## 3 Passive Radar Array Antenna Design
### Design of a Non-Uniform Array for Optimizing the SLL Value
\(SLL_{max}(dB)\) is defined as the difference between the main beam level in decibels and the highest sidelobe level in decibels: \(SLL_{max}=-20\times\log(sl_{max})\), if the radiation pattern is normalized. * \(BW_{3 dB}(\mathbf{d})\) is the main beam width for the considered distances vector \(\mathbf{d}\). As the array must be pointed towards a set of azimuth directions ranging from \(-\phi_{max}\) to \(\phi_{max}\), a vector of \(N_{\phi}\) steering angles was defined, \(\mathbf{\Phi}_{sa}=[\phi_{sa,1},\phi_{sa,2},...,\phi_{st,N_{\phi}}];\phi_{sa,i} \in[-\phi_{max},\phi_{max}]\), \(i=1,...N_{\phi}\), and a radiation pattern was generated for each pointing direction using conventional beamforming weights, \(\tilde{\mathbf{E}}_{\phi_{sa,i}}(\phi,\mathbf{d};\phi_{sa,i})\). The final cost function was defined as the average of the preliminary one, throughout the azimuth sector of pointing angles defined by \(\mathbf{\Phi}_{sa}\), Equation (4):  
\[C_{average}(\mathbf{d},k,sl_{max},\mathbf{\Phi}_{sa})=\frac{1}{N_{\phi}}\sum_{ i=1}^{N_{\phi}}\left(\sum_{ \begin{subarray}{c}\phi\
otin BW_{max}\\ |\mathbf{\Phi}_{sa,i}(\phi,\mathbf{d},\phi_{sa,i})|>sl_{max}\end{subarray} } \left|\left|\mathbf{E}_{\phi_{sa,i}}(\phi,\mathbf{d},\phi_{sa,i})\right|-sl_{ max}\right|+k\cdot BW_{3 dN_{sa,i}}(\mathbf{d},\phi_{sa,i})\right) \tag{4}\]  
Once the cost function was defined, the optimization problem to be solved for an array of \(N\) elements was formulated with Equation (5), where \(d_{min}\) and \(d_{max}\) are the limits of the allowed distances between elements. \[ \begin{gathered}\min_{\mathbf{d},k}\hskip 56.905512 ptC_{average}( \mathbf{d},k;sl_{max},\mathbf{\Phi}_{sa})\\ \text{subject to:}\\ d_{min}\leqslant d_{i}\leqslant d_{max},i=1,\cdots,N-1\end{gathered}  \tag{5}\]  
The optimization problem was solved using a genetic algorithm [28], using MATLAB(r), with the default options:  
* Initial population size: 20 individuals. * Number of best individuals that survive to next generation without any change: two. * The fraction of genes swapped between individuals: 0.8. * Migration direction: forward. * The number of generations between the migration of the fittest individuals to other sub-populations: 20. * Fraction of those individuals scoring the best that will migrate: 0.2. * Number of generations to be simulated: 100. * Stopping criteria based on the maximum number of generations; if after this number of generations there is no improvement, the simulation will end. The maximum SLL value was defined to be \(SLL_{max}=14\) dB. A set of pointing directions \(\mathbf{\Phi}_{sa}=\{-30^{\circ},-15^{\circ},0^{\circ},15^{\circ},30^{\circ}\}\), was defined. Distances should be greater than \(d_{min}=210\) mm (the minimum value defined by the antenna width after removing the radome and the reflector) and lower than \(d_{max}=270\) mm, in order to avoid higher grating lobes in the desired azimuth coverage. The optimization procedure was applied to estimate the optimal inter-element distances of a five-element array antenna, for different \(k\) values. Good results were obtained for \(0.5\leqslant k\leqslant 1\), and \(k=1\) was chosen for simplicity. The final values obtained for the design parameters (inter-element distances) were: \(d_{1}=243\) mm, \(d_{2}=210\) mm, \(d_{3}=210\) mm and \(d_{4}=243\) mm (Figure 9). Although \(d_{1}=d_{4}\), \(d_{2}=d_{3}\), this symmetry was not imposed in the optimization algorithm. Beam patterns generated with the resulted NULA are presented in Figure 10. The HPBW (Half Power Beamwidth), SLL (dB), GLL (dB) and directivity, \(D\) (dB), were estimated for steering angles \(\phi=[-30^{\circ},-15^{\circ},0^{\circ},15^{\circ},30^{\circ}]\) and are presented in Table 2. The worst results of SLL (dB) and GLL (dB) are associated with steering angles of \(30^{\circ}\) and \(-30^{\circ}\). For these angles, the obtained SLL (dB) is lower than \(SLL_{max}(dB)\) due to the condition imposed in the optimization cost function related to maintaining a narrow beamwidth, controlled with \(k=1\).