# The Extended SLM Combined Autoencoder of the PAPR Reduction Scheme in DCO-OFDM Systems
## 2 Methods  
### An Overview of the DCO-OFDM System Model  
Figure 1 shows an overview of the DCO-OFDM transmitter and receiver structure based on the proposed ESLM-AE scheme. Different from the conventional DCO-OFDM system, an AE is applied in the whole model. AE is a special neural network which can represent the mapping from the input to itself because of universal approximation theorem [27; 28]. The neurons in the hidden layer of autoencoder can be considered as a coding representation of input. In our scheme, the input signals are firstly fed into the encoder and phase rotator modules before Hermitian symmetry and inverse fast Fourier transform (IFFT) to get the in-phase and quadrature (I-Q) constellation mapping and generate the alternative output sequence. The outputs of IFFT are then converted into unipolar by adding DC bias and clipping. Signal clipping is performed in order to fit the real time-domain OFDM symbols into a limited range of the LED. In the optical channel, the transmitted signals are influenced by the noise sources in a real scenario. At the receiver side, phase recovery and the decoder part aim to recover the distorted signals. The network can be trained with the lowest PAPR and BER. It is assumed that the OFDM signal is transformed by \(2N\) subcarriers. Let \(x,\ f(x)\) and \(g(x)\) be the input of encoder, encoder, and decoder of the AE, respectively. According to the DCO-OFDM system, the transmitted data stream is mapped into complex-value symbols. As shown in Figure 1, after serial to parallel conversion, the input data sequence is divided into \(2N\) messages \(x\), where \(x=\left[x_{0},x_{1},\ldots,x_{2N-1}\right]^{T}\). Then \(x\) is mapped into I-Q constellation according to the encoder part which will be detailed discussed in Section 2.2. We define the output of the encoder \(X=f(x)\), where \(X\in\Re^{2N}\) consists of \(2N\) real values and among them pairwise combination in a certain order forms \(N\) complex symbols \(A\), \(A=\left[A_{0},A_{1},\ldots,A_{N-1}\right]^{T}\), \(A\in\mathbb{C}^{N}\). However, the classic AE is only designed to minimize the BER. In practice, the transceiver usually suffers from a high PAPR. To mitigate the high PAPR, each \(A_{\rm k}\), \({\rm k}=0,1,\cdots,N-1\), is multiplied by a phase factor \({\rm a}_{k}\) which can be expressed as \(\overline{A_{k}}=A_{\rm k}\cdot{\rm a}_{k}\), where \({\rm a}_{\rm k}=e^{i\psi_{k}}\) and \(\psi_{k}\in[0,2\pi)\)[29]. For the VLC-OFDM system, the intensity modulation requires the transmitted signals of the LED to be nonnegative and real-valued [5]. Thus, Hermitian symmetry is imposed on \(\overline{A_{k}}\) to form the frequency domain OFDM symbols \(S=\left[\overline{A_{0}},\overline{A_{1}},\cdots,\overline{A_{N-1}}, \overline{A_{N-2}}^{*},\cdots,\overline{A_{1}}^{*}\right]\) and the DC components are \(\overline{A_{0}}=\overline{A_{N-1}}=0\). This results in a \(2N\)-point IFFT output of the OFDM symbols. Subsequently the time domain OFDM signal can be obtained by feeding \(S(k)\), \({\rm k}=0,1,\cdots,2N-1\), into an inverse discrete Fourier transform using Equation (1). \[s(n)=\frac{1}{\sqrt{2N}}\sum_{k=0}^{2N-1}S(k)e^{j2\pi nk/2N}, \tag{1}\]  
Figure 1: An overview of the DC-biased optical orthogonal frequency division multiplexing (DCO-OFDM) system with the autoencoder network combined with extended selected mapping methods (ESLM-AE) structure, where S/P, P/S, and PD denote serial-to-parallel converter, parallel-to-serial converter, and photodetector, respectively. where \(\mathrm{n}=0,1,\cdots,2N-1\). Consequently, the PAPR of the DCO-OFDM signal is calculated using Equation (2). \[PAPR\{s(n)\}=\frac{\max_{0\leq n\leq 2N-1}\abs{s(n)}^{2}}{E\abs{\abs{s(n)}^{2}}}, \tag{2}\]  
where \(\mathrm{n}=0,1,\cdots,2N-1\). When the high amplitudes of different subcarriers with the same phase appear at the same time, the high PAPR will appear.