526 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 3, MARCH 1997 Antenna Array Pattern Synthesis via Convex Optimization Herv´ e Lebret and Stephen Boyd Abstract— We show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems, which can be (numerically) solved with great effi- ciency by recently developed interior-point methods. The syn- thesis problems involve arrays with arbitrary geometry and element directivity, constraints on far- and near-field patterns over narrow or broad frequency bandwidth, and some important robustness constraints. We show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays. I. INTRODUCTION A NTENNA arrays provide an efficient means to detect and process signals arriving from different directions. Compared with a single antenna that is limited in directivity and bandwidth, an array of sensors can have its beampattern modified with an amplitude and phase distribution called the weights of the array. After preprocessing the antenna outputs, signals are weighted and summed to give the antenna array beampattern. The antenna array pattern synthesis problem consists of finding weights that satisfy a set of specifications on the beampattern. The synthesis problem has been studied quite a lot. From the first analytical approaches by Schelkunoff [1] or Dolph [2] to the more general numerical approaches such as mentioned in the recent paper by Bucci et al. [3], it would be impossible to make an exhaustive list. An important comment in [3] is that in many minimization methods, there is no guarantee that we can reach the absolute optimum unless the problem is convex. In this paper, we emphasize the importance of convex optimization in antenna array design. Of course, not all antenna array design problems are convex. Examples of nonconvex problems include those in which the antenna weights have fixed magnitude (i.e., phase-only weights), problems with lower bound constraints (contoured beam antennas), or prob- lems with a limit on the number of nonzero weights. Nevertheless, other important synthesis problems are con- vex and can thus be solved with very efficient algorithms that have been developed recently. Even nonlinear convex optimization problems can be solved with great efficiency using new interior-point methods that generalize Karmarkar’s Manuscript received August 28, 1995; revised July 26, 1996. This work was supported in part by AFOSR under Grant F49620-92-J-0013, NSF under Grants ECS-9222391 and EEC-9420565, and ARPA under Grant F4920-93-1- 0085. The associate editor coordinating the review of this paper and approving it for publication was Dr. A. Lee Swindlehurst. H. Lebret is with ENSTA, DFR/A, Paris, France. S. Boyd is with the Information Systems Laboratory, Electrical Engineering Department, Stanford University, Stanford, CA 94305 USA. Publisher Item Identifier S 1053-587X(97)01868-0. linear programming method (see [4] and [5]). Moreover, by “solve” here we mean a very strong form: Global solutions are found with a computation time that is always small and grows gracefully with problem size. Of course, the computation time is not as small as that required by an “analytical” method, but the number and variety of problems that can be handled is much larger. At the other extreme, we find nonconvex optimization, which is completely general—essentially, all synthesis problems can be posed as general optimization prob- lems. The disadvantage is that such methods cannot guarantee global optimality, small computing time, and graceful growth of computing time with problem size. In our opinion, convex optimization is an excellent tradeoff in efficiency/generality between the (fast but limited) analytical methods and the (slow but comprehensive) general numerical techniques. Convexity of problems arising in engineering design has been exploited in several fields, e.g., control systems [6], mechanical engineering [7], signal and image processing [8], [9], circuit design [10], and optimal experiment design [11]. To our knowledge, however, it has not been used very much in antenna array design. As the main objective of our paper is to illustrate the importance and utility of convex optimization for antenna array pattern synthesis problems, we will limit most of our examples to simple arrays. In Section II, we formulate the gain pattern for two examples, which are then generalized. In Section III, we will briefly describe the basic properties of convex optimization and of the algorithms mentioned above. In Section IV, we introduce some design problems and show how they can be recast or reduced to convex optimization problems. Finally, in Section V, we show some numerical examples. II. THE ANTENNA ARRAY PATTERN FORMULATION A. The Linear Array Pattern Consider a linear array of isotropic antennas at locations A harmonic plane wave with frequency and wavelength is incident from direction and propagates across the array (which, we assume for simplicity does not change the field). The signal outputs are converted to baseband (complex numbers), weighted by the weights , and summed to give the well-known linear array beampattern (1) 1053–587X/97$10.00  1997 IEEE