Synthesis of unequally Spaced Antenna Arrays by ... - Semantic Scholar
20 International Journal of Communication Networks and Information Security (IJCNIS)
Vol. 1, No. 1, April 2009
Synthesis of unequally Spaced Antenna Arrays by a new Differential Evolutionary Algorithm Chuan Lin1, 2, Anyong Qing2, Quanyuan Feng1 1
School of Information Science and Technology, Jiaotong University, Chengdu, China 610031 2 Temasek Laboratories, National University of Singapore, 5 Sports Drive 2, Singapore 117508 {tslqay, tslv6}@nus.edu.sg
Abstract: The differential evolution (DE) algorithm with a new differential mutation base strategy, namely best of random, is applied to the synthesis of unequally spaced antenna arrays. In the best of random mutation strategy, the best individual among three randomly chosen individuals is used as the mutation base while the other two are for the vector difference. Hence a good balance of diversity and evolution speed can be obtained. An indirect coding method is utilized to convert the constrained synthesis problem to an unconstrained one. The effect of angle resolution of the radiation pattern is also discussed. Simulation results demonstrate that the new DE algorithm is promising. Keywords: Antenna arrays, pattern synthesis, differential evolution, best of random, angle resolution
1. Introduction In array pattern synthesis, the main objective is to find an appropriate weighting vector and layout of the elements to yield the desired radiation pattern. Over the past few decades, the synthesis of uniformly spaced antenna arrays has been extensively studied. In recent years, the unequally spaced array, also termed as aperiodic array, has attracted increasing attention. Compared with the equally spaced array, the unequally spaced array has more degree of freedom and is able to lower the peak sidelobe level with smaller number of elements. The difficulties in the design of unequally spaced arrays are mainly caused by the nonlinear and non-convex dependency of the array factor to the weights and the sensor positions [1][2][3]. Various analytical and numerical array pattern synthesis techniques have been developed to meet the challenge[1]-[8]. Recently, evolutionary algorithms (EAs) including genetic algorithms (GAs)[5][6][7], simulated annealing (SA)[8], particle swarm optimization (PSO)[3], and so on, have become more popular as they are applicable not only to regular arrays (such as linear arrays and circular arrays) but also to arrays with complicated geometry layout and radiation pattern. Moreover, they can simultaneously optimize the weight vector and sensor positions. Differential evolution (DE) algorithm, introduced by Price and Storn, is a simple, efficient and robust evolutionary algorithm [9][10][11]. It has also been successfully applied to many electromagnetic fields, including array synthesis problems [2][12] and electromagnetic inverse problems[13][14]. A DE variant is usually marked as DE/x/y/z, where x denotes how the differential mutation base is chosen, y denotes the number of vector differences added to the base vector and z indicates the crossover method. Best and
random are the two most often implemented differential mutation bases. Although DE performs well as compared to many other methods, it still remains a challenge in DE to properly balance the exploration and exploitation. Through a comprehensive parametric study on DE, it has been observed that the balance of exploration and exploitation is greatly affected by differential mutation base strategies [11][15][16]. For example, guidance provided by the best differential mutation base often leads to higher efficiency at the cost of diversity while blind search by using random differential mutation bases gains diversity and sacrifices efficiency. Hence, using differential mutation base providing both guidance and diversity simultaneously might be a more promising solution to effectively balance exploration and exploitation. A new differential mutation base strategy, named “best of random” differential mutation base, is accordingly proposed. The new differential mutation strategy is very simple and introduces no additional intrinsic control parameter. Like random differential mutation, best of random differential mutation also randomly chooses three mutually different individuals. However, unlike random differential mutation, best of random differential mutation uses the best individual among the three individuals as differential mutation base while the other two worse individuals are donors for vector difference. DE algorithm with best of random mutation base is denoted as DE/BoR/1/*. Its advantages have been verified by our comprehensive simulations over a set of benchmark functions. In this paper, DE/BoR/1/* is applied to the unequally spaced array synthesis. Its efficiency in the array synthesis is also demonstrated by the numerical results. The rest of this paper proceeds as follows. Section II briefly describes differential evolution. Details of the new differential mutation base strategy are given in Section III. In Section IV, DE with best of random mutation base is applied to the array synthesis problem. Conclusion is given in Section V.
