## 2 Theoretical Analysis
### SVM-based Detection  
For coherent optical communications systems, the modulation format \(M\)-QAM is usually selected. To be able to process the signals at the receiver by using an SVM, each cluster of the signal constellation represents one class, for example, 16-QAM consists of 16 classes. Since the SVM is fundamentally a binary classifier, an extension of the SVM structure is required. Various methods have been proposed for combining multiple binary SVMs in order to build a multi-class SVM. Common methods to extend the SVM are the one-vs-one (OVO), one-vs-all (OVA) and binary-coding SVM (BCSVM) or also called M-ary SVM [15]. The OVO principle is based on the comparison of two different classes from the entire set of all classes. If a data set contains \(N\) different classes, then \(N(N-1)/2\) different binary SVMs are trained. For the concluding decision a voting procedure is used and the class with the highest vote is detected. In the OVA scheme the data of one class is separated from the complete data of all remaining \(N-1\) classes. Thus, \(N\) SVMs are trained. At the end, the class with the highest vote is detected. For communication systems, the BCSVM is the most appropriate choice. The symbols or classes are already labelled in binary format, enabling each individual bit to be modelled with one conventional SVM. For \(M\)-QAM \(\log_{2}(M)\) SVMs are required.  
Figure 1a shows the principle of the BCSVM for 16-QAM. The respective SVMs are color-coded with the corresponding bits. Figure 1b illustrates the processing structure. The complex input vector \(\mathbf{x}_{\text{rx}}\) contains the received symbols divided into real and imaginary parts. Training is done according to the transmitted bit sequence \(\mathbf{y}\). The separating hyperplane of each SVM is determined by quadratic programming during training [3].  
The output of the binary SVM array is an estimation of the transmitted bit sequence \(\mathbf{\hat{y}}\ =\ \left[\mathbf{\hat{y}}_{1},\mathbf{\hat{y}}_{2},\mathbf{ \hat{y}}_{3},\mathbf{\hat{y}}_{4}\right].\) Consequently, the received signal is classified and demodulated at the same time.  
Figure 1: Binary Coding Support Vector Machine (BCSVM) nonlinear classification using four support vector machines (SVMs) [3]: (**a**) Coding and classification scheme for BCSVM based detection and (**b**) the processing structure for BCSVM used for 16-QAM signal detection.  
Figure 2 exemplarily shows an iteration of the training process of the presented methods. The data points of the two opposite classes are colored red or blue. It can be seen, that the OVA and BCSVM method, in contrast to the OVO, take into account all data points in each iteration step. This may result in a significant computational complexity, if too much training data is used. However, the OVO method requires significantly more iterations steps than the other methods [17].  
During the training process it is necessary to adapt certain parameters of the optimization problem to the characteristics of the input data. The aim is to avoid over- or underfitted systems, which leads to a significantly reduced classification accuracy [12]. The adaptation and verification is implemented using the two optimization algorithms Grid-Search and Cross Validation [12; 18]. In the optimization process, 70% of the training data is used for training and 30% for validation.