提出了一种新型的基于幅度分档的赋形波束方向图综合算法。该算法共分为3步。首先，使用传统方向图综合方法如交替投影得到波束的无幅值限制的阵元激励；然后，使用概率密度理论对得到的阵元激励幅度进行处理得到量化的阵元激励幅度，最后，通过量化阵元激励幅度，使用半正定松弛（SDR）方法得到阵元激励的相位分布。上述步骤中，如何使用概率密度理论得到量化的阵元激励幅度是3步中较为重要的一步。将阵元激励幅度用概率密度变量进行替代，通过事先设定的阵元激励幅度档位个数，以及每个阵元激励幅度落在相应档位时取值的概率，可以得到含有概率密度变量的综合方向图表达式。最小化含有概率密度变量的综合方向图与理想方向图的功率之差即可得到量化的阵元激励幅度。使用概率密度理论得到量化阵元激励幅度的优势在于，可以根据任意形状的阵面和阵元栅格排布来划分幅度的档位区间，从而有着更广泛的适用性。在例证部分，通过多组算例的仿真，以及与一些对算法性能的分析，所提算法验证了其在综合效果上的优越性。
In this paper, a novel algorithm for shaped-beam pattern synthesis is proposed based on quantization of element excitation amplitude. The algorithm is divided into three steps. First, traditional pattern synthesis, such as alternative projection, is performed to obtain the element excitation of the beam without amplitude constraints. Second, the probability density method is used to acquire the quantized amplitude of element excitations. Finally, the Semi-Definite Relaxation (SDR) method is employed to obtain the element excitation phase of each beam. In the above procedures, how to use the probability density method to obtain the quantized amplitude of the element excitation is the most important step. Each element excitation amplitude is replaced by the probability density variable. By setting the element excitation amplitude levels in advance and the probability of each element excitation amplitude when falling in the corresponding level, the probability density pattern is obtained. The difference between the probability density pattern and the ideal pattern is then minimized to acquire the quantized element excitation amplitude. The probability density method has the advantages in that the quantization levels can be acquired through footprints and arrays with arbitrary shapes, and hence can be applied in a wider background. Simulation verifies the superiority of the proposed algorithm.
[1] WARREN L, STUTZMAN A, THIELE G. Antenna theory and design[M]. New York:John Wiley & Sons, 1998:76-171.
[2] STRAIT B J. Antenna arrays with partially tapered amplitudes[J]. IEEE Transactions on Antennas & Propagation, 1967, AP15(5):611-617.
[3] HANSEN R C. Effect of field amplitude taper on measured antenna gain and sidelobes[J]. Electronics Letters, 1981, 17(7):260-261.
[4] HAUPT R L. Reducing grating lobes due to subarray amplitude tapering[J]. IEEE Transactions on Antennas & Propagation, 1985, 23(8):119-122.
[5] HAUPT R L. Optimization of subarray amplitude tapers[C]//Antennas and Propagation Society International Symposium. Piscataway, NJ:IEEE Press, 1995, 4:1830-1833.
[6] OGG F C. Steerable array radars[J]. IRE Transactions on Military Electronics, 1961, MIL-51(2):80-94.
[7] MAFFETT A. Array factors with nonuniform spacing parameter[J]. IRE Transactions on Antennas & Propagation, 1962, 10(2):131-136.
[8] LO Y. A spacing weighted antenna array[C]//1958 IRE International Convention Record. Piscataway,NJ:IEEE Press, 1966:191-195.
[9] WILLEY R. Space tapering of linear and planar arrays[J]. IRE Transactions & Antennas and Propagation, 1962, 10(4):369-377.
[10] ISHIRU A. Theory of unequally-spaced arrays[J]. IRE Transactions on Antennas & Propagation, 1962, 10(6):691-702.
[11] SKOLNIK M L, SHERMAN J, OGG F. Statistically designed density-tapered arrays[J]. IEEE Transactions & Antennas and Propagation, 1964, 12(4):408-417.
[12] HAUPT R L. Thinned arrays using genetic algorithms[J]. IEEE Transactions on Antennas & Propagation, 1994, 42(7):993-999.
[13] NUMAZAKI T, MANO S, KATAGI T, et al. An improved thinning method for density tapering of planar array antennas[J]. IEEE Transactions on Antennas & Propagation, 1987, 35(9):1066-1070.
[14] BALL G J. Statistical density and amplitude tapering[C]//1989 Sixth International Conference on Antennas and Propagation. Coventry:IET, 1989:249-253.
