# Optimized Sonar Broadband Focused Beamforming Algorithm
## 2 Broadband Focused Beamforming Algorithm
### Principle of an Optimized Broadband Focused Beamforming Algorithm  
The correlation matrix estimated from the sample values in frequency domain is:  
\[ \begin{array}{l}R_{l}=\frac{1}{K}\sum\limits_{k=1}^{K}X_{k}(f_{l})X_{k}{}^{H} (f_{l})\\ l=1,\cdots,L\end{array} , \tag{12}\]where \(K\) is the data snapshot in each frequency sub-band and \(f_{0}\) is the optimal focusing frequency according to the determination method. According to the given range of frequencies and optimal focusing frequency, the beamforming algorithm based on RCB becomes:  
\[\left\{ \begin{array}{ll}\max\limits_{a_{0}}~{}{\sigma_{0}}^{2}\\ subject~{}~{}\text{to}~{}~{}R_{0}{-\sigma_{0}}^{2}~{}~{}a_{0}a_{0}{}^{H}\geq 0 \\ and~{}~{}~{}\|a_{0}-\overline{a}_{0}\|^{2}\leq\varepsilon\end{array} \right., \tag{13}\]  
where \({\sigma_{0}}^{2}\) is the power of the optimal focusing frequency sub-band; \(R_{0}\) and \(a_{0}\) are the covariance matrix and the steering vector of the optimal focusing frequency sub-band, respectively; \(\overline{a}_{0}\) is presumed steering vector; and \(\varepsilon\) is the custom error range of steering vector. The correction vector is the closest to the real value as follows:  
\[\hat{a}_{0} =\left(\frac{R^{-1}}{\kappa}+I\right)^{-1}\overline{a}_{0} \tag{14}\] \[=\overline{a}_{0}-(I+\kappa R)^{-1}\overline{a}_{0}\]  
where \(\kappa\) is determined by Newton iteration method [12; 13]. The correction vector \(\hat{a}_{l}\) of the other reference frequency can be determined. \(f_{l}\) is the other reference frequency. The correction focusing matrix can be determined by the steering vectors \(\hat{a}_{0}\) and \(\hat{a}_{l}\). \[\hat{T}_{l}(\varphi,\theta)=\hat{V}(f_{l})\hat{U}^{H}(f_{l}) \tag{15}\]  
Each of the sub-band data is focused on the optimal frequency sub-band by the focusing matrix. The focusing data of the \(l\)th frequency sub-band are [14]:  
\[\widetilde{X}(f_{l}) =\hat{T}_{l}(\varphi,\theta)X(f_{l}) \tag{16}\] \[=\hat{T}_{l}(\varphi,\theta)A_{l}(\varphi_{0},\theta_{0})s(f_{l}),\] \[+\hat{T}_{l}(\varphi,\theta)n(f_{l})\]  
where \(\widetilde{X}(f_{l})\) for \(l=1,\cdots,L\), and for each \(l\), \(\widetilde{X}(f_{l})\) is the narrowband signal [15]. The narrowband signals can be beamformed by second-order cone programming. The robust low-sidelobe narrowband beamforming ensures that the response of the beam to the direction of observation is 1, and the sidelobe level is minimized. The weighted vector of beamforming should be constrained for beamforming robustness. \[ \begin{array}{ll}\min\limits_{W}~{}~{}\max\limits_{|\theta-\theta_{0}|\geq \Delta}|p(\theta)|,\\ subject~{}~{}\text{to}~{}~{}~{}~{}~{}p(\theta_{0})=1,\\ \|W\|\leq\sigma\end{array} , \tag{17}\]  
where \(\theta_{0}\) is the observation direction, \(\Delta\) is the half width of the main lobe, \(\sigma\) is the constraint limit of the weight vector norm, and \(y_{1}\) is the nonnegative real variable. Let \(\theta_{s}\) be directions of the sidelobe for \(s=1,2,\cdots,S\), Equation (17), which can be written as:  
\[ \begin{array}{ll}\min\limits_{y_{1},W}~{}~{}y_{1},\\ subject~{}~{}~{}\text{to}~{}~{}~{}~{}~{}~{}V^{H}(\theta_{0})W=1,\\ \|V^{H}(\theta_{s})W\|\leq y_{1}\\ \left(|\theta_{s}-\theta_{0}|\geq\Delta,~{}~{}~{}s=1,2,\cdots,S\right)\\ \|W\|\leq\sigma\end{array} . \tag{18}\]Given the value of \(\Delta\), the vector \(y\) can be calculated through the above formula, and the weighted vector \(W\) is obtained. Each of the sub-band weighted vectors is satisfied:  
\[W_{l}=W_{0} \tag{19}\]  
The array output beam in frequencies \(f_{0}\) and \(f_{l}\) is:  
\[p_{0}(\varphi,\theta)=\Big{|}W_{0}^{H}A_{0}(\varphi,\theta)\Big{|}, \tag{20}\]  
\[p_{l}(\varphi,\theta)=\Big{|}W_{0}^{H}\hat{T}_{l}(\varphi,\theta)A_{l}(\varphi,\theta)\Big{|}. \tag{21}\]  
To make \(p_{0}(\varphi,\theta)\) equal to \(p_{l}(\varphi,\theta)(l=1,\cdots,L)\), then:  
\[\Big{|}W_{0}^{H}A_{0}(\varphi,\theta)\Big{|}=\Big{|}W_{0}^{H}\hat{T}_{l}( \varphi,\theta)A_{l}(\varphi,\theta)\Big{|}. \tag{22}\]  
Therefore, the beam width formed by each narrow band beam is constant.