## 4 Array Processing  
Beamforming can be applied with weight vectors if the narrowband constraint expressed in Equation (6) is verified, \(B\) being the signal bandwidth, \(da_{max}\) the maximum physical distance between two elements in the array and \(TBWP\) the time-bandwidth product [15]. \[TBWP=\frac{B\cdot da_{max}}{c}<1 \tag{6}\]  
Snapshots for array signal processing can be generated in the time domain (outputs of the analog-to-digital converters associated with each sensor) or in the frequency domain (outputs of the CAF processing stage calculated for each sensor). In PRSs, the CAF implements a time-frequency mapping with a time (bistatic delay) resolution that depends on the signal bandwidth and a frequency (Doppler shift) resolution that depends on the integration time. If beamforming is applied in this transformed domain, we could take advantage of the integration gain and the range-Doppler mapping of the targets and interferences present in the input signal [6]. Interferences are expected to concentrate around the zero Doppler line of the CAF, while targets will concentrate in the cells associated with their position and speed (range, Doppler), respectively, giving rise to sparse matrices from the point of view of targets. Therefore, if snapshots are defined in the CAF domain, a specific beam space could be generated for each range-Doppler (time-frequency) cell. The CAF will separate targets according to range and Doppler, and in each range-Doppler cell, each beam space will be only responsible for the azimuth discrimination. Figure 13 shows the processing chain previous to the array signal processing stage. If the snapshots were defined in the time domain, the resulting vector would be: \(\mathbf{s}_{s}[n]=[s_{1}[n],s_{2}[n],...,s_{N}[n]]^{T}\). In this paper, we rely on the definition of the snapshots in the frequency domain, and an independent CAF must be calculated from each \(s_{i}[n]\), \(i=1,...,N\). The snapshots are obtained for each \((m,p)\) in the CAF domain: \(\mathbf{s}_{s}[m,p]=[s_{CAF1}[m,p],s_{CAF2}[m,p],...,s_{CAFN}[m,p]]^{T}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pointing Angle** & **Array** & **BW** & **SLL (dB)** & **GLL (dB)** & **D (dBi)** \\ \hline \multirow{4}{*}{\(-30^{\circ}\)} & ULA (\(d=210\) mm) & \(9^{\circ}\) & 10.98 & x & 15.93 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(7^{\circ}\) & 11.25 & 11.68 & 16.60 \\ \cline{2-6}  & No ULA & \(9^{\circ}\) & 15.06 & x & 16.14 \\ \hline \multirow{4}{*}{\(-15^{\circ}\)} & ULA (\(d=210\) mm) & \(9^{\circ}\) & 12.25 & x & 16.07 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(7^{\circ}\) & 12.32 & x & 17.09 \\ \cline{2-6}  & No ULA & \(8^{\circ}\) & 16.25 & x & 16.50 \\ \hline \multirow{4}{*}{\(0^{\circ}\)} & ULA (\(d=210\) mm) & \(8^{\circ}\) & 13.62 & x & 16.15 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(6^{\circ}\) & 13.41 & x & 17.18 \\ \cline{2-6}  & No ULA & \(8^{\circ}\) & 17.44 & x & 16.61 \\ \hline \multirow{4}{*}{\(15^{\circ}\)} & ULA (\(d=210\) mm) & \(9^{\circ}\) & 12.33 & x & 16.09 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(7^{\circ}\) & 12.38 & x & 17.11 \\ \cline{2-6}  & No ULA & \(8^{\circ}\) & 16.32 & x & 16.52 \\ \hline \multirow{4}{*}{\(30^{\circ}\)} & ULA (\(d=210\) mm) & \(9^{\circ}\) & 11.09 & x & 15.92 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(7^{\circ}\) & 11.35 & 12.52 & 16.58 \\ \cline{1-1} \cline{2-6}  & No ULA & \(9^{\circ}\) & 15.16 & x & 16.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main characteristics of azimuth plane radiation patterns using conventional weighting of the ULAs with inter-element distances of 210 mm and 270 mm and an NULA with \(N=11\) elements. The considered steering angles are \(\phi=[-30^{\circ},-15^{\circ},0^{\circ},15^{\circ},30^{\circ}]\). The processing scheme requires the division of the acquisition time into Coherent Processing Intervals (CPIs), of duration \(T_{int}\). A set of \(N\) CAFs is generated for each CPI (Figure 13).