[15] MAILLOUX R J, COHEN E. Statistically thinned arrays with quantized element weights[J]. IEEE Transactions on Antennas & Propagation, 1991, 39(4):436-447.
[16] LEE J J. Sidelobes control of solid-state array antennas[J]. IEEE Transactions on Antennas & Propagation, 1988, 36(3):339-344.
[17] XUE F, GUO Y, FANG N. Analysis and computation of circular aperture arrays with multi-step amplitude quantization[C]//IEEE Antennas & Propagation Society International Symposium. Piscataway, NJ:IEEE Press, 1992:470-473.
[18] QIN L J, SHAO J D. Design of multi-step amplitude quantized weights for solid-state planar phased array antennas[C]//Aerospace & Electronics Conference. Piscataway, NJ:IEEE Press, 1995:77-80.
[19] TIE G, JIAN X L, YAN C G. Design and analysis of multi-step amplitude quantization weighted 2-D solid-state active phased array antennas[J]. Journal of Electronics, 1994, 11(1):71-78.
[20] HUAN Z, WANG Y X, XU X W, et al. Direct amplitude optimization and quantization for ultralow sidelobe phased arrays[C]//IEEE International Conference on Computational Electromagnetics and Its Applications. Piscataway, NJ:IEEE Press, 1999:183-186.
[21] HUAN Z, WANG Y X, XU X W, et al. Multi-step amplitude quantization for ultralow sidelobe phased arrays by direct optimization synthesis[J]. Journal of Systems Engineering and Electronics, 2001, 12(1):65-69.
[22] XU H. Research on phased array antenna multi-step mixed feed[C]//Microwave and Millimeter Wave Technology. Piscataway, NJ:IEEE Press, 2010:2003-2006.
[23] XU F M, MENG L Q, XIE Y N. Realizing stepped amplitude quantization of low sidelobe linear array using particle swarm optimization algorithm[C]//IET International Communication Conference on Wireless Mobile and Computing. Coventry:IET, 2009:570-573.
[24] LUO Z Q, MA W K, SO A M C, et al. Semidefinite relaxation of quadratic optimization problems[J]. IEEE Signal Processing Magazine, 2010, 27(3):20-34.
[25] BALANIS C A. Antenna theory:Analysis and design[M]. New York:Harper & Row, Publishers, Inc.,1982:212-216.
[26] POZAR D M. The active element pattern[J]. IEEE Transactions on Antennas & Propagation, 1994, 42(8):1176-1178.
[27] HADDADI A, GHORBANI A, RASHED M J. Cosecant-squared pattern synthesis using a weighted alternating reverse projection method[J]. IET Microwaves, Antennas & Propagation, 2011, 5(15):1789-1795.
[28] BOYD S, VANDENBERGHE L. Convex optimization[M]. New York:Cambridge University Press, 2004:51-53
[29] HUANG Y, PALOMAR D P. Rank-constrained separable semidefinite programming with applications to optimal beamforming[J]. IEEE Transactions on Signal Processing, 2010, 58(2):664-678.
[30] ZHANG Z H, HU W D, YU W X. Design of quantized amplitude weights for low sidelobe phased array antennas[J]. Acta Electronica Sinica, 2007, 35(3):580-584.
[31] 徐锋明, 孟令琴, 谢亚楠. 基于改进的粒子群算法实现阶梯幅度量化相控阵天线的低副瓣[J]. 上海大学学报(自然科学版), 2010, 16(4):361-366. XU F M, MENG L Q, XIE Y N. Realization of low sidelobe for step-quantized amplitude phased array antennas based on modified particle swarm optimization[J]. Journal of Shanghai University (Natural Science Edition), 2010, 16(4):361-366(in Chinese).
[32] 虞泓波, 冯大政, 解虎. 采用序列二次规划求解的稳健波束形成新算法[J]. 西安电子科技大学学报, 2016, 43(2):41-45. YU H B, FENG D Z, XIE H. Novel robust beamforming algorithm using sequential quadratic programming[J]. Journal of Xidian University, 2016, 43(2):41-45(in Chinese).
[33] KENNEDY J. Particle swarm optimization[M]. Encyclopedia of machine learning. Berlin:Springer, 2011:760-766.
[34] LIU Y Q, LIU S Y, GU M T. Application of powell algorithm in linear support vector machine[J]. Computer Engineering, 2011, 37(12):161-163.
CJA