# Low Discrepancy Sparse Phased Array Antennas
## 3 Sparse Planar Phased Array Antennas
### Element Distributions on the Aperture  
Examples of element distributions for uniform, random, and multiple LDS methods appear in Figure 3.  
The LDS element placements in Figure 3d-o do not correspond to a periodic grid. Neither are they random. In comparison of the techniques, higher order Hammersley methods, Figure 3f,g with prime bases of 5 and 7 have shades of a periodic placement that potentially raise the peak sidelobe levels. This will be discussed further in the next section. Other LDS methods however appear more random and are suitable candidates for sparse arrays. Here we provide some metrics on the element distributions.  
As discussed earlier, from a practical fabrication perspective, we need elements distributed on the aperture in a manner that the physical antenna elements do not touch. Bar plots in Figure 4 show the number of elements that fall within a minimum range of element spacing for the different distributions. As expected, for the uniform case, Figure 4a, all elements have the same minimum separation between them. The random distribution, Figure 4b, on the other hand almost produces a Gaussian-like distribution of elements but has several elements that are placed too close to each other. Random distribution with jitter, Figure 4c, has a similar Gaussian-like distribution, but provides a slightly larger minimum element spacing. The distribution however looks too regular. Hammersley distributions, Figure 4d-g, place many of the elements at a certain minimum spacing, which avoids the issue with minimum element spacing, but the distributions appear more periodic. This is primarily due to the hierarchical problem with this sampling method. Halton samplings, Figure 4h-m, avoid both issues, i.e., too close placement of elements and a periodic distribution, although it appears that the performance improves as the base prime numbers are picked further away from each other, e.g., 2 and 7. Sobol and Poisson samplings also show a Gaussian distribution of elements and avoid small element spacings. Notably, the Poisson distribution here was set to half the element spacing of the uniform array, i.e., \(4\lambda/6\), and it can be seen that the minimum element spacing is exactly \(4\lambda/6\) as designed. Note that Halton, Sobol, and Poisson distributions provide a non-uniform and well distributed placement of the elements, while avoiding small element distances.  
Figure 3: Position of elements on the aperture with different distributions: (**a**) uniform, (**b**) random, (**c**) random with jitter, (**d**) [PERSON] (base 2), (**e**) [PERSON] (base 3), (**f**) [PERSON] (base 5), (**g**) [PERSON] (base 7), (**h**) [PERSON] (bases 2, 3), (**i**) [PERSON] (bases 2, 5), (**j**) [PERSON] (bases 2, 7), (**k**) [PERSON] (bases 3, 5), (**l**) [PERSON] (bases 3, 7), (**m**) [PERSON] (bases 5, 7), (**n**) [PERSON], (**o**) Poisson disk.  
Figure 4: Bar plot of minimum element spacings with different distributions: (**a**) uniform, (**b**) random, (**c**) random with jitter, (**d**) [PERSON] (base 2), (**e**) [PERSON] (base 3), (**f**) [PERSON] (base 5), (**g**) [PERSON] (base 7), (**h**) [PERSON] (bases 2, 5), (**j**) [PERSON] (bases 2, 7), (**k**) [PERSON] (bases 3, 5), (**l**) [PERSON] (bases 3, 7), (**m**) [PERSON] (bases 5, 7), (**n**) [PERSON], (**o**) Poisson disk.