# Low Discrepancy Sparse Phased Array Antennas
## 3 Sparse Planar Phased Array Antennas
### Quantitative Analysis of Sparse Array Performances  
As discussed earlier in this work, for sparse arrays we are interested in removing the grating lobes, while at the same time having sampling that avoids too sparse or too dense element distributions. In order to quantitatively analyze these arrays, here we look at some metrics for element distributions and radiation pattern performances.  
For element distributions, we look at an important statistical parameter, i.e., average minimum element spacing. This average value is obtained by computing the minimum element spacing for each element of the array, and then averaging the sum of those numbers over the total number of elements. These results are given in Table 1 for all 15 cases studied here. All LDS techniques provide a larger value of average minimum element spacing compared to the random technique, with Hammersley technique yielding the largest in comparison. From an element distribution perspective, it can be seen that LDS methods are very effective in providing a physically realizable distribution and outperform the random approach. For a quantitative study of radiation performance, we look at peak sidelobe level (SLL), as well as directivity and aperture efficiency. Peak SLL is the ratio of the pattern of the sidelobe peak (\(F_{\text{SLL}}\)), to the pattern value of the main lobe (\(F_{\text{max}}\)). We note that for a boresight beam \(F_{\text{max}}\) is the value of the far-field radiation pattern at \(F(0,0)\). The directivity is defined as  
\[D=\frac{4\pi F_{\text{max}}}{\sqrt{u^{2}+v^{2}}\leq 1\sqrt{u^{2}+v^{2}}\leq 1} \tag{16}\]  
\(\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}F(u,v)\ du\ dv\)Figure 5: Normalized power patterns of the antenna arrays with different element distributions in the u-v space: (**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], (**e**) Poisson disk.  
Aperture efficiency is defined as directivity divided by maximum aperture directivity, where the maximum aperture directivity is given by  
\[D_{aperture}=\frac{4\pi A}{\lambda^{2}} \tag{17}\]  
Here, \(A\) is the size of the array aperture, which in our study in this section is \(32\lambda\times 32\lambda\), and \(\lambda\) is the wavelength.  
These results are also given in Table 1 for all 15 cases. It can be seen that the uniform case has the poorest performance in terms of SLL as well as directivity and efficiency. Random distribution can notably improve these, but as discussed earlier, the element distribution is undesirable. Random distribution with jitter improves directivity and efficiency but degrades SLL. In comparison between the LDS methods, Hammersley sampling has the poorest SLL performance, which degrades as the prime base number is increased. Halton sampling shows a better performance in comparison. The prime bases of 2 and 7 yield the best performance for this technique. Sobol sampling shows a similar performance to Hammersley with a prime base of 2. Poisson disk sampling shows the best performance of all LDS techniques, with a SLL of \(-12.28\) dB and close to 15% aperture efficiency. While all these LDS techniques outperform the uniform case, the best performances come from Halton and Poisson disk methods that outperform the random technique.