YANG Kai, ZHAO Zhi-qin, and NIE Zai-ping

(School of Electronic Engineering, University of Electronic Science and Technology of China Chengdu 610054)

Narrow mainlobe and low sidelobe level (SLL)are required in many applications, such as radar, sonar and w ireless communications etc[1]. The mainlobe w idth is essentially determined by the aperture size of an array rather than by the number of the elements.However, the SLL is critically related w ith the number of the elements. Nevertheless the number of the elements w ill determ ine the complexity and the cost of a system. Compared w ith common uniform ly spaced arrays, thinned arrays have the advantages of covering larger aperture but w ith less number of elements. And due to larger inter-element space, thinned arrays may have smaller mutual couplings between the elements.Furthermore, if the inter-element space of an uniform linear arrays is larger than ( is the wavelength),grating lobe w ill rise. When the inter-elements space is between λ / 2 and λ, the generation of grating lobe w ill depend on the steering angle. Unequally spaced arrays can avoid grating lobe even when the average space is larger than . Therefore, how to thin an array has received lots of attentions. In antenna analysis, peak side lobe level (PSLL) is always an important parameter. This paper aims to thin the arrays whose positions are restricted by regular linear lattices to minimize the PSLL w ith or w ithout the restraint of the mainlobe w idth.

Particle swarm optim ization (PSO) is a popular stochastic, population-based algorithm modeled on swarm intelligence. It was originally described by Ref.[2]. Compared w ith common optimization methods, PSO has few parameters and easier to be implemented. It has shown better performance in certain instances than other evolutionary algorithms,such as genetic algorithm[3]. PSO is good at settling continuous problems. But how to efficiently deal w ith discrete problems has still received little attention yet.A discrete binary version of the PSO was introduced by Ref.[4]. The main idea is to set the velocity on a one or a zero as a probability. Nevertheless this method is hard to escape from local best position. Ref. [5]developed a fuzzy discrete particle swarm optimization(FDPSO). Then, this method was induced to solve traveling salesman problem by Ref. [6]. Some attractive results were shown.

The FDPSO method is also possible to be trapped in local optima which w ill lead to premature convergence for complicated multimodal problems. In order to tackle this problem, a chaos mapping[7]method is induced in the FDPSO in this paper.Simulation results show that global optima and convergence rate have been greatly improved by the method.

1 Beam Pattern

2 FDPSO

Where the superscript k and the subscript i represent the k-th iteration time and the i-th particle,respectively, Ä and Å denote the multiplication and corresponding elements addition of two N´M matrices, respectively,iP and Pgare the best position of the i-th particle searched and the global best position searched respectively, C1and C2are the acceleration coefficients, rand1() and rand2() are two N´M matrices, each of which is composed of copies of a N´1 vector, and the two vectors are independent uniform distributions w ith random number between 0 and 1,w is the inertia weight which controls the impact of previous velocity of particle on its current one. w controls the exploration and exploitation in PSO.

It is important to maintain a balance between global exploration and local exploitation. At the beginning of the PSO, a larger inertia weight is preferred to explore the whole search space, but a smaller inertia weight is needed to exploit around the local area. According to the analysis above, a variable inertia weight is given by[8]:

Where k and k max are current iteration time and the total iteration time respectively.

3 Im plementation of FDPSO And Its App lication in Antenna Analysis

The algorithm w ill jump into a chaotic dynam ics process if the FDPSO gets trapped into local optimum.Trapping into the local optimum is judged by whether the historic global best position keeps unchanged or changes a little in a certain number of iteration steps or the particles aggregate heavily.

Suppose that each dimension value of X is chosen from the integers from 0 to M-1. In order to make the best use of the characteristic of sensitive dependence on initial condition of the chaotic dynamics, a random vector is added to:

6) Convert the new position obtained above to fuzzy matrix, and random ly replace one fifth of the particles fuzzy matrices by it.

In the follow ing, the FDPSO w ill be used in the optimization of unequally spaced antenna arrays analysis.

The mainlobe w idth and the PSLL are two very important parameters in antenna analysis. The mainlobe w idth is mainly connected w ith the aperture size. While the number of the elements and their positions w ill greatly affect the SLL. The mainlobe w idth plays an important role in some applications,such as direction of arrival (DOA) estimation. A smaller mainlobe w idth is more important than a lower PSLL when the MUSIC algorithm is applied in DOA estimation[9]. Therefore the constraint of the mainlobe w idth is also considered in the follow ing section.Usually it would be very difficult to get the lowest SLL and the narrowest mainlobe w idth simultaneously.

In order to comprehensively consider the mainlobe and the sidelobe, a penalty function is used to constrain the mainlobe w idth rather than a certain value. The constraint is given by:

4 Simulation

Consider a array w ith 25 elements located on linear lattices w ith an aperture of 50 . For a continuous problem, the array elements can be located at any possible place. But the problem we investigated here is a discrete one, which means that the elements can only be set at the lattices w ith a grid spacing of. The first and the last elements are fixed at the two ends of the array. It should be pointed out that the lattices space can also be set to other values, such as, here it is set tojust for the comparison w ith the results of other literatures.

4.1 Symmetrically Alignment W ith the Constraint of M ainlobe

Firstly, assume the array elements are aligned symmetrically w ith respect to the origin. At this condition, the dimension of the particle is equal to 11.In order to compare w ith the results obtained by Ref. [11], the mainlobe w idth is restrained to be less than 0.015, and the iteration time is set as 30 000. The algorithm described in Sec. 3 is applied in the array analysis. The simulated beam pattern is shown in Fig.1. And the resulted particle position is (0.5, 1, 3, 4.5,6.5, 7.5, 8, 9, 14.5, 25). The mainlobe w idth is 0.0143,which meets the constraint of 0.015. The PSLL is-10.55 dB, versus the -10.43 dB given by Ref. [11].

