# Low Discrepancy Sparse Phased Array Antennas
## 5 Beam-Scanning Performance of Sparse Phased Array Antennas  
In this section, we study the beam-scanning performance of sparse phased array antennas. The studies conducted in Sections 3 and 4 showed that the performance of LDS distributions are not impacted by the type/shape of the array aperture, so for brevity we only consider the square aperture configuration in Section 3. By adding a progressive phase shift to the elements on the aperture, the sparse arrays can provide 2D beam scanning. Mathematically, the phase shift needed to scan the beam is given by  
\[\varphi(x_{n},y_{n})=k(x_{n}\sin\theta_{\mathrm{s}}\cos\phi_{\mathrm{s}}+y_{n }\sin\theta_{\mathrm{s}}\sin\phi_{\mathrm{s}}) \tag{18}\]  
where, \(k=2\pi/\lambda\), \(\lambda\) is the wavelength, \((x_{n},y_{n})\) is the location of element \(n\), \(\theta_{\mathrm{s}}\) and \(\phi_{\mathrm{s}}\) are the elevation and azimuth angle of the desired scanned beam direction, respectively. Here we investigate 1-D and 2-D scanning performance of the arrays by studying scanning in the elevation plane at \(\phi=0^{\circ}\) and \(\phi=45^{\circ}\) directions, but similar results are observed for other scan directions. We note that scanning is essentially shifting the obser  
Figure 8: Normalized power patterns of sparse antenna arrays with rectangular, circular, and elliptical apertures and different element distributions in the u-v space: (**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). vation window. With sparse arrays this means that if grating lobes are created, scanning may result in more grating lobes appearing in the visible space. Here we compare the scan performance of two LDS sparse arrays that showed the best performance, namely Halton (with bases of 2 and 7), and Poisson distributions, along with uniform and random distributions. Scanned patterns of each of these arrays are given in Figure 9 (\(\phi\) = 0\({}^{\circ}\) direction) and 10 (\(\phi\) = 45\({}^{\circ}\) direction), for 20\({}^{\circ}\), 40\({}^{\circ}\), and 60\({}^{\circ}\) elevation scans. We note that the array elements are isotropic point sources. For the uniform array, four grating lobes are observed when the array beam is at boresight, Figure 5a. When the array is scanned in 1-D, Figure 9a-c, the number of grating lobes increases to five at 20\({}^{\circ}\) and to seven when pointing at 40\({}^{\circ}\) and 60\({}^{\circ}\). When the array is scanned in 2-D, Figure 10a-c, the number of grating lobes first reduces to three at 20\({}^{\circ}\) and 40\({}^{\circ}\), and then increases to five when pointing at 60\({}^{\circ}\). This change in the number of grating lobes degrades directivity, and in general the uniform array is not suitable for beam-scanning. The other three arrays however, i.e., random, Halton, and Poisson, show a good beam-scanning performance. These arrays do not have grating lobes, and scanning does not change that. In particular, we note that Poisson distribution, Figures 9 and 10j-l, shows a performance better than random distribution, i.e., Figures 9 and 10d-f, with the added advantage of having a physically realizable element distribution. It is important to note that while Halton sampling does have a higher SLL in the visible space, it does not have grating lobe issues, and again provides a notably better element distribution compared to the random case. Nonetheless, these studies also confirm that the Poisson disk sampling provides the best performance.