## 3 Application to Reflectarray for Compact Antenna Test Range
### Comparison with Other Techniques in the Literature  
Table 3 shows a comparison of the proposed optimization technique for spatially fed arrays with others described in the literature in terms of the type of constraints, capabilities and computational performance. The other techniques are based on the definition of upper and lower templates in the region of interest that are used to trim the near field according to the given specifications [47]. Although it is claimed that figures of merit, such as pointing direction, ripple, secondary lobes, etc., are optimized [33], in reality, they are used to define the templates which are the ones actually used in the algorithm. This is true for the studies cited in Table 3. Only the present study directly optimizes the relevant figures of merit, as described in previous sections. In addition, all techniques implement at least NF magnitude constraints, although this is not the case for phase constraints that are necessary for applications such as PWG for CATR. However, it would be possible to extend the techniques with proper definition of upper and lower phase templates. In the area of NF-field-shaping for spatially fed arrays, the present study is the first in which a multi-frequency optimization with both magnitude and phase constraints is implemented, which enables a wideband array to be obtained when performance would otherwise deteriorate at frequencies different from the designated frequency. A more direct comparison can be performed with the synthesis technique employed in [28], since it also implements both magnitude and phase constraints. To this end, upper and lower templates [47] were defined in three regions, \(\Omega_{1}\), \(\Omega_{2}\) and \(\Omega_{3}\), as shown in Figure 3. \(\Omega_{1}\) is still the region where the plane wave is to be generated and so the templates impose ripples of 1 dB for the magnitude and \(10^{\circ}\) for the phase. \(\Omega_{2}\) defines a transition zone only for the magnitude, while \(\Omega_{3}\) is the non-coverage area. To ease the optimization, a maximum side-lobe level of \(-10\) dB is imposed. In the case of the phase, in regions \(\Omega_{2}\) and \(\Omega_{3}\), no constraints are specified. The same procedure described in Section 3.3 is applied, increasing the number of optimizing variables in six successive stages, but, \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Frequency (GHz)**} & \multicolumn{3}{c}{**Compliance (\%)**} & \multicolumn{3}{c}{**Ripple**} \\ \cline{2-9}  & \multicolumn{2}{c}{**Pol. X**} & \multicolumn{2}{c}{**Pol. Y**} & \multicolumn{2}{c}{**Pol. X**} & \multicolumn{2}{c}{**Pol. Y**} \\ \cline{2-9}  & **Mag.** & **Phase** & **Mag.** & **Phase** & **Mag. (dB)** & **Phase (\({}^{\circ}\))** & **Mag. (dB)** & **Phase (\({}^{\circ}\))** \\ \hline
27 GHz & 100 & 100 & 100 & 100 & 0.98 & 5.81 & 0.77 & 5.52 \\
28 GHz & 100 & 100 & 100 & 100 & 0.86 & 6.13 & 0.54 & 6.51 \\
29 GHz & 100 & 100 & 100 & 100 & 0.75 & 6.87 & 0.98 & 6.39 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance of the reflectarray after the wideband multi-frequency optimization at a plane 0.5 m away from the antenna. Compliance (%) indicates percentage of surface in \(\Omega_{1}\) that complies with specifications. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Reference**} & **Constraint** & **Magnitude** & **Phase** & **Multi-Frequency** & **Memory** & **Computational** \\  & **Type** & **Constraints** & **Constraints** & **Optimization** & **Footprint** & **Efficiency** \\ \hline
[28] & Upper \& lower & Yes & Yes & No & High & Medium \\
[32] & Upper \& lower & Yes & No & No & High & Medium \\
[33] & Upper \& lower & Yes & No & No & Low & High \\
[34] & Upper \& lower & Yes & No & No & High & Medium \\  & Templates & & Yes & Yes & Low & Medium \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the proposed algorithm with other techniques described in the literature in terms of type of constraints, capabilities and computational performance.