# Low Discrepancy Sparse Phased Array Antennas
## 1 Introduction
A random distribution of elements in the sparse array lowers the grating lobes to a level of the surrounding sidelobes. Low sidelobes are not an option for sparse arrays. Figure 1 distinguishes between dense and sparse arrays. Given this definition, our literature review of sparse arrays emphasizes papers that present arrays with an average element spacing of at least one wavelength. Sparse arrays (by our definition) in the literature generally have random element spacing [15; 16; 17]. Sparse arrays are widely used in radio telescopes and MIMO systems, as well as other sensing systems. The Long Wavelength Array Station 1 (LWA1) is an aperture synthesis imaging array (20-80 MHz) for radio astronomy [18]. It contains 512 antenna elements inside an ellipse that has a 110 m major axis and a 100 m minor axis. To reduce grating lobes in this sparse array at 80 MHz, the elements are pseudorandomly distributed with a 5 m minimum spacing constraint that results in an average element spacing of about 5.4 m  
Figure 1: Categories of antenna array grids. or 1.44\(\lambda\) at 80 MHz. The LOw Frequency ARray (LOFAR) Low Band Antenna (LBA) in the Square Kilometer Array (SKA) consists of 96 dual polarized crossed dipole active antennas operating from 30 to 80 MHz [19]. An inner array of 46 randomly spaced dual-polarized elements have average element spacings between 0.4\(\lambda\) and 0.8\(\lambda\) within a 30 m radius, while the outer array of 48 randomly spaced dual-polarized elements have average element spacings between 0.8\(\lambda\) and 1.7\(\lambda\) within an annulus having an inner diameter of 30 m and outer diameter of 85 m [20]. Some examples of very sparse radio telescopes include the Very Long Baseline Array (VLBA) that has 10 parabolic reflectors that are 25 m in diameter forming a total collecting area of 19,635 m\({}^{2}\)[21], The Atacama Large Millimeter/submillimeter Array (ALMA) that has 10 parabolic reflectors that are 12 m in diameter forming a total collecting area of 6600 m\({}^{2}\)[22], and the Very Large Array (VLA) that has 27 parabolic reflectors that are 25 m that extend in a \"Y\" shape with each arm over 15 km long [23]. Microwave imaging experiments were performed using an array aperture of 320 \(\times\) 320 mm operating over 17-20 GHz [24]. The first model was a fully populated planar array of 64 \(\times\) 64 elements on a square grid. The second model was a sparse array of randomly placed elements. An x-y positioner moved a single antenna to the designated element positions in the aperture to form the dense and sparse arrays. A compressive sensing algorithm (average sampling rate that is less than the Nyquist rate) outperformed other reconstruction algorithms for sparse arrays having 1024, 400, and 160 antennas which correspond to 25%, 10%, and 4% of the elements in the fully populated array. Sparse antenna arrays with random element spacing significantly improve the sum rate capacity (maximum aggregation of all the users' data rates) of a MIMO base station antenna system [25]. The sparse array aperiodicity spreads the grating lobe (GL) energy over all the lower sidelobes [26]. The difference between the uniform element grid and the randomized element grid is mathematically known as the discrepancy. Using a low discrepancy sequence (LDS) to place elements in an array aperture ensures that elements do not overlap while keeping a uniform sampling of the aperture. An LDS produces a random-like equidistribution of elements using a deterministic generating formula. Equidistributed means that if the aperture is divided into equal subareas, then the number of elements in all subareas is the same. The aperture discrepancy approaches zero as the number of elements approaches infinity. Random element spacing has the highest discrepancy, because large areas of empty space as well as high densities of elements exist within the aperture. In contrast, elements on a regular grid have the lowest possible discrepancy.