# The Extended SLM Combined Autoencoder of the PAPR Reduction Scheme in DCO-OFDM Systems
## 2 Methods
### An Overview of the DCO-OFDM System Model
Complementary cumulative distribution function (CCDF) is used to denote the probability that the PAPR of signals will exceed a given threshold value, \(PAPR_{0}\), i.e., \(CCDF=\Pr(PAPR>PAPR_{0})\). After the parallel to serial conversion and adding a cyclic prefix (CP), the DC bias and clipping are added to the time-domain discrete signals \(s(n)\) to ensure all the signal amplitudes are nonnegative. In VLC systems, the transmitted signal has to be constrained in the linear range due to the nonlinear characteristics of LED [30]. Accordingly, \(s(n)\) is subjected to amplitude clipping at given upper (\(\xi_{upper}\)) and lower (\(\xi_{lower}\)) levels before fed into LED. Assume the linear range of LED is \(\left[0,2\xi_{upper}\right]\) and the symmetric clipped signal of \(s(n)\) is given in Equation (3) [31]. \[x_{Clipped}(n)=\left\{ \begin{array}{ll}\xi_{upper},\;s(n)>\xi_{upper}\\ s(n),\;\xi_{lower}\leq s(n)\leq\xi_{upper}\\ \xi_{lower},\;s(n)<\xi_{lower}\end{array} \right., \tag{3}\]  
where \(\mathrm{n}=0,1,\cdots,2N-1\). We define \(\xi_{upper}{=-\xi_{lower}=\gamma\sqrt{E\abs{\abs{s(n)}^{2}}}}\) and \(\gamma\) is referred to the clipping ratio. To assure a steady light intensity and maximize the modulation depth, the DC bias is set as \(B_{DC}=\xi_{upper}\), Then the DCO-OFDM signals fed into LED are expressed as Equation (4). \[x_{DCO}(n)=x_{Clipped}(\mathrm{n})+\mathrm{B}_{DC}=\left\{ \begin{array}{ll}2 \xi_{upper},\;\mathrm{s}(n)>\xi_{upper}\\ s(n)+\xi_{upper},\;-\xi_{upper}\leq s(n)\leq\xi_{upper}\\ 0,\;\mathrm{s}(n)<-\xi_{upper}\end{array} \right., \tag{4}\]  
where \(\mathrm{n}=0,1,\cdots,2N-1\). Essentially, clipping noise will degrade the performance of the DCO-OFDM system due to the non-linear distortion. Subsequently the unipolar signal drives the LED to converts the electrical signals to the optical signals. In the optical channel, the line-of-sight (LOS) links are assumed to dominate over all multipath components from the wall and ceiling reflections. The received signal is influenced by the noise sources in a real scenario. The dominant noise source in an indoor wireless optical channel is the ambient light induced shot noise [32], which is modeled as the additive white Gaussian noise (AWGN) given by \(Z=\left[Z_{0},Z_{1},\ldots,Z_{2N-1}\right]^{T}\). It has been considered that there is no additional clipping introduced by the photodetector (PD). Thus, after converting the optical signals to electrical signals using PD, the received signal, \(y=\left[y_{0},y_{1},\ldots,y_{2N-1}\right]^{T}\) can be computed using Equation (5). \[y=x_{DCO}\otimes q+Z, \tag{5}\]  
where \(x_{DCO}\) is the vector of DCO-OFDM signal \(x_{DCO}(n)\), \(q\) is the channel response, \(Z\) is AWGN with zero mean and variance of \(\delta_{Z}^{2}\) and \(\otimes\) denotes the convolution operation. We assume that the channel state information is perfectly known in advance. Then a reverse process can be implemented to demodulate the data. After removing the DC bias, the corresponding vector \(y\) passes through the fast Fourier transform (FFT) operation. A simplified representation of the output \(Y\) is shown in Equation (6). \[Y=FFT\{Q\circ IFFT[f(x)]+\epsilon\}, \tag{6}\]where \(Q\) denotes the effects of the optical channel and \(\varepsilon\) is the noise at the receiver. Finally \(Y\) is transformed to the decoder \(g(Y)\), which functions as the constellation mapper to get the recovered symbol \(\hat{x}\).