# Low Discrepancy Sparse Phased Array Antennas
## 4 Aperture Shape Effects on the Performance of Sparse Phased Array Antennas  
To see the impact of aperture shape on the performance of these LDS arrays, in this section we study three different types of arrays with rectangular, circular, and elliptical apertures. All apertures are designed with the same surface area of \(1024\lambda^{2}\) as the square aperture studied in Section 3. The rectangular aperture has an aspect ratio of 9/4, i.e., the ratio of its longer side to its shorter side, corresponding to longer and shorter side lengths of \(48\lambda\) and \(64\lambda/3\), respectively. The circular aperture has a radius of \(18\lambda\). To maintain the same aspect ratio as the rectangle and the same aperture size as the other arrays, the elliptical array major and minor axis are \(27\lambda\) and \(12\lambda\), respectively. Here we compare the performance of two LDS sparse arrays that showed the best performance, namely Halton (with bases of 2 and 7) and Poisson distributions, with uniform distribution. The element distributions for these arrays are given in Figure 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Average Minimum** & **Peak SLL** & **Directivity (dB)** & **Aperture Efficiency** \\  & **Element Spacing (\(\lambda\))** & **(dB)** & **(\%)** & **(\%)** \\ \hline Uniform & 1.3333 & 0 & 31.5478 & 11.168 \\ Random & 0.6667 & \(-10.90\) & 31.9619 & 12.209 \\ Random with Jitter & 0.9325 & \(-9.36\) & 32.5775 & 14.068 \\ Hammersley (base 2) & 1.0037 & \(-7.0\) & 32.7075 & 14.166 \\ Hammersley (base 3) & 1.1688 & \(-2.69\) & 32.6226 & 14.215 \\ Hammersley (base 5) & 1.2624 & \(-0.55\) & 31.6892 & 11.466 \\ Hammersley (base 7) & 0.7538 & \(-0.25\) & 33.8949 & 19.054 \\ Halton (bases 2, 3) & 0.8436 & \(-10.10\) & 32.4609 & 13.696 \\ Halton (bases 2, 5) & 0.8115 & \(-4.33\) & 32.3534 & 13.361 \\ Halton (bases 2, 7) & 0.9172 & \(-12.35\) & 32.8188 & 14.872 \\ Halton (bases 3, 5) & 0.8430 & \(-8.10\) & 32.8611 & 15.018 \\ Halton (bases 3, 7) & 0.7663 & \(-5.90\) & 32.2561 & 13.065 \\ Halton (bases 5, 7) & 0.8633 & \(-5.60\) & 32.9511 & 15.332 \\ Sobol & 0.9307 & \(-6.58\) & 32.7127 & 14.513 \\ Poisson Disk & 0.9031 & \(-12.28\) & 32.8212 & 14.880 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance metrics for sparse arrays with different element distributions. Bar plots in Figure 7 show the number of elements that fall within a minimum range of element spacing for the different distributions. Similar to Figure 4, for the uniform case, all elements have the same minimum separation between them. Halton sampling avoids too close placement of elements and distributes them in the range around \(\lambda/2\) to \(2\lambda\), while Poisson sampling provides a Gaussian distribution of elements with and minimum element spacing of \(4\lambda/6\). We note that these observations are similar to the square aperture studies given in Section 3. The radiation patterns of these arrays in the u-v space are given in Figure 8, where it can be seen that similar to the square aperture array, the uniform case has the worst performance and four grating lobes with sidelobe level of 0 dB appear in the visible space. Both LDS techniques remove the grating lobes; however, it can be seen that the Poisson technique outperforms Hammersley. Nonetheless, these studies show that the performance of LDS sampling techniques are not impacted by the shape of the array aperture, and in general these sampling approaches can be used for arbitrary shaped arrays. Figure 6: Position of elements on rectangular, circular, and elliptical apertures with different distributions: (**a**) uniform (rectangular), (**b**) Halton (rectangular), (**c**) Poisson disk (rectangular), (**d**) uniform (circular), (**e**) Halton (circular), (**f**) Poisson disk (circular), (**g**) uniform (elliptical), (**h**) Halton (elliptical), (**i**) Poisson disk (elliptical). Table 2 summarizes the performance factors for these three types of arrays.