1152 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 43, NO. 10, OCTOBER 1995 with the variation of one parameter was not influenced considerably by the values of the other parameters. Therefore, for engineering purposes, such an approach appeared to be satisfactory.) The number of strips was adopted to be odd with the middle strip of constant length (dl + d2) = (0.5 + 2.7) cm. In this manner, it was found that for the 4-GHz antenna, the conditionally optimal interstrip spacing was p = 4 mm, the number of strips N =13, and the taper angle cy = 18.4 degrees. In the process of optimization, a set of six parameterized graphs were obtained for the frequency dependence of the VSWR and efficiency. In these graphs, the number of elements, the interstrip spacing, and the taper were considered as parameters one at time. In addition to yielding the approximate optimal values of the parameters, this set of computed graphs pointed to some important steps in the design of such antennas. We mention here one of these as an example. Fig. 6 shows the computed efficiency as a function of frequency for an antenna with N = 13 and Q = 18.4 degrees with the inter- strip spacing, p, as parameter. In addition to the conclusion that the optimal value of p is between 3 mm and 4 111111, we see that, for p = 5 mm, the efficiency exhibits an abrupt drop at around 3.9 GHz to only about 20% (off-scale in Fig. 6). This strange behavior is due to an undesirable resonance that a good design must avoid. This resonance is best understood if one considers the current distribution along the strips at 3.9 GHz and a close frequency with normal antenna behavior. Fig. 7 shows the computed distributions at 3.9 GHz and 3.95 GHz. It is evident that at 3.9 GHz two neighboring strips form a two-wire line with large, almost equal currents in opposite direction. There is, consequently, only small radiation from these two strips. Since only these two strips are in resonance, the antenna as a whole radiates poorly. On the other hand, at 3.95 GHz only one strip is in resonance which results in efficient radiation of the antenna as a whole. All three antennas presented here were designed with this in mind. Their behavior was verified numerically in small frequency increments, but no signs of such an anomaly were observed. For a properly designed antenna, the current distribution within the operation range must look like the one in Fig. 7(b). At higher frequencies within the bandwidth of the antenna, the resonant peak moves to shorter strips (Le., to the bottom of the figure) and at lower frequencies toward the longer strips (i.e., to the top of the figure). REFERENCES A. G. Demeryd and I. Karlsson, “Broadband microstrip antenna element and array,” IEEE Trans. Antennas Propagat, vol. AP-29, no. 1, pp. 140-144, Jan. 1981. W. C. Chew, “A broadband annular-ring microstrip antenna,” IEEE Trans. Antennas Propagat., vol. AP-30, no. 5, pp. 918-922, Sep. 1982. K. C. Gupta and G. Kumar, “Directly coupled multiple resonator wideband microstrip antennas,” IEEE Trans. Antennas Propagat., vol. AP-33, pp. 853-855, 1985. H. Pues, J. Bogaers, R. Pieck, and A. Van de Capelle, “Wideband quasi- log-periodic microstrip antennas,” IEE Proc., vol. 128, no. 3, June 1981, F. Croq and D. M. Pozar, “Multifrequency operation of microstrip antennas using aperture coupled parallel resonators,” IEEE Trans. Antennas Propagat. vol. 40, no. 11, pp. 1367-1374, 1992. P. S. Hall, C. Wood, and C. Garrett, “Wide bandwidth microstrip antennas for circuit integration,” Electronics Lett., vol. 15, pp. 458-459, 1979. R. R. DeLyser, D. C. Chang, and E. F. Kuester, “Design of a log periodic strip grating microstrip antenna,” Int. J. Microwave Millimeter- Wave Computer-Aided Eng., vol. 3, no. 2, pp. 143-150, 1993. pp. 159-163. [SI Z. B. PopoviC, E. Kuester, and B. D. Popovif, “Broadband quasi- microstrip anisotropic antennas,” in IEEE APS In?. Symp. Dig., pp. 2073-2076, Chicago, July 1992. [9] B. D. PopoviC, CAD of Wire Antennas and Related Radiating Structures. New York: Wiley, 1991. [lo] B. D. PopoviC and A. NesiC, “Generalization of the concept of equiv- alent radius of thin cylindrical antennas,” Proc. IEE, Part H, vol. 131, pp. 153-158, 1984. Unit Circle Representation of Aperiodic Arrays Randy L. Haupt Abstract-Traditiona~y, unit circle analysidsynthesis techniques were only applied to amplitude tapered arrays. This paper extends the method to uniformly weighted thinned and aperiodic arrays. I. INTRODUCTION. An antenna array is a finite impulse-response (FIR) digital filter that samples an incident plane wave at discrete points in space. A myriad of FIR filter designs are available in which the signals at the taps (antenna elements) are weighted and combined to give a desired response. These designs have been applied to uniformly spaced antenna arrays to optimize sidelobe levels and beamwidths [l], [2]. The antenna array factor is t-transformed into a polynomial. Then, the zeros of the polynomial are represented on the unit circle in the complex plane [3]. A zero on the unit circle corresponds to a null in the antenna pattern. The placement of these zeros determines the antenna’s response. 2-transform analysis/synthesis methods rely upon an amplitude- phase representation of the array factor written as F(2) =ao + a12 + .-. + aN-lZN--l = “I(Z - Z1) . (2 - 22). . . (Z - ZN) (1) where a, z e’+. $ kdu. k 27r/wavelength. d Distance between elements. U sin$. $ N Amplitude weight at element n. Angle of arrival measured from broadside. Number of elements in the array. Equation (1) is the basis for synthesizing array factors with desired directional characteristics. Relating the zeros of (1) to the zeros of a Chebychev polynomial [ 11 produces the familiar Dolph-Chebychev array factor with equal level sidelobe levels. Many other amplitude tapers may be derived using this method [l]. Manuscript received September 14, 1994; revised January 30, 1995. The author is with the Department of the Air Force, USAF Academy, CO IEEE Log Number 9414201. 8084M236 USA. 0018-9261(/95$04.00 0 1995 IEEE