## 2 Principle of the Proposed Scheme
### Principle of M-ary SVM  
As a binary classifier, the basic idea of SVM is to find the separating hyperplane which can correctly divide the training data set and has the maximum margin. Figure 2 shows the example of linearly separating data from two groups in two dimensions. The separating hyperplane can be expressed as:  
\[y(\mathbf{x})=\mathbf{w}^{\mathsf{T}}\mathbf{x}+b, \tag{3}\]  
where the vector \(\mathbf{w}=(w_{1};w_{2};\ldots;w_{d})\) and the scalar \(b\) are the parameters of the hyperplane. The hyperplane function \(y(\mathbf{x})\) is learned from \(N\) training vectors \(\mathbf{x}_{1},\cdots,\mathbf{x}_{N}\) with the  
Figure 1: The probability distribution of PS-64 QAM.  
corresponding labels \(l_{1},\cdots,l_{N}\), where \(l_{n}\in\{-1,+1\}\), and the testing data are classified according to the sign of \(y(\mathbf{x})\).  
However, the data cannot always be separated linearly. For the linearly inseparable situation, a kernel function can be used to map the data from a lower dimensional feature space to a higher dimensional feature space, which is shown in Figure 3. In the higher dimensional space, a hyperplane can be obtained to linearly separate the data. In this paper, the Gaussian radial basis function (RBF) was chosen as the kernel function:  
\[\kappa(\mathbf{x}_{i},\mathbf{x}_{j})=exp(-\frac{\left\|\mathbf{x}_{i}- \mathbf{x}_{j}\right\|^{2}}{2\sigma^{2}}). \tag{4}\]  
SVM is a binary classifier which can only divide the data into two classes. For PS-64 QAM signals, each constellation cluster should be considered as a class. Therefore, it is inevitable to build a multiclass classifier by combining many binary SVMs to divide the data into different classes. In order to solve the problem, a \"one-against-all\" approach is proposed in the Ref. [21], which is a multiclass classifier whose number of SVMs is the same as the number of the classes. Therefore, the PS-64 QAM signal requires 64 SVMs to divide the data into 64 classes. Another commonly used scheme for the multiclass classifier is called \"one-against-one\" [22]. In this scheme, \(K(K-1)/2\) (\(K\) is the number of classes) different binary classifier models are trained on all possible pairs of classes. Then the data are detected according to which class has the highest number of \"votes\". However, the above approaches are time-consuming because they need more training data to obtain a large number of classifiers.  
A M-ary SVM scheme for the \(K\)-class signal was proposed in the Ref. [16]. In this scheme, only \(log_{2}K\) binary SVMs are required for the \(K\)-class signal. For PS-64 QAM, each signal carries 6 bits and each bit is modeled by a binary SVM. The classification strategy for PS-64 QAM is shown in Figure 4. The six SVMs from SVM1 to SVM6 are used to detect  
Figure 3: The linearly inseparable data from a low-dimensional feature space mapped to the linearly separable high-dimensional feature space through the kernel function.  
Figure 2: Example of binary SVM.  
the first bit to the sixth bit of the PS-64 QAM constellation points. For PS-64 QAM, each symbol is detected by six SVMs. We take one constellation point (010010) of PS-64 QAM as an example to illustrate the classification strategy. Firstly, the test data (010010) are detected by SVM1, and the first bit \"0\" is labeled as \"\(-\)1\". Next, the SVM2 is used to detect the second bit and the second bit \"1\" is labeled as \"\(+\)1\". Then SVM3, SVM4, SVM5 and SVM6 are used to detect the third, fourth, fifth and sixth bits, respectively. The third to the sixth bits of the example constellation point (010010) were labeled as \"\(-\)1\", \"\(-\)1\", \"\(+\)1\" and \"\(-\)1\", respectively. Therefore, PS-64 QAM can be detected accurately by only six SVMs aided by the M-ary SVM scheme. This scheme can effectively reduce the training time and calculation quantity of the multiclass classification SVM model.