.\end{cases}  \tag{39}\]  
where  
\[\beta_{1}=\alpha I_{\text{s}}(\xi),\ \ \text{and}\ \ \beta_{2}=I_{\text{s}}(\xi)P_{ \text{bias}}+I_{\text{n}}(\xi). \tag{40}\]  
and \(I_{\text{s}}(\xi)\) with \(I_{\text{n}}(\xi)\) can be calculated from (5). According to (27), the nonlinear distortion in an OFDM-based system can be quantified by a gain factor (\(\eta\)) and an additional distortion-induced noise (\(w^{\text{clip}}[n]=w_{d}[n]\)). The gain factor(\(\eta\)) can be expressed as [47]  
\[ \begin{split}\eta_{\text{DCO}}=&\frac{\beta_{1}^{2} \tau_{\text{d}}T_{\text{s}}}{\sqrt{2\pi}N_{\text{a}}}\left\{\exp\biggl{[}-\frac {\kappa_{t}^{2}}{2}-\frac{\tau_{\text{d}}}{N_{\text{a}}}(\beta_{1}\kappa_{t}+ \beta_{2})\biggr{]}-\exp\biggl{[}-\frac{\kappa_{b}^{2}}{2}-\frac{\tau_{\text{ d}}}{N_{\text{a}}}(\beta_{1}\kappa_{b}+\beta_{2})\biggr{]}\right\}\\ &+\beta_{1}T_{\text{s}}e^{-\frac{\beta_{1}\tau_{\text{d}}}{N_{ \text{a}}}\frac{\beta_{1}^{2}\tau_{\text{d}}^{2}}{2N_{\text{s}}}}\left[1+\frac {\beta_{1}^{2}\tau_{\text{d}}^{2}}{N_{\text{a}}^{2}}-\frac{\beta_{2}\tau_{ \text{d}}}{N_{\text{a}}}\right]\times\biggl{[}Q\biggl{(}\kappa_{b}+\frac{ \beta_{1}\tau_{\text{d}}}{N_{\text{a}}}\biggr{)}-Q\biggl{(}\kappa_{t}+\frac{ \beta_{2}\tau_{\text{d}}}{N_{\text{a}}}\biggr{)}\biggr{]}.\end{split}  \tag{41}\]The variance of the distortion induced noise, \(\sigma^{2}_{w_{\rm d}}\), can be calculated by  
\[\sigma^{2}_{w_{\rm d}}=\mathbb{E}\{\mu^{2}_{\rm a}(x)\}-\mathbb{E}^{2}\{\mu_{\rm a }(x)\}-\eta^{2}_{\rm DCO}, \tag{42}\]  
where \(\mathbb{E}\{\mu^{2}_{\rm a}(x)\}\) and \(\mathbb{E}^{2}\{\mu_{\rm a}(x)\) can be retrieved from ([49] Equation (33)) and ([49], Equation (37)), respectively. After serial-to-parallel (S/P) mapping, the detected photon count at the output of the SPAD array during the sample period \(T_{\rm s}\) is denoted as \(y[n]\). As the SPAD array size increases, CLT is valid causing \(y[n]\) to be approximately Gaussian distributed with mean and both variances given by (6), (7), and (42), respectively. Hence, taking into account (27), \(y[n]\) can be represented as  
\[y[n]=\eta_{\rm DCO}x[n]+w_{\rm a}[n]+w_{\rm d}[n], \tag{43}\]  
where \(w_{\rm a}[n]\) depicts signal-dependent shot noise that is Gaussian distributed with zero mean and signal-dependent variance (7), and \(w_{\rm d}[n]\) is the distortion-induced noise according to the [PERSON] theorem. The photon-counting signal \(y[n]\) is transformed back to the frequency domain using FFT as in (34) and can be expressed as  
\[Y[k]=\eta_{\rm DCO}X[k]+W_{\rm a}[k]+W_{\rm d}[k], \tag{44}\]  
where \(W_{\rm a}[k]\) and \(W_{\rm d}[k]\) denote the FFT of \(w_{\rm a}[n]\) and \(w_{\rm d}[n]\), respectively.