# Efficient Support Vector Regression for Wideband DOA Estimation Using a Genetic Algorithm
## 3 The Proposed Method  
### Data Preprocessing  
CSM is an effective method for handling broadband signals, as it has the advantages of low computational complexity, superior estimation accuracy, etc. The core of the CSM algorithm is the construction of a focusing matrix. In this paper, the TCT algorithm is adopted to design the focusing matrix to satisfy the minimization problem with the following constraints:  
\[\left\{ \begin{array}{l}\min\limits_{\mathbf{T}(f_{j})}\left\|\mathbf{P}(f_{0})- \mathbf{T}(f_{j})\mathbf{P}(f_{j})\mathbf{T}^{\mathbf{H}}(f_{j})\right\|_{F^{ \prime}}\,j=1,2,\cdots,J\\ \mathbf{T}(f_{j})\mathbf{T}^{\mathbf{H}}(f_{j})=\mathbf{I}\end{array} \right. \tag{5}\]  
where \(\mathbf{P}(f_{0})=\mathbf{A}(f_{0},\theta)E\left[\mathbf{s}_{l}(f_{0})\mathbf{ s}_{l}^{\mathbf{H}}(f_{0})\right]\mathbf{A}^{\mathrm{H}}(f_{0},\theta)\) represents the denoised covariance matrix of the array output data at the reference frequency point, \(f_{0}\) is the reference frequency point, \(\mathbf{P}\big{(}f_{j}\big{)}=\mathbf{A}\big{(}f_{j},\theta)E\left[\mathbf{s} _{l}(f_{j})\mathbf{s}_{l}^{\mathbf{H}}(f_{j})\right]\mathbf{A}^{\mathrm{H}}(f _{j},\theta)\) is the denoised covariance matrix of the array output data at the frequency point \(f_{j}\), and \(\left\|\cdot\right\|_{F}\) denotes the Frobenius norm of the matrix. In practice, \(P\big{(}f_{j}\big{)}\) is estimated using \(\mathbf{P}\big{(}f_{j}\big{)}=\hat{\mathbf{R}}_{j}-\delta_{j}^{2}\mathbf{I}\), where \(\hat{\mathbf{R}}_{j}=\frac{1}{L}\sum\limits_{l=1}^{L}\mathbf{x}_{l}\big{(}f_{ j}\big{)}\mathbf{x}_{l}^{\mathrm{H}}(f_{j})\) is the estimated value of \(\mathbf{R}_{j}\), and \(\delta_{j}^{2}\) is the estimated noise power, i.e., the average of the small eigenvalues of the covariance matrix \(\hat{\mathbf{R}}_{j}\) at the frequency point \(f_{j}\). Reference [44] gives a solution to Equation (5), i.e.,  
\[\mathbf{T}\big{(}f_{j}\big{)}=\mathbf{Q}(f_{0})\mathbf{Q}^{\mathrm{H}}(f_{j}) \tag{6}\]  
where \(\mathbf{Q}(f_{0})\in\mathbf{C}^{M\times M}\) and \(\mathbf{Q}\big{(}f_{j}\big{)}\in\mathbf{C}^{M\times M}\) are the eigenvectors of \(\mathbf{P}(f_{0})\) and \(\mathbf{P}\big{(}f_{j}\big{)}\), respectively, and the columns are mutually orthogonal. When multi-frequency data from the wideband signal are directly input into the SVR model, the training time increases significantly. In this paper, we employ a focusing technique to map the multi-frequency data to the reference frequency point. This approach reduces the input dimensionality of the model without compromising the effective information across all frequency bands. By applying Equation (6) to Equation (3), we obtain the focused data  
\[\mathbf{x}_{l}^{0}\big{(}f_{j}\big{)}=\mathbf{T}\big{(}f_{j}\big{)}\cdot \mathbf{x}_{l}\big{(}f_{j}\big{)}=\mathbf{A}(f_{0},\theta)\cdot\mathbf{s}_{l} (f_{0})+\mathbf{n}_{l}^{0}\big{(}f_{j}\big{)} \tag{7}\]  
where \(\mathbf{x}_{l}^{0}\big{(}f_{j}\big{)}\) and \(\mathbf{n}_{l}^{0}\big{(}f_{j}\big{)}=\mathbf{T}\big{(}f_{j}\big{)}\cdot \mathbf{n}_{l}\big{(}f_{j}\big{)}\) denote the receive data and the noise vector after focusing at frequency \(f_{j}\) for the \(l\)-th snapshot, respectively. Since the focusing matrix \(\mathbf{T}(f_{j})\) is a unitary matrix and satisfies the constraint of \(\mathbf{T}\big{(}f_{j}\big{)}\mathbf{T}^{\mathrm{H}}(f_{j})=\mathbf{I}\), the noise vector after focusing follows a Gaussian distribution with 0 mean and variance \(\sigma_{n}^{2}\). Using Equations (4) and (7), the focused covariance matrix can be calculated:  
\[ \begin{split}\mathbf{R}_{0}&=\frac{1}{J}\sum \limits_{j=1}^{J}E\left[\mathbf{x}_{l}^{0}\big{(}f_{j}\big{)}\mathbf{x}_{l}^{0 }\big{(}f_{j}\big{)}^{\mathrm{H}}\right]\\ &=\frac{1}{J}\sum\limits_{j=1}^{J}\Big{(}E\Big{[}\mathbf{T}\big{(} f_{j}\big{)}\cdot\mathbf{x}_{l}\big{(}f_{j}\big{)}\mathbf{x}_{l}^{\mathrm{H}}(f_{j} \big{)}\mathbf{T}^{\mathrm{H}}(f_{j})\Big{]}\Big{)}\\ &=\frac{1}{J}\sum\limits_{j=1}^{J}\Big{(}\mathbf{T}\big{(}f_{j} \big{)}\mathbf{R}_{j}\mathbf{T}^{\mathrm{H}}\big{(}f_{j}\big{)}\Big{)}\end{split}