## 4 Array Processing
### Data Independent Beamforming  
Data independent beamforming is usually based on the maximization of array directivity or the minimization of the squared error between the beamformer output and the desired radiation pattern at defined directions for sidelobes control [15; 29; 30]. In this subsection, both approaches are summarized to provide the theoretical basis of the two-stage array signal processing scheme for improving detection and tracking in PRSs, which is presented in Section 5. * Maximization of array directivity: Taking into consideration the snapshots defined in the CAF domain, array directivity is calculated as \(D(p,\phi_{sa,i})=\left(\mathbf{w}(p,\phi_{sa,i})^{T}\cdot\mathbf{A}(p)\cdot \mathbf{w}(p,\phi_{sa,i})\right)^{-1}\), where matrix \(\mathbf{A}(p)\) is calculated in Equation (9). The element at the m-th row, and n-th column can be expressed with Equation (10). \[\mathbf{A}(p)=\frac{1}{2}\int_{-\pi/2}^{\pi/2}-\mathbf{a}(p,\phi_{sa,i})\cdot \mathbf{a}(p,\phi_{sa,i})^{T}cos(\phi)d\phi\] (9)  
Figure 13: Signals throughout the processing chain previous to the array signal processing stage. \[\mathbf{A}(p)[m,n]=\text{sinc}\left(\frac{2\pi}{c}(f_{s}\frac{p}{N_{CAF}}+f_{p})|d_{ m}-d_{n}|\right)\] (10) The vector \(\mathbf{w}(p,\phi_{sa,i})\), which maximizes \(D(p,\phi_{sa,i})\), is expressed in Equation (11). In the special case of ULA, \(\mathbf{A}(p)\) tends to the identity matrix, and the maximum directivity weight vector is close to the uniform weight vector steered to the desired direction. For the special case of a ULA with inter-element spacing equal to \(\lambda/2\), \(\mathbf{A}(p)\) is equal to the identity matrix. \[\mathbf{w}(p,\phi_{sa,i})=\mathbf{A}(p)^{-1}\cdot\mathbf{a}(p,\phi_{sa,i}) \cdot\left(\mathbf{a}(p,\phi_{sa,i})^{H}\cdot\mathbf{A}(p)^{-1}\cdot\mathbf{a }(p,\phi_{sa,i})\right) \tag{11}\]
* Sidelobes reduction: If lower sidelobes are desired, sacrificing some directivity, a methodology based on [31] can be developed. The \(\phi\) space is divided into \(R_{\delta}\) sectors, \(\delta_{i}=\left(\frac{2\pi}{R_{\delta}}(i-1),\frac{2\pi}{R_{\delta}}i\right)\), \(i=1,...,R_{\delta}\), and a desired radiation pattern, \(|\bar{\mathbf{E}}_{t,i}(\phi)|\), \(\phi\in\delta_{i}\), is defined for each sector. Let \(\mathbf{w}_{t,i}^{opt}\), \(i=1,...,R_{\delta}\), be the set of optimum weight vectors to generate the desired radiation pattern, \(|\bar{\mathbf{E}}_{t,i}(\phi)|=(\mathbf{w}_{t,i}^{opt})^{H}\mathbf{a}(\phi)\), \(\phi\in\delta_{i}\), \(i=1,...,R_{\delta}\). The objective is to find a single weight vector, \(\mathbf{w}_{SLL}\), to approximate the radiation pattern obtained as the concatenation of the set of defined target radiation patterns \(|\bar{\mathbf{E}}_{t,i}(\phi)|\), \(\phi\in\delta_{i}\), \(i=1,...,R_{\delta}\). This is equivalent to maximize the directivity controlling the SLL level in the whole \(\phi\) interval, \((0,2\pi]\). If the radiation pattern obtained using \(\mathbf{w}_{SLL}\) is \(|\bar{\mathbf{E}}_{SLL}(\phi)|=\mathbf{w}_{SLL}^{H}\mathbf{a}(\phi)\), \(\phi\in(0,2\pi]\), the squared error to be minimized is defined in Equation (12), being \(\mathbf{Q}_{i}=-\int_{\delta_{i}}\mathbf{a}(\phi)\mathbf{a}(\phi)^{H}\cos( \phi)d\phi\). \[\varepsilon^{2}=\sum_{i=1}^{R_{\delta}}\int_{\delta_{i}}-||\bar{\mathbf{E}}_{ SLL}(\phi)|-|\bar{\mathbf{E}}_{t,i}(\phi)||^{2}\cos(\phi)d\phi=\sum_{i=1}^{R_{ \delta}}(\mathbf{w}_{SLL}-\mathbf{w}_{t,i})^{H}\cdot\mathbf{Q}_{i}\cdot( \mathbf{w}_{SLL}-\mathbf{w}_{t,i})\] (12) If snapshots are defined in the CAF domain, the methodology described so far must be applied for every Doppler shift. Assuming that the target radiation pattern is the same for all Doppler shifts, a weight vector must be determined for each \(p\) value.