IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 6, 2007 313 On the Orthogonal Nonuniform Synthesis From a Set of Uniform Linear Arrays Sotirios K. Goudos, Member, IEEE, George S. Miaris, Katherine Siakavara, Member, IEEE, and John N. Sahalos, Fellow, IEEE Abstract—Synthesis of uniform linear arrays by using the or- thogonal method is presented. Composing functions similar to that of Woodward–Lawson technique are used. In Woodward–Lawson technique, sampling of the desired pattern is made. In this letter, in- stead of sampling, the orthogonal method is applied. The number of composing functions can be the same as or different than the elements of the arrays. Also, the progressive phase of each func- tion can be derived in several ways. Numerical examples for the synthesis of different array patterns are presented, and the results show the usefulness of the method. Index Terms—Analog beamforming, array synthesis, linear array design, orthogonal method. I. INTRODUCTION AND FORMULATION S YNTHESIS of linear antenna arrays has been extensively studied in the last five decades; the existence of a long se- ries of papers on this subject is enough to emphasize the im- portance of the area. Most of the procedures allow the synthesis of narrow-beam or low sidelobe patterns or the maximization of an index (gain, signal-to-noise ratio) subject to one or sev- eral constraints. Excellent textbooks in the international litera- ture [1]–[4] present several treatments on the synthesis problem. Among them we note the Fourier transform [2], the Schelkunoff procedure [5], the Dolph–Chebyshev and the Riblet synthesis [6], [7], the Woodward–Lawson method [8]–[10], and the Or- chard et al. synthesis [11]. The orthogonal method (OM) was introduced by Unz [12] and has been extensively used by Sa- halos [13] in many antenna synthesis problems. Our effort in this letter is the application of the OM for nonuniform synthesis of a set of uniform linear arrays. In the examples that follow, it is shown that our method can solve synthesis problems with less elements than those required by the classical methods. Also several patterns like shaped beam patterns with different desired sidelobes can be synthesized. The procedure starts from the nonorthogonal progressively phased composing functions [9], which are orthonormalized and overlapped. The orthonormalized functions are weighted to form the desired pattern. Our method could be called or- thosynthesis from the composition of the words “orthogonal synthesis.” Manuscript received October 31, 2006; revised May 2, 2007. The authors are with the Radiocommunications Laboratory, Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece (e-mail: sgoudo@physics.auth.gr; gmiar@physics.auth.gr; skv@auth.gr; sahalos@auth.gr). Digital Object Identifier 10.1109/LAWP.2007.899915 Fig. 1. Geometry of an -element uniformly spaced linear array. Let a linear array (Fig. 1) be composed of equidistant point sources. The array factor is given by (1) where is the excitation coefficient of the th element, , and is the equidistance of the array elements. In this letter, we suppose that the array factor comes from a set of different patterns of uniformly illuminated element arrays. In this case, can be expressed in the following form: (2) where (3) The functional base of (3) is not orthogonal and the nonzero inner product can be found by transforming the independent variable to . We have (4a) 1536-1225/$25.00 © 2007 IEEE