# Efficient Support Vector Regression for Wideband DOA Estimation Using a Genetic Algorithm
## 3 The Proposed Method
### Multiband Joint Direction-Finding Method Based on SVR
\begin{array}{l}\mathcal{L}=\underset{\mathbf{z}_{p}\mathbf{z}_{p}^{\prime}}{\min} -\frac{1}{2}\sum\limits_{p=1}^{p}\sum\limits_{i=1}^{p}\left(\alpha_{p}^{ \prime}-\alpha_{p}\right)\left(\alpha_{i}^{\prime}-\alpha_{i}\right)\kappa \left(\mathbf{z}_{0}^{p},\mathbf{z}_{0}^{i}\right)+\\ \sum\limits_{p=1}^{p}\theta^{p}\left(\alpha_{p}^{\prime}-\alpha_{p}\right)- \sum\limits_{p=1}^{p}\varepsilon\left(\alpha_{p}+\alpha_{p}^{\prime}\right) \\ s.t.\quad\sum\limits_{p=1}^{p}\left(\alpha_{p}-\alpha_{p}^{\prime}\right)=0 \\ 0\leq\alpha_{p},\alpha_{p}^{\prime}\leq\lambda,p=1,2,\cdots,P\end{array}  \tag{16}\]  
where \(\alpha_{p},\alpha_{p}^{\prime}\) are Lagrange multipliers, and \(\kappa(\mathbf{z}_{0}^{p},\mathbf{z}_{0}^{i})=e^{-\gamma\left\|\mathbf{z}_{0}^{p}-\mathbf{z}_{ 0}^{i}\right\|^{2}}=\left[\mathbf{\Phi}(\mathbf{z}_{0}^{p})\right]^{T}\mathbf{\Phi}(\mathbf{z}_ {0}^{i})\) is the kernel function. Obviously, Equation (16) is a quadratic programming problem, which can be solved by the sequential minimum optimization algorithm for the parameter \(\left\{\alpha_{p},\alpha_{p}^{\prime}\right\}_{p=1}^{p}\). The final DOA estimation result of the proposed algorithm in this paper can be obtained as  
\[\hat{\theta}=\mathbf{w}^{T}\mathbf{\Phi}(z_{0})+b=\sum\
olimits_{p=1}^{p}\left(\alpha_ {p}^{\prime}-\alpha_{p}\right)\kappa\left(\mathbf{z}_{0}^{p},\mathbf{z}_{0}\right)+b \tag{17}\]  
As can be seen from Equation (17), the algorithm proposed in this paper takes the single-frequency-point data as the input feature of the network and the signal incidence angle as the output. This design enables the joint estimation of wideband signals across multiple frequency bands without significantly increasing the model complexity. By avoiding the direct input of multiband data, the algorithm effectively reduces the dimensionality and scale of the network model, thereby decreasing the training time and computational cost. Moreover, this approach ensures that the essential features of the wideband signal are preserved, thereby achieving high estimation accuracy alongside improved computational efficiency. An algorithm block diagram of the proposed method is given by Algorithm 1. \(N_{epochs}\) represents the number of epochs used for the GA. \(Model_{SVR}(C*,\gamma*)\) represents the trained SVR model. \(C*\) and \(\gamma*\) are the optimal parameter estimation results. \(upper(\cdot)\) denotes taking the upper triangular part, and \(vec(\cdot)\) is the vectorization operator. ```
Inputs:\(x_{m}(t)\), \(L\), \(\{\theta_{i}\}\), \(N_{epochs}\) Outputs:\(Model_{SVR}(C*,\gamma*)\)
1. Sub-band decomposition using Equations (2) and (3): \(x_{m}(t)\rightarrow\{x_{l}(f_{1}),x_{l}(f_{2}),\cdots,x_{l}(f_{l})\}\);
2. Estimate the covariance matrix at frequency \(f_{j}:\ \hat{R}_{j}=\frac{1}{L}\sum\limits_{l=1}^{L}\mathbf{x}_{l}(f_{j}) \mathbf{x}_{l}^{\text{H}}(f_{j})\);
3. Construct the focusing matrix using Equation (6);
4. TCT focusing: ;
5. Extract the feature vector: \(\mathbf{z}_{0}=\frac{\left[\Re(\mathbf{u}^{\text{T}}),\Im(\mathbf{u}^{\text{T} })\right]^{\text{T}}}{\|\mathbf{z}\|_{2}}\), \(\mathbf{\eta}=vec(upper(\hat{\mathbf{R}}_{0}))\);
6. Genetic algorithm optimizes SVR parameters: for\(n_{epochs}=1\) to \(N_{epochs}\)  Initialization \((C,\gamma)\); \(\overset{\wedge}{\theta}=Model_{SVR}(\mathbf{z}_{0};C,\gamma)\);  Calculating adaptation using Equation (12);  Selection; Crossover; Mutation; \(\left(C_{n_{epochs}},\gamma_{n_{epochs}}\right)=\text{argmin}_{(C,\gamma)}f= \sqrt{\frac{1}{NK}\sum\limits_{n=1}^{N}\sum\limits_{k=1}^{K}\left[\theta_{k}( n)-\hat{\theta}_{k}(n)\right]^{2}}\); \(n_{epochs}=n_{epochs}+1\);  end \((C*,\gamma*)=\text{arg}\min_{(C_{n_{epochs}},\gamma_{n_{epochs}})}f\);
7. \(\overset{\wedge}{\theta}=Model_{SVR}(\mathbf{z}_{0};C*,\gamma*)\); Return\(Model_{SVR}(C*,\gamma*)\). ```  
**Algorithm 1**Broadband signal processing and parameter optimization