# Highly Sensitive SPAD-Based Receiver for Dimming Control in LiFi Networks +
## 4 Multi-Carrier Transmission Schemes with Dimming Control
### Dco-Ofdm  
In a DCO-OFDM system, the \(M\)-quadrature amplitude modulation (QAM) modulator at the transmitter converts the input bit stream into a complex symbol stream, where \(M\) is the constellation size. The symbol stream is then transformed from serial to parallel (S/P) to create vectors that can be used with the inverse fast Fourier transform (IFFT).We consider a \(K\)-point fast Fourier transform (FFT) operation, in which the information-carrying subcarriers fill the first half of the OFDM frame while leaving the 0-th and \(K/2\)-th subcarriers empty leading to a total \(K^{\prime}=K/2-1\) information-carrying subcarriers [44]. Denote the symbol allocated to the \(k\)th subcarrier as \(X[k]=1,\ldots,K\), the second half of the OFDM frame is subjected to Hermitian symmetry of the subcarriers, \(X[l]=X^{*}[K-l]\), for \(l=1,2,\ldots K/2-1\), in order to ensure a real-valued time domain signal at the cost of a 50% reduction in the spectral efficiency compared to conventional radio frequency based OFDM (RF-OFDM). The time domain signal \(x[n]\), that represents the \(n\)-th time-domain OFDM sample emitted from the source, can be obtained after the IFFT as  
\[x[n]=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1}X[k]\,e^{2\pi\frac{kn}{K}}\ ;\qquad 0\leq n \leq K-1 \tag{25}\]  
where \(j=\sqrt{(-1)}\) is the imaginary number. As long as \(K\) is fairly high (\(K\geq 64\)), the central limit theorem (CLT) states that the amplitude of \(x[n]\) is approximately zero-mean Gaussian distributed [35]. Therefore, the variance of \(X[k]\) is adjusted to \(\sigma_{X}^{2}=K/(K-2)\) in order to ensure that \(x[n]\) is with unit variance while taking into account the uniform power distribution over the subcarriers [34]. Since actual light sources have constrained dynamic ranges \([P_{\min},P_{\max}]\) that correspond to the minimum and maximum optical power of the linear dynamic range of the LED, respectively, and the relatively high signals produced, \(x[n]\) must be correctly clipped to efficiently utilize that dynamic range.The clipping would ideally be within the range \([\kappa_{\rm b},\,\kappa_{\rm t}]\) which denote the predefined normalized bottom and top clipping levels, respectively. Therefore, the clipped signal, \(\hat{x}[n]\), can be expressed as  
\[\hat{x}[n]= \begin{cases}\kappa_{t}&\text{if}\ \ \ x[n]\geq\kappa_{t},\\ x[n]&\text{if}\ \ \kappa_{b}<x[n]<\kappa_{t},\\ \kappa_{b}&\text{if}\ \ \ x[n]\leq\kappa_{b}.\end{cases}  \tag{26}\]According to the Bussgang theorem [45], the nonlinear distortion effects caused by the clipping operation can be approximated by  
\[\hat{x}[n]=\eta x[n]+w^{\text{clip}}[n], \tag{27}\]  
where \(\eta\) denotes the clipping attenuation factor and \(w^{\text{clip}}[n]\) is the clipping noise. It is noted that the clipping noise \(w^{\text{clip}}[n]\) is not correlated with the signal \(x[n]\), hence, based on the [PERSON] theorem, we can get \(\eta=\mathbb{E}[x[n]\hat{x}[n]]/\sigma_{x}^{2}\) and \(\mathbb{E}[x[n]w^{\text{clip}}[n]]=0\) where \(\sigma_{x}^{2}\) is the variance of \(x[n]\) which is unity throughout the paper, and \(\mathbb{E}[. ]\) is the expectation operator. Therefore, the moments of the clipping noise can be expressed as  
\[\mathbb{E}[w^{\text{clip}}[n]]=\mathbb{E}[\hat{x}[n]],\quad\sigma_{\text{clip }}^{2}=\mathbb{E}[\hat{x}^{2}[n]]-\eta\sigma_{x}^{2}-\mathbb{E}^{2}[\hat{x}[n ]], \tag{28}\]  
To achieve a maximal signal SNR and to transform the electrical signal into an optical one, the linear dynamic range 0 and \(P_{\text{max}}\) should be mapped by the normalised clipping levels, \(\kappa_{b}\) and \(\kappa_{t}\). Hence, to form such mapping, scaling, DC biasing, and digital-to-analogue conversion are required stages before the light source can be driven.