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1、1,Chapter 4阵列综合,Dolph-Chebyshev综合,乌特沃特综合法Fourier 级数法Schelkunov方法,泰勒综合法,非均匀幅度分布的直线阵,ne为偶数时，根据直线阵场强相加可以得到：式中,2,非均匀幅度分布的直线阵,no为奇数时式中,3,非均匀幅度分布的直线阵,例如一9单元点源阵列，间距/2，等幅同相馈电。,4,5,Dolph-Chebyshev综合（最优分布）,对于指定的旁瓣电平，其第一零点波束宽度为最窄；反之，对于指定的波束宽度，其旁瓣电平最低。综合得到的方向图为(NT为阵元数量),由于主瓣与副瓣之比r1，因此,其中M=NT -1,6,将阵列多项式与Chebysh
2、ev多项式进行匹配，使阵列的副瓣占据 的区域，阵列的主瓣位于z0 1的区域，有,当NT 为偶数、阵元间距dx/0.5时，所需激励如下:,7,设计步骤：选取与阵列如下多项式同幂次（m = n-1）的切比雪夫多项式对于偶数个阵元对于奇数个阵元,8,选取主瓣与副瓣之比r，并从下式中解出x0 . 引入新的总量w，使得此时 。以w取代 中的变量x，令故波瓣图多项式 和 便可表示为w的多项式。,9,使切比雪夫多项式和阵列多项式相等，即由此可解出阵列多项式的系数，然后得到阵列的口径电平分布。,详见J. D. Kraus天线Dolph-Chebyshev分布的八源阵举例阵列综合的实质是以Chebyshev多项
3、式表示阵列多项式。,10,乌特沃特综合法,一个均匀照射的阵列方向图有着如下的形式：,均匀照射的阵列方向图是一组正交波束的叠加，因此可以用来综合所需要的方向图。,一个长度为 L=Ndx的阵列，在u空间中将有N个波束覆盖大小为 (N-1)/L的扇区，,11,第i个波束由如下的相位步进 激励：,其中n取值与i相同，方向图函数如下：,给定的方向图函数E(u)可以由在ui上的N个取样近似：,在每个阵元上的总电流即是形成所有波束的电流之和。对于第n个单元有：,12, 正交波束,平顶方向图的综合,13,由乌特沃特法综合得到的64个点源阵列的脉冲形方向图,sinc基函数（i = -13）,14,泰勒综合法,对
4、于大型阵列，Dolph-Chebyshev 综合方法得出的是单调的口径分布，因此该方法会导致口径tapered efficiency降低.,泰勒指出，由于Chebyshev方向图的所有副瓣电平均相等，因此导致tapered效率的损失。对于大型阵列，这就意味着更多的能量将集中于副瓣内。,15,泰勒建议，可以设计这样的方向图函数，使得靠近主瓣的方向图零点类似于Chebyshev 方向图，但远离主瓣的零点位置对应于均匀分布的情况。,由泰勒综合法得到的64个点源阵列的方向图,16,副瓣比 r 即是F0 在z = 0的值：,以上的理想方向图对应于另 一类Chebyshev方向图，其零点位置在：,17,为
5、了匹配两类零点，泰勒引入尺度因子 ，通过调整零点的位置zn 来拉伸空间因子，以使其中一个零点对应于 。新的方向图函数变为：,所需要的口径分布可以展开为有限项的傅里叶级数，且该口径分布函数在阵列的边缘处导数为零。,18,口径分布函数可以表示为,19,Bayliss Line Source Difference Patterns,该方法通常用于脉冲系统. 参数A 和 通常用于控制副瓣及其下降的情况.,阵列的激励由如下公式给出：,此处,20,由傅里叶级数可以算出各系数的值：,在此阵列中，方向图的零点位于：,21,For A and n, Elliott presented a table of th
6、e coefficients themselves for SLLs from -15dB to -40dB in increments of 5dB.,22,Fourier 级数法,以上求和的结果即是有限项傅里叶级数，它在u空间是周期性的。对于一个期望的F(u)，所需激励条件可由正交性质得到：,该方法常用于赋形波束的综合.,23,由Fourier级数法综合得到的64个点源阵列方向图。脉冲形方向图（ 0.4 u 0.4 ，F(u)=1，其他F(u)=0）,24,Schelkunov方法,阵因子可以写为关于复变量z的多项式形式，其中,以上为(N-1)阶多项式，它有(N-1)个零点，因此,对于均匀
7、照射的阵列有：,25,基于优化方法的方向图综合,GA; (R.L. Haupt, Y. Rahmat-Samii, D.H. Werner, ) SA; (F. Ares, ) ANN; (F. Ares, ) TACO; (N. Karaboga, ). PSO; (Y. Rahmat-Samii, D.H. Werner, ). DE. (S. Yang, A. Rydberg, ) ,Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms. New York: Wiley
8、, 1999.,26,Pattern Synthesis Using Measured Element Patterns,Where e0(u) is the isolated element pattern and Cmn is an unknown coupling coefficient. The radiated signal from the whole array is,Steyskal, and J. S. Herd, “Mutual coupling compensation in small array antennas,” IEEE Trans. Antennas Prop
