# On the Trade-Off between the Main Parameters of Planar Antenna Arrays
## 1 Introduction  
There is no doubt that antenna arrays, thanks to their capability to provide a dynamical manipulation of the received and radiated field, have a strong potential for most communication applications. Antenna arrays have been studied for half of the last century, and all crucial high-performance applications, (radar [1], satellite communication [2], and astronomical observation [3]), have made use of them. In the last decades, we have also seen the extension of antenna arrays to civil, low-cost applications (Wi-Fi networks, 4G [4] and 5G [5, 6] networks); as a matter of fact, the achievement of the demands of modern communication systems (in terms of the number of terminals served, band, latency) require the aforementioned potential [7]. This trend is not going to stop, particularly now that we are moving towards the use of mm-wave frequencies for communications [8], which require the use of arrays to obtain electronically focusable beams with good directivity. Roughly speaking, the more radiating elements, the more directivity is possible to achieve. Unfortunately, antenna arrays have a non-trivial cost, usually not related to the antenna itself, but to all the amplifiers, phase shifters, beam forming networks that are required to operate the array. All these devices have a weight, occupy a volume, and require maintenance. In some applications, like satellite ones, payload reduction is a mandatory requirement; so, in general, the reduction of the elements' number is fundamental to lower the overall cost [9]. The choice of the number of radiators needs to be done with judiciousness. Furthermore, in many applications, we need to verify specific constraints in terms of beam-width (BW) and side-lobe-level (SLL), but their realization requires a tapering of the source, which is known to reduce the directivity of the array [10]. Moreover, the design of an antenna array capable of scanning the beam in a certain angular range requires particular care in the inter-elements distance otherwise grating lobes could appear in the visible range of the antenna. This could be a very negative effect for most of the application cases. The difficulty of the array synthesis problem, particularly when non-regular lattices are used, is testified by a large number of papers on this topic that have been published in the last 60 years; many deterministic, evolutionary, and hybrid algorithms have been recently proposed [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51]. Even if a large literature exists, the array dimensioning is still a non-trivial topic for antenna engineers; one may think that using the \"right number\" of radiators, half-wavelength equispaced, is the only thing to do, but this simplistic approach is not the best choice [52; 53; 54]. So, an antenna engineer dealing with the synthesis of an array radiating a pencil beam needs to take into account at least five parameters: the number of radiating elements that is possible to use, the required beam-width and side-lobe level, the necessary scanning range of the beam, and the desired directivity. Obviously, those five parameters are not the only ones that could be taken into account in the synthesis, but a more general analysis would be beyond the scope of these notes. In this paper, in particular, the issue of the directivity decrease in sparse arrays will be analyzed, showing that with a novel approach to the synthesis it is possible to achieve sparse layouts that can challenge with equispaced lattices in terms of directivity, with a significant reduction of the number of radiators. To achieve this task, the present manuscript is structured in the following way. In the first part (Section 2), the relationship between the relevant parameters is reviewed and simple ready-to-use formulas for the preliminary dimensioning of equispaced planar arrays are provided.