IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II:ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 48, NO. 3, MARCH 2001 225 A Semi-Infinite Quadratic Programming Algorithm with Applications to Array Pattern Synthesis Sven Nordebo, Member, IEEE, Zhuquan Zang, Member, IEEE, and Ingvar Claesson Abstract—This paper presents a new extended active set strategy for optimizing antenna arrays by semi-infinite quadratic programming. The optimality criterion is either to maximize the directivity of the antenna or to minimize its sidelobe energy when subjected to a specified peak sidelobe level. Additional linear constraints are used to form the mainlobe. The design approach is applied to a numerical example that deals with the design of a narrow-band circular antenna array for the far field. Index Terms—Antenna arrays, optimization methods. I. INTRODUCTION I N SIGNAL-PROCESSING applications, the design of an- tenna arrays and digital filters is governed by two basic norm criteria: the -norm (least squares) criterion and the -norm (minimax) criterion (see, e.g., [1]–[10]). In this context, it is em- phasized that many optimization techniques developed for finite impulse response (FIR) filter design in the complex domain are easily extended to the case with arbitrary antenna arrays (see, e.g., [8], [9]), whereas some techniques are restricted to FIR fil- ters only (see, e.g., the complex Remez algorithm [10]). Except for the optimization-based algorithms referred to above, there also exist another family of array pattern synthesis approaches based on adaptive array theory (see, e.g., [11]). In this paper, we focus on the optimization approach and emphasize that optimization can be used as a means to obtain the physical limits for an antenna array processor that is implemented adaptively. The and the -norm criteria are commonly related to noise gain and magnitude response specifications, respectively. However, the most flexible design approach in a practical situ- ation is to consider the combination of these methods, i.e., the tradeoff between the least squares and the minimax errors. A new optimal window was defined in [12] providing the op- timum tradeoff between the peak sidelobe level and the side- lobe energy for FIR filters. It was demonstrated that the classical minimax and least squares methods (Dolph–Chebyshev/prolate- spheriodal windows) are both fundamentally inefficient with re- spect to the other design criterion (the minimax window has the highest sidelobe energy, etc.). The optimum window [12] minimizes the sidelobe energy subjected to a specified peak sidelobe level together with some Manuscript received October 1999; revised February 2001. This paper was recommended by Associate Editor Y.-C. Lim. S. Nordebo and I. Claesson are with the Department of Telecommunications and Signal Processing, Blekinge Institute of Technology, Ronneby S-372 25 Sweden. Z. Zang is with the Australian Telecommunications Research Institute (ATRI), Curtin University of Technology, Perth, WA 6845, Australia. Publisher Item Identifier S 1057-7130(01)04200-8. additional linear constraints (which can be used to form the mainlobe, etc.). The solution for linear-phase FIR filters (sym- metric windows) can be found by conventional quadratic pro- gramming methods [13] since the magnitude-response can be represented by a real amplitude function [4]. In this paper, we extend the definition in [12] and consider the design of general antenna arrays with complex response [1]. The performance measure used here is either the directivity of the antenna [14] or its sidelobe energy, both quadratic functions of the array weights. The design criterion is to optimize the per- formance when subjected to a specified peak sidelobe level and a linear mainlobe constraint. By employing the real rotation the- orem [4], the optimum array design is formulated as a semi-in- finite quadratic program. A new extended active set strategy is proposed for the solution of the optimization problem (see [15]). We emphasize that the presented design technique may be useful for complex approximation with any filter having linear structure such as the digital Laguerre networks [16], digital FIR equalizers [6], and narrow-band as well as broadband and three- dimensional beamformers [1], [17], [18]. Thus, the semi-infinite quadratic programming technique may be used as a means for interpolation between the - and -norms when the filter response is complex, the corresponding basis is finite, and the specification is given in terms of amplitude and phase. Fur- thermore, this optimization approach is advantageous due to its flexibility with respect to additional specifications in frequency, time, and space (group delay, envelope constraints, null con- straints, etc.); see [9]. A number of numerical methods exist for the solution of general semi-infinite programming problems (see, e.g., [19]–[27]). However, most methods are computationally involved and employ approximation by discretization in some step of the optimization procedure. Finitization can in principle give an arbitrarily accurate approximation of semi-infinite programming problems. However, the computation time and memory requirements with finitization approaches become extensive as the number of filter parameters increases and the domain grid spacing decreases [8]. The proposed extended active set strategy is the first simple and numerically efficient technique exploiting a result from Caratheodory dimensionality theory to avoid the need of discretization [15]. The advantages of this procedure are as follows. 1) The constraint set can be represented in functional form rather than stored in memory as numerical values. 2) A finite active set having (at most) the same size as the number of filter parameters needs to be considered at any one step in the optimization process. 1057–7130/01$10.00 © 2001 IEEE