Fig. 1 The optimum pattern for symmetrical arraysw ith mainlobe w idth constraint

4.2 Asymmetrically Alignment W ithout the Constraint of M ainlobe

Figure 1 is the result w ith the restraint that the elements are arranged symmetrically. It w ill have more degrees of freedom w ithout that restraint. Better results can be expected.

Now the dimension of the particle increases to 23.The pattern obtained using the method proposed in this paper w ithout mainlobe constraint is shown in Figure 2.The PSLL is equal to -12.63 dB. The element positions are given by 0, 12, 12.5, 13, 13.5, 15.5, 16,16.5, 17, 17.5, 18, 18.5, 19, 19.5, 20.5, 21.5, 23, 23.5,24.5, 25, 26, 26.5, 28, 47.5, 50. To our know ledge, the best result shown in the literature is in Ref. [12], whose PSLL is equal to -12.07 dB. Therefore, the result here by using FDPSO get a 0.56 dB lower PSLL than the result using simulated annealing[12].

Fig. 2 The beam power pattern w ithout mainlobe and symmetry constraint

The result obtained in this simulation is aimng to get a low PSLL but w ithout the constraint of mainlobe w idth. In the follow ing subsection, the constraint of mainlobe w idth w ill be involved in the consideration.

4.3 Asymmetrically Alignment W ith the Constraint of M ainlobe W idth

To make trade-offs between the mainlobe w idth and the PSLL, mainlobe constraint is used. We set the upper bound of the mainlobe w idth as 0.01, and the algorithm is to find the lowest PSLL under this constraint. The other parameters are chosen as:

The result is shown in Fig. 3. The PSLL is -11.42 dB, and the mainlobe w idth is 0.009 6. The exact element positions are at 0, 1, 3.5, 5.5, 6, 13.5, 16.5,17.5, 20.5, 21.5, 22, 24, 24.5, 25.5, 27.5, 30.5, 31, 32.5,33, 34, 34.5, 39, 40.5, 45.5, 50. The result obtained by constraint the mainlobe w idth is a pencil beam.Generally, the optimal arrays w ith narrower mainlobe tend to have higher peak side lobe levels according to the simulation results that constraint the mainlobe w idth to a specific set of values. Compared w ith Fig. 2,Fig. 3 has a better mainlobe but pays the price of a little bit higher PSLL. In engineering, usually both mainlobe and sidelobe must be taken into account simultaneously.

Fig. 3 The beam power pattern w ith mainlobe constraint.

5 Conclusion

In this paper, a FDPSO algorithm is proposed to synthesize unequally spaced arrays for obtaining low peak side lobe level. The result is better than that of other methods, such as simulated annealing. The algorithm can easily combine w ith the constraints to obtain a specific mainlobe w idth.

[1] KIRKEBO J E, AUSTENG A. Sparse cylindrical sonar arrays[J]. IEEE Journal of Oceanic Engineering, 2008, 33(2):224-231.

[2] KENNEDY J, EBERHART R C. Particle swarm optim ization[C]//Proc IEEE Conference Neural Networks IV.Perth: IEEE, 1995.

[3] DANIEL W B, DOUGLAS H W. Particle swarm optim ization versus genetic algorithms for phased array synthesis[J]. IEEE Trans Antennas and Propagation, 2004,52(3): 771-779.

[4] KENNEDY J, EBERHART R C. A discrete binary version of the particle swarm algorithm[C]//The 1997 Conference on Computational Cybernetics and Simulation. Orlando:IEEE, 1997.

[5] CLERC M. Discrete particle swarm optim ization: a fuzzy combinatorial black box[EB/OL]. [2010-02-25]. http://clerc.maurice.free.fr/pso/Fuzzy_Discrete_PSO/ Fuzzy_DPSO.htm.

[6] PANG W, WANG K, ZHOU C, et al. Fuzzy discrete particle swarm optimization for solving traveling salesman problem[C]//Proceeding of the Fourth International Conference on Computer and Information Technology.Wuhan: IEEE, 2004.

[7] LIU B, WANG L, JIN Y, et al. Improved particle swarm optimization combined w ith chaos[J]. Chaos, Solution and Fractals, 2005, 25(5): 1261-1271.

[8] LEI K Y, WANG F, QIU Y H, et al. An adaptive inertia weight strategy for particle swarm optim izer[C]//The 3rd International Conference on Mechatronics and information technology. Chongqing: SPIE, 2005.

[9] VIBERG M, ENGDAHL C. Element position consideration for robust direction finding using sparse arrays[C]//Conf Rec Thirty Third Asilomar Conf Signals, Syst, Comput.Pacific Grove: IEEE, 1999.

[10] PARSOPULOS K, VRAHATIS M. Particle swarm optimization method for constrained optimization problems[C]//Proceedings of the Euro-International Symposium on Computational Intelligence. Slovakia: ISO Press., 2002.

[11] LEE K C, JHANG J Y. Application of particle swarm algorithm to optim ization of unequally spaced antenna arrays[J]. Journal of Electromagnetic Waves and Applications, 2006, 20(14): 2001-2012.

[12] MURINO V, TRUCCO A, REGAZZONI C S. Synthesis of unequally spaced arrays by simulated annealing[J]. IEEE Trans Signal Processing, 1996, 44(1): 119-123.

编 辑 税 红