## 3 Passive Radar Array Antenna Design
### Design of a Non-Uniform Array for Optimizing the SLL Value  
In this subsection, the design of the NULA for the considered five-element array antenna is described. The objective was to minimize the importance of the sidelobe level, so that the absolute value of the difference (\(\ell_{1}\) norm) between the normalized radiation pattern and a specific value of sidelobe level was calculated for \(\phi\) outside the main beam width (\(BW_{3 dB}\)), which defines the system azimuth resolution, and only for the values of \(\phi\) where the normalized radiation pattern is higher than the specific value of sidelobe level. When the sidelobe level is minimized, \(BW_{3 dB}\) increases, and the resolution decreases. In order to control \(BW_{3 dB}\), an additional term was added to the cost function, to get a trade-off between sidelobe level and main beam width. The preliminary cost function to be minimized is presented in Equation (3):  
\[C(\mathbf{d},k,sl_{max})=\sum_{ \begin{subarray}{c}\phi\
otin BW_{3 dB}\\ |\mathbf{E}(\phi,\mathbf{d})|>sl_{max}\end{subarray} }\left|\left|\mathbf{E}( \phi,\mathbf{d})\right|-sl_{max}\right|+k\cdot BW_{3 dB}(\mathbf{d}) \tag{3}\]  
\(\bullet\)\(\mathbf{d}\) and \(k\) are the parameters to be adjusted: \(\mathbf{d}=[d_{1},d_{2},d_{3},\cdots d_{N-1}]\) is the inter-element distances vector (\(\mathbf{d}\in\mathbb{R}^{N-1}\), for an array with N elements), and has been adjusted using a genetic algorithm; \(k\) is the parameter that allows us to control the main beam width, \(BW_{3 dB}\), and has been adjusted experimentally. \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) & \(20^{\circ}\) & 9.43 & x & 12.73 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(17^{\circ}\) & 9.5 & 9.16 & 13.21 \\ \cline{2-6}  & No ULA & \(19^{\circ}\) & 12.01 & 12.18 & 12.90 \\ \hline \multirow{4}{*}{\(-15^{\circ}\)} & ULA (\(d=210\) mm) & \(19^{\circ}\) & 12.12 & x & 12.86 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(15^{\circ}\) & 11.57 & x & 13.78 \\ \cline{2-6}  & No ULA & \(18^{\circ}\) & 13.94 & x & 13.13 \\ \hline \multirow{4}{*}{\(0^{\circ}\)} & ULA (\(d=210\) mm) & \(18^{\circ}\) & 15.06 & x & 12.93 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(14^{\circ}\) & 13.84 & x & 13.87 \\ \cline{2-6}  & No ULA & \(17^{\circ}\) & 16.62 & x & 13.20 \\ \hline \multirow{4}{*}{\(15^{\circ}\)} & ULA (\(d=210\) mm) & \(19^{\circ}\) & 12.29 & x & 12.88 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(15^{\circ}\) & 11.7 & x & 13.80 \\ \cline{2-6}  & No ULA & \(18^{\circ}\) & 14.09 & x & 13.15 \\ \hline \multirow{4}{*}{\(30^{\circ}\)} & ULA (\(d=210\) mm) & \(20^{\circ}\) & 9.68 & x & 12.72 \\ \cline{2-6}  & ULA (\(d=270\) mm) & \(17^{\circ}\) & 9.64 & 9.75 & 13.21 \\ \cline{1-1} \cline{2-6}  & No ULA & \(19^{\circ}\) & 11.66 & 12.67 & 12.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main characteristics of azimuth plane radiation patterns using conventional weighting of the ULAs with inter-element distances of 210 mm and 270 mm, and a Non-ULA (NULA) with \(N=5\) elements. The considered steering angles are \(\phi=[-30^{\circ},-15^{\circ},0^{\circ},15^{\circ},30^{\circ}]\). SLL, Sidelobe Level; BW, Bandwidth; GLL, Grating Lobe Level. * \(|\tilde{\mathbf{E}}(\phi,\mathbf{d})|=\frac{|\mathbf{E}(\phi,\mathbf{d})|}{|\mathbf{ E}(\phi,\mathbf{d})|_{max}}\) is the magnitude of the normalized radiation pattern of the array along the y-axis expressed in natural units, particularized in \(\theta=90^{\circ}\), the azimuth plane, for the considered distances vector \(\mathbf{d}\). Note that for the evaluation of \(C(\mathbf{d},k,sl_{max})\), only azimuth values outside the main beam and with an associated radiation pattern magnitude higher than \(sl_{max}\) are considered in the summation. * \(sl_{max}\) is the specified maximum value of the sidelobe level in natural units.