## 2 Principe of Proposed End-to-End Optical Fiber System with the Adaboost Algorithm Receiver Design
### The Design of the Adaboost Algorithm Receiver  
As the receiver, since the waveform is generated by ANN, ordinary algorithms, such as FFE, can only be used to improve the ISI between blocks and cannot identify the waveform generated by ANN. The usual demodulation method is not applicable, so it is impossible to identify the waveform and demodulate the information carried by the waveform. Therefore, this paper uses the Adaboost algorithm as the receiver. In the receiver side, we use the Adaboost algorithm to combine the Decision Trees as the receiver. The Adaboost training method is shown in Figure 2. At the beginning, the weights of all Decision Trees are initialized for the same parameters. The common training method of the Adaboost algorithm is to average all the parameters, and then train each Decision Tree. The output weight of the \(k\)-th weak learner of the training set is:  
\[D(k)=(w_{k1},w_{k2},w_{k3},\ldots,w_{kn});\,w_{1i}=\frac{1}{m}\,i=1,2,3, \ldots,m. \tag{5}\]  
The error rate of classification problems is well understood and calculated. Since multivariate classification is a generalization of binary classification, assuming that we are a binary classification problem, the output is \(\{0,1\}\), and then the weighted error rate of the k-th weak classifier \(G_{k}(x)\) on the training set is  
\[e_{k}=P(G_{k}(x_{i})\
eq y_{i})=\sum_{i=1}^{m}w_{k1}(G_{k}(x_{i})\
eq y_{i}). \tag{6}\]  
The Adaboost algorithm generally uses a single-layer Decision Tree as its weak classifier. The single-layer Decision Tree is the most simplified version of the Decision Tree, which has only one decision point. It can be seen from the above formula that, if the classification error rate \(e_{k}\) is larger, the corresponding weak classifier weight coefficient \(\alpha_{k}\) is smaller. In other words, a weak classifier with a small error rate has a larger weight coefficient. Espe  
Figure 2: Training process of the Adaboost algorithm. cially, the reason why this weight coefficient formula is used will be discuss when we investigate the loss function optimization of the Adaboost algorithm. Assuming that the weight coefficient of the sample set of the \(k\)-th weak classifier is \(D(k)=(w_{k1},w_{k2},w_{k3},\ldots,w_{km})\), the weight coefficient of the sample set of the corresponding \(k+1\) weak classifier is  
\[w_{k+1,i}=\frac{w_{ki}}{Z_{K}}\exp\left(-\frac{1}{2}log\frac{1-e_{k}}{e_{k}}y_ {i}G_{k}(x_{i})\right). \tag{7}\]  
From the calculation formula of \(w_{k+1,i}\), it can be seen that, if the \(i\)-th sample is classified incorrectly, \(y_{i}G_{k}(x_{i})<0\), which causes the weight of the sample to increase in the \((k+1)\)-th weak classifier. If the classification is correct, then the weight is reduced in the \((k+1)\)-th weak classifier. Only one weak classifier is trained for each iteration, and the trained weak classifier will be used in the next iteration. That is to say, in the N-th iteration, there are a total of N weak classifiers, of which (N \(-1\)) weak classifiers are previously trained, and their various parameters are no longer changed. Among them, the relationship of the weak classifiers is that the N-th weak classifier is more likely to match the data that the other weak classifiers did not match, and the final classification output depends on the comprehensive effect of these N classifiers. Adaboost classification uses a weighted voting method, and the final strong classifier is  
\[f(x)=sign\left(\sum_{k=1}^{K}\frac{1}{2}log\frac{1-e_{k}}{e_{k}}G_{k}(x) \right). \tag{8}\]  
For the updated sample weight, the sample set weight coefficient of the \(k+1\) weak classifier is  
\[w_{k+1,i}=\frac{w_{ki}}{\sum_{i}^{m}w_{ki}\alpha_{k}^{1-e_{ki}}}\alpha_{k}^{1 -\frac{|y_{i}-G_{k}(x_{i})|}{e_{k}}}\ i=1,2,3,\ldots,m. \tag{9}\]  
Finally, we obtain a strong classifier:  
\[f(x)=G_{k^{*}}(x). \tag{10}\]  
Among them, \(G_{k^{*}}(x)\) is the weak classifier corresponding to the serial number \(k^{*}\) corresponding to the median value of all \(ln\frac{1}{\alpha_{k^{*}}},\ k=1,\ 2,\ 3,\ldots,K\).