# The Extended SLM Combined Autoencoder of the PAPR Reduction Scheme in DCO-OFDM Systems
## 2 Methods
### The ESLM-AE PAPR Reduction Scheme
#### 2.2.1 Autoencoder Network  
In our scheme, an AE network is applied to the DCO-OFDM system to optimize the end-to-end performance. Usually a feedforward neural network(NN) with \(L\) layers describes a mapping \(f(r_{0};\theta):R^{N_{0}}\mapsto R^{N\ell}\) of an input vector \(r_{0}\in R^{N_{0}}\) to an output vector \(r_{L}\in R^{N_{L}}\) through \(L\) iterative processing steps [33]:  
\[r_{\ell}=f_{\ell}(r_{\ell-1};\theta_{\ell}),\ell=1,\ldots,L \tag{7}\]  
Where \(f_{\ell}(r_{\ell-1};\theta_{\ell}):R^{N_{\ell-1}}\mapsto R^{N\ell}\) is the mapping carried out by the \(\ell\)th layer and \(\theta=\{\theta_{1},\ldots,\theta_{L}\}\) is used to denote the set of all parameters of the network. The \(\ell\)th layer is called dense or fully-connected (FC) layer if \(f_{\ell}(r_{\ell-1};\theta_{\ell})\) has the form  
\[f_{\ell}(r_{\ell-1};\theta_{\ell})=\rho(W_{\ell}r_{\ell-1}+b_{\ell}), \tag{8}\]  
Where \(W_{\ell}\in R^{N_{\ell}}\times R^{N\ell-1},\ b_{\ell}\in R^{N\ell}\), are the weights and bias for the \(\ell\)th layer respectively. \(\rho(\cdot)\) is an activation function and the set of parameters for this layer is \(\theta_{\ell}=\{W_{\ell},b_{\ell}\}\). For the classic AE, its expected output is the input, which is different from other neural networks. Therefore, the AE can be trained from scratch without supervision, and a multilayer network can represent a mapping from the input to the expected output, identity as the input. The AE has been applied in many communication fields such as channel encoding and decoding, channel compensation and modulation recognition [22; 23; 24]. A brief illustration of the proposed AE system is shown in Figure 2[20]. The transmitter part is called the encoder and it maps the input signals into the I-Q constellation. We assume both encoder and decoder are composed of \(L_{f}=L_{g}=3\) sub-blocks. Each of the sub-blocks is composed of dense layer, batch normalization (Batchnorm), activation function and dropout. Let \(r_{\ell}^{f}\) be the input of the \(\ell\)th dense layer of the encoder and the output can be expressed as \(h_{\ell}^{f}=W_{\ell}^{f}r_{\ell}^{f}+b_{\ell}^{f}\), where \(W_{\ell}^{f}\) and \(b_{\ell}^{f}\) are the weights and bias for the \(\ell\)th layer. The output of each dense  
Figure 2: Illustration of the proposed autoencoder system. layer passes through the batch normalization (Batchnorm) layer to minimize the internal covariate shift. The Batchnorm can be mathematically expressed as \(\left|h_{\ell}^{f}\right|_{\text{norm}}=\alpha\frac{h_{\ell}^{f}-\text{E}\left[h_ {\ell}^{f}\right]}{\sqrt{var\left[h_{\ell}^{f}\right]+\
u}}+\beta\), where \(\alpha\) and \(\beta\) are the scaling and shift factors, respectively. \(\
u=0.001\) is a constant which prevents the division by zero [22]. Then the normalized value is fed into the activation function to make the data features nonlinear. The activation functions used in our scheme are the rectifier linear unit (ReLU) [33] and sigmoid [34] which are defined as \(\max\Bigl{(}\left|h_{\ell}^{f}\right|_{\text{norm}},\;0\Bigr{)}\) and \(\frac{1}{1+e^{-\left|h_{\ell}^{f}\right|_{\text{norm}}}}\) respectively. In the encoder part the activation function used in each of the sub-blocks is ReLU. Finally, dropout is used for addressing the overfitting problem for the proposed AE network, which has large number of parameters. The key idea is to randomly drop units from the neural network during training, which significantly reduces overfitting and gives major improvements over other regularization methods [35]. As is shown in Figure 2, the output of the encoder is then transmitted to the simulated channel and decoder. Decoder has a similar structure as the encoder network. The only difference is the activation function of the last sub-block is sigmoid. It aims to recover the original binary information from the distorted signal after the complicated channel transmission. Mathematically, the output of the encoder can be expressed as Equation 9 [22].