9、agat., vol. 38, no. 12, pp. 1971-1975, Dec. 1990.,Corresponding to an incident signal An at the nth element , the radiation,27,Tseng-Cheng Pattern Synthesis Technique,For planar array with rectangular grid, conventional synthesis approach has been to assume separable and independent current distribu
10、tions in the two dimensions and to employ the method of pattern multiplication.,If a Dolph-Chebyshev excitation is used, the radiation pattern of a rectangular array is optimum only in two principal sections; the patterns in other cross sections have a much broader main beam and greatly reduced side
11、lobes.,D.K. Cheng and F.I. Tseng proposed a synthesis approach for scanning rectangular arrays, which will produce a Chebyshev pattern in any cross section with the same specified SLL.,28,F. I. Tseng, and D. K. Cheng, “Optimum scannable planar arrays with an invariant sidelobe level ,” IEEE Proc., v
12、ol. 56, no.11, pp. 1771-1778, 1968. Y. U. Kim, and R. S. Elliott, “Extensions of the Tseng-Cheng pattern synthesis technique,” J. Electromag. Waves Appl. , vol. 2, pp. 255-268, 1988.Rivas A, Rodriguez JA, Ares F, Moreno E. Planar arrays with square lattices circular boundaries: sum patterns from dis
13、tributions with uniform amplitude or very low dynamic-range ratio. IEEE Antennas and Propagation Magazine 2001; 43(5):90-93.,29,Consider a rectangular planar array with a rectangular boundary, and assume a quadrantal symmetry in the excitation of the array elements. With 2N2N elements, the array sum
14、 pattern is given by,The pattern was connected to a polynomial of one variable by using the Baklanov transformation,30,Since,With The same formula applying for . Then a general polynomial of degree 2N-1 can be written in the form,31,If one desired that the pattern S(u,v) have the characteristics of
15、the polynomial , comparing these two we obtain,where,Tseng and Cheng Chose as the Chebyshev Polynomial . By Comparison, a2s-1 is obtained. Thus, Imn is given by,32,In which b2s-1 is a substitution for . Again,Alternatively, using the concept of collapsed distribution, one can deduce the excitation o
16、f a linear array laid out along the X-axis that will give the same XZ pattern as does the planar array:,33,The synthesis procedure:,Design a linear array which has the same pattern as the desired pattern in any -cut plane of the planar array;Obtain the coefficients b2s-1 from the excitation distribution Im of the linear array; Obtain the planar distribution Imn from b2s-1 .,Example:,A 2020 square grid planar array with /2 element spacing. Target: -50dB SLL Chebyshev pattern in every -cut plane.,34,
