# The Extended SLM Combined Autoencoder of the PAPR Reduction Scheme in DCO-OFDM Systems
## 2 Methods
### The ESLM-AE PAPR Reduction Scheme
#### 2.2.2 Extended Selected Mapping Technique  
For the DCO-OFDM, a high signal peak value implies a need for a large DC bias that causes serious degradation of the system's power efficiency. Therefore, inspired by SLM, a reduction scheme, named extended Selected Mapping method is given in this section.  
The Selected Mapping (SLM) method is one popular scheme to reduce the PAPR, because it is simple to implement without introducing any distortion to the signal and it can be used with any subcarrier number and modulation style [36]. The principle is that \(u\) copies of the complex data vector \(\mathbb{C}^{u}=[C_{0}^{u},C_{1}^{u},\cdots,C_{N-1}^{u}]\) are multiplied by \(u\) distinct phase vectors \(B^{u}=[B_{0}^{u},B_{1}^{u},\cdots,B_{N-1}^{u}]\), \(u=1,2,\cdots,U\). The corresponding time domain data vector after IFFT is shown in Equation (13).  
\[\mathbb{c}_{n}^{u}=IFFT\{\mathbb{C}_{k}^{u}\cdot B_{k}^{u}\}, \tag{13}\]  
where \(\mathrm{n}=0,1,\cdots,N-1\).  
The objective of the SLM technique is to determine the transmitted \(\mathbb{c}_{n}^{u}\) using Equation (14) [29].  
\[\min_{(B^{u})}\left(\frac{\max\Bigl{\{}|c_{n}^{u}|^{2},n=0,1,\ldots N-1\Bigr{\}} }{\frac{1}{N}\sum_{n=0}^{N-1}|c_{n}^{u}|^{2}}\right), \tag{14}\]  
To improve the PAPR reduction performance of the SLM scheme, the SLM technique requires an increase in the number of phase sequences. However, the computational complexity of the SLM scheme linearly increases as the number of phase sequences increases. The alternative method is an extensive search for the optimal sequence that achieves the minimum PAPR. Consequently, in the SLM, the PAPR reduction is primarily dependent on the chosen phase sequence candidates [37].  
In this paper, we extended the SLM technique to AE to get the adaptive phase sequence shown in Figure 3. Each phase factor \(a_{k}\) of \(A_{k}\) no longer needs to be artificially arranged since it can be trained and continuously optimized in the deep learning network. Simultaneously, in the test, once the phase sequence is determined, the calculation of the IFFT is needed only once.  
In our proposed scheme, the network is trained to reduce the PAPR without reducing the BER performance [38]. Consequently, two distinct factors must be taken into the account at the same time. To reduce PAPR value, we define the second loss component \(Loss_{2}(x)\) which is given as Equation (15).  
\[Loss_{2}(x)=PAPR\{s(n)\}, \tag{15}\]  
Based on simulation, the \(Loss_{2}(x)\) is helpful to reduce the high PAPR and lower distortion will in turn improve the BER performance in the training process.  
Considering the two factors, we use a hyperparameter \(\eta\) to balance the two different loss components. Thus, the total loss function can be expressed in Equation (16).  
\[Loss(x,\hat{x})=Loss_{1}(x,\hat{x})+\eta Loss_{2}(x), \tag{16}\]  
Notice that each phase factor \(a_{k}\) of \(A_{k}\) can be trained in terms of \(\frac{\partial Loss}{\partial a_{k}}\) based on the propagation algorithm [39].  
Figure 3: Partial block diagram of an OFDM transmitter with the extended Selected Mapping (SLM) technique.