# Low Discrepancy Sparse Phased Array Antennas
## 1 Introduction  
Sensing applications, such as radio telescopes, satellite communications, sonars, and defense radars, require large antenna arrays. Physically large arrays offer high resolution as well as high directivity (as long as the element spacing, or sampling remains small). Since the array cost is proportional to the number of elements in the array, designers try to minimize the number of elements in the aperture. However, if the array has a uniform grid of elements that undersamples the aperture (large element spacing), then grating lobes (extra main beams) result due to aliasing and are predictable from theory [1]. Most arrays are designed with an element spacing of \(\lambda/(1+\sin\theta_{\max})\) or less, where \(\lambda\) is the wavelength and \(\theta_{\max}\) is the maximum scan angle from broadside. Thinned and aperiodic arrays have fewer elements than dense periodic arrays [2; 3; 4]. For a dense array, thinning (removing elements from a regular grid [5]) and aperiodic spacing (spacing between elements are not constant [6]) mimic low sidelobe amplitude distributions through an amplitude density across the aperture. These arrays have far field patterns with low sidelobes near the main beam and increased sidelobe levels farther from the main beam. Sparse arrays fill an antenna aperture with elements that are widely separated from each other in order to reduce the cost but maintain a narrow beamwidth. The definition of asparse array in both the antenna and signal processing literature is vague. For instance, the IEEE standard defines a sparse antenna array as [7]: \"An array antenna that contains substantially fewer driven radiating elements than a conventional uniformly spaced array with the same beamwidth having identical elements. Interelement spacings in the sparse array can be chosen such that no large grating lobes are formed and sidelobes are reduced.\" On the other hand, mathematics has a different definition: a sparse matrix has most elements equal to zero [8]. Sparsity of a matrix equals the number of zero valued elements divided by the total number of elements. The words \"fewer\" and \"most\" do not specify a sharp dividing line between dense and sparsity. It is important to note here that while these two definitions of sparsity are different, both have been used in designing sparse arrays. For the latter, compressive sensing (CS) approaches aim at solving a system of linear equations, forcing the solution to be maximally sparse, namely, to have the minimum number of nonzero coefficients, with respect to an expansion basis [9]. Accordingly, CS-based methods have been applied to the synthesis of sparse arrays by properly reformulating the design problem as a pattern matching one [10]. The problem unknowns are the set of complex (amplitude and phase) excitation coefficients of the \"candidate\" array elements, the positions of these latter obtained through a dense sampling of the array aperture. The CS solution is the sparse complex-valued vector of the excitations, and the positions of the array elements are obtained as a byproduct and correspond to the candidate locations having non-null coefficient [10]. CS has been applied to the design of both linear [11; 12] and planar [13; 14] sparse arrays, considering symmetric [11; 13] as well as asymmetric [12; 14] pattern shapes. The obtained results have shown achieving up to 40% elements reduction with respect to regular/uniform array arrangements. It is important to note that CS approaches do not change the definition of sparse arrays; but deal with minimizing the number of nonzero coefficients in the system of equations in designing the array. The authors of this paper propose the following definition of sparsity, based on the IEEE definition [7], but we add that: a sparse array has an average element spacing greater than \(\lambda\). If the array has a uniform square grid of isotropic elements, then grating lobes exist when the main beam points at broadside. Periodic sparse arrays have grating lobes with the same gain as the main beam.