JOURNAL OF IEEE TRANS. ON ANTENNAS AND PROPAGATION 1 Sparse Antenna Array Optimization with the Cross-Entropy Method Pierre Minvielle, Emilia Tantar, Alexandru-Adrian Tantar and Philippe B´ erisset Abstract—The interest in sparse antenna arrays is growing, mainly due to cost concerns, array size limitations, etc. Formally, it can be shown that their design can be expressed as a constrained multi-dimensional nonlinear optimization problem. Generally, through lack of convex property, such a multi-extrema problem is very tricky to solve by usual deterministic optimiza- tion methods. In this article, a recent stochastic approach, called Cross-Entropy method, is applied to the continuous constrained design problem. The method is able to construct a random sequence of solutions which converges probabilistically to the optimal or the near-optimal solution. Roughly speaking, it per- forms adaptive changes to probability density functions according to the Kullback-Leibler cross-entropy. The approach efficiency is illustrated in the design of a sparse antenna array with various requirements. Index Terms—Antenna design, phased array, stochastic opti- mization, Cross-Entropy. I. I NTRODUCTION Electronically controlled antenna radiation patterns dates back to the forties when it became possible to substitute mechanical steering of the antenna main beam for electronics. Afterwards, arrays of phased antennas have been developed in order to better the antenna capabilities: reduction of the mechanical movement, fast scan of the field of view, control and possible reconfiguration of the radiation pattern (e.g., beamwidth, sidelobes), etc. Fig. 1 shows a basic phased array, used for Radar Cross Section (RCS) measurements. Yet, the realization of fully sampled arrays is awkward [1]. It is mainly due to conflicting dimensional requirements, to tech- nical limitations and finally to the cost concern. For example, considering a high resolution purpose, a fully populated phased antenna array require at a time a large spatial dimension (to provide a narrow beamwidth), a low inter-element spacing close to half of the minimum wave length (to prevent the appearance of grating lobes due to spatial under-sampling) and miniaturized antennas (to avoid mutual coupling between adjoining antennas) [2]. These problems account for the growing interest in antenna arrays with fewer elements. From this viewpoint, different kinds of arrays have been proposed [3]–[5]. The two main ones are the thinned arrays [2], [5], [6] (regular arrays where a certain number of elements have been withdrawn) and the sparse arrays, also called non-uniform arrays. For sparse antenna arrays, the number of antennas is reduced and the P. Minvielle and Ph. B´ erisset are with CEA, DAM, CESTA, F-33114 Le Barp, FRANCE e-mail: {pierre.minvielle,philippe.berisset}@cea.fr. E. Tantar and A.-A. Tantar are with INRIA Bordeaux Sud-Ouest, 351 cours de la Lib´ eration 33405 Talence cedex, FRANCE e-mail: {emilia.tantar,alexandru-adrian.tantar}@inria.fr. Fig. 1. A phased antenna array illuminating a metallic biconical target. inter-element spacing is increased. Besides, they can lead to improvements in weight, heat dissipation, power consumption, etc. A known pitfall of sparse arrays is high level sidelobes in the radiation pattern. Designing a non-uniformly spaced array consists in controlling the number of antennas, their positions and their associated weights in the aim to reduce the sidelobes and fulfill an expected radiation pattern requirement. Unfortunately, there are no closed form solutions [7]. It can be shown that the sparse antenna array design, according to some application requirements, corresponds to the resolution of a constrained multi-dimensional nonlinear optimization problem [1]. Usually, through lack of convex property, such a multi- extremal problem is awkward to solve. Deterministic gradient- based methods, as Steepest Descent or Quasi-Newton methods, do not cope well with many local optima. As they determine their search direction from locally evaluated features, they might get trapped in a locally convex zone, around a local extremum. In such a situation, global stochastic optimization ap- proaches, are known to be more efficient [4]. Their search strategy is based not only on one point but on a large number of random points; it tends to provide a global point of view of the objective function. Well-known random search meth- ods are simulated annealing, based on Markov Chain Monte Carlo (MCMC), evolutionary algorithms (EAs), Taboo meta- heuristic search, etc. Some of them have theoretic foundations, associated to asymptotic convergence results; others are just efficient heuristics. In any case, the global random search must be guided in order to isolate progressively the global extremum. These search methods have been widely applied to Electromagnetic (EM) optimization problems. Thus, [4] and [8] give a large review of EAs applications, from the design