# Efficient Support Vector Regression for Wideband DOA Estimation Using a Genetic Algorithm
## 3 The Proposed Method
### Multiband Joint Direction-Finding Method Based on SVR  
From Equation (4), it can be seen that the incident signal angle space \(\Theta\subset R^{K}\) and the array covariance matrix space \(\mathbf{R}\subset C^{M\times M}\) are connected via a mapping function. After the data preprocessing in Section 3.1, the angle space is mapped to the feature vector \(\mathbf{z}_{0}\). There exists the mapping relation \(\mathbf{G}:\mathbf{\theta}\rightarrow\mathbf{z}_{0}\), and the DOA estimation can be regarded as the mapping of the angle space of sources from the input feature space of the array, \(\mathbf{F}:\mathbf{z}_{0}\rightarrow\mathbf{\theta}\), i.e., it is the inverse mapping of the mapping relation \(\mathbf{G}\). Building on this, the SVR parameters are optimized using the GA, and an SVR model for multiband joint direction finding is subsequently established, as illustrated in Figure 4. It is important to point out that the detail of the signal features varies with different numbers of array elements and bandwidths, which consequently leads to different optimization results of \(C\) and \(\gamma\). This model achieves the inverse mapping from the input feature vector to the angle space. Firstly, the SVR model is trained with sample data to obtain \(\hat{\mathbf{F}}\). Subsequently, during the testing phase, the trained model \(\hat{\mathbf{F}}\) is employed to estimate the angles of the input wideband signal test samples with unknown directions. The objective of the SVR model in the proposed algorithm is to learn a functional relationship between the input features used for training \(\mathbf{z}_{0}\) and the output \(\hat{\theta}\):  
\[\hat{\theta}=\hat{F}(\mathbf{z}_{0})=\mathbf{w}^{T}\mathbf{\Phi}(\mathbf{z}_{0})+b \tag{13}\]  
where \(\hat{\theta}\) denotes the estimation of \(\theta\), \(\mathbf{\Phi}(\cdot)\) is a nonlinear mapping function that maps the input feature \(\mathbf{z}_{0}\) to a higher-dimensional space, and \(\{\mathbf{w},b\}\) are the model parameters to  
Figure 4: SVR model for multiband joint direction finding. be trained. Through the optimization of the objective function, the problem presented in Equation (13) can be equivalently transformed into  
\[\left\{\hat{\mathbf{w}},\hat{b}\right\}=\underset{\{\mathbf{w},b\}}{argmin}\frac{1}{2} \|\mathbf{w}\|^{2}+C\sum\
olimits_{p=1}^{p}L_{\varepsilon}\left(\theta^{p},\hat{F} \left(\mathbf{z}_{0}^{p}\right)\right) \tag{14}\]  
where \(C\) is a regularization parameter, \(L_{\varepsilon}(\cdot)\) denotes the loss function, \(L_{\varepsilon}\left(\theta^{p},\hat{F}\left(\mathbf{z}_{0}^{p}\right)\right)= \left\{ \begin{array}{l}\left|\theta^{p}-\hat{F}\left(\mathbf{z}_{0}^{p}\right) \right|-\varepsilon,\left|\theta^{p}-\hat{F}\left(\mathbf{z}_{0}^{p}\right)\right| >\varepsilon\\ 0,\text{\emph{otherwise}}\end{array} \right.\), and \(\left\{\left(\mathbf{z}_{0}^{1},\theta^{1}\right),\left(\mathbf{z}_{0}^{2},\theta^{2 }\right),\cdots,\left(\mathbf{z}_{0}^{p},\theta^{p}\right)\right\}\) denotes \(p\) sets of wideband signal training data. To accommodate allowable prediction errors in the SVR model, the slack variables \(\xi_{p}\) and \(\xi_{p}^{\prime}\) are introduced, leading to the reformulation of Equation (14) as follows:  
\[ \begin{array}{l}\underset{\{\mathbf{w},b,\xi_{p}\xi_{p}^{\prime}\}}{\min}\frac{1 }{2}\|\mathbf{w}\|^{2}+C\sum\limits_{p=1}^{p}\left(\xi_{p}+\xi_{p}^{\prime}\right) \\ s.t.\quad\left(\mathbf{w}^{T}\mathbf{\Phi}\left(\mathbf{z}_{0}^{p}\right)+b\right)-\theta^ {p}\leq\varepsilon+\xi_{p}\\ \theta^{p}-\left(\mathbf{w}^{T}\mathbf{\Phi}\left(\mathbf{z}_{0}^{p}\right)+b\right)\leq \varepsilon+\xi_{p}^{\prime}\\ \xi_{p}\geq 0,\xi_{p}^{\prime}\geq 0,p=1,2,\cdots,P\end{array}  \tag{15}\]  
Similar to the derivation process in [38], the dual problem of Equation (15) can be obtained as follows:  
\[