2. Differential evolution 2.1 Evolution Mechanism Classic DE optimizes an objective function with a population of Np individuals. It involves two stages, namely, initialization and evolution. Initialization generates initial population P0. Then the population evolves from one generation (Pn) to the next (Pn+1) until the termination conditions are met. While evolving from Pn to Pn+1, the three
21 International Journal of Communication Networks and Information Security (IJCNIS)
evolutionary operations, namely, differential mutation, crossover and selection are executed in sequence. The Fortran-style pseudo-code of classic differential evolution is shown in Fig. 1 where xn, i is the vector of optimization parameters of the ith individual pn,i in population Pn, xn+1,v,i is the vector of optimization parameters of mutant vn+1,i, xn,b,i is the vector of optimization parameters of differential mutation base bn,i, p1y and p2y are integer numbers, usually random, Fy, a constant usually in [0,1], is the mutation intensity for the yth vector difference n , p1 y
n, p
n , p2 y
n, p
, x 1 y and x 2 y are vectors of optimization n, p n, p parameters of donors p 1 y and p 2 y from Pn, xn+1,c, i is the vector of optimization parameters of child (or trial individual) cn+1,i. αji is a real random number in [0,1], [biL,biU] is the search space of the ith optimization parameter, xi. Initialization n=0 do i = 1, Np do j = 1, N
x
−x
x
0, i j
(
= b +α b −b L j
i j
U j
L j
)
(
y ≥1
n , p1 y
−x
n , p2 y
),
1 ≤ i ≠ p1 y ≠ p 2 y ≤ N p regularize infeasible mutant vn+1,i do j = 1, N n +1, v , i
do while ( x j
< b Lj )
x nj +1, v ,i = x nj +1, v ,i + bUj − b Lj end do n +1, v , i
do while ( x j
2.2 Differential Mutation Base Differential mutation base can be chosen in a variety of ways. Best and random are the two most often implemented differential mutation bases. DE/best/1/* uses the best individual pn,best in Pn as the differential mutation base while DE/rand/1/* randomly chooses the differential mutation base from Pn. 2.3 Binomial Crossover Binomial crossover and exponential crossover are two most often implemented crossover methods in DE. In this paper, binomial crossover is used. In binomial crossover scheme, cn+1,i is generated as follows:
x
n +1,c ,i j
where
x nj+1,v ,i β jn+1,i ≤ C r = n ,i otherwise x j
β jn +1,i
(1)
is a real random number uniform in the range
[0,1] and Cr, a constant in [0,1], is the crossover probability. There is an extreme case. The child duplicates vn+1,i. In this n,i
end do evaluate f(x0,i) end do Evolution do while termination conditions are not satisfied n=n+1 do i = 1, Np differential mutation to obtain mutant vn+1,i choose differential mutation base bn,i, n, p n, p donors p 1 y and p 2 y
x n +1, v ,i = x n , b ,i + ∑ F y x
Vol. 1, No. 1, April 2009
> bUj )
x nj +1, v ,i = x nj +1, v ,i − bUj + b Lj end do end do crossover mutant vn+1,i with pn,i to deliver child cn+1,i evaluate child f(xn+1,c, i) selection to get individual pn + 1,i if f(xn+1,c, i) < f(xn, i) pn + 1,i = cn+1,i else pn + 1,i = pn,i end if end do end do Figure 1 Fortran-style Pseudo-code of Classic Differential Evolution
case, a randomly chosen parameter of pn,i, x j , will replace n +1, c , i
the corresponding parameter of the child cn+1,i, x j
.
n+1,i
There is another extreme case. c inherits no parameter from vn+1,i and hence no evolution happens. In this case, a n +1, c , i
randomly chosen parameter of the child cn+1,i, x j
, will
be replaced by the corresponding parameter of the mutant n +1, v , i
vn+1,i, x j
.
2.4 Dynamic Differential Evolution Classic differential evolution is static in nature from the point of view of population updating and is therefore inherently inefficient. Inspired by the advantage of GaussSeidel method against Jacobi method for linear equations, the dynamic differential evolution (DDE) [14] was developed. The dynamic updating of population in DDE leads to a larger virtual population size and quicker response to change of population status. Thus in general, dynamic differential evolution outperforms classic differential evolution: faster and more reliable, just as the Gauss Seidel method does over the Jacobi method. For more details of dynamic differential evolution, please refer to [14].
3. Best of Random Differential Mutation Base How to balance exploration and exploitation is a crucial problem for most evolutionary algorithms including DE. In fact, this problem can also be understood from another point of view: the balance of diversity and evolution of the population. Diversity here means the dispersing of the individuals in the population, while evolution means constant improvement of the (average) performance of the population as they evolve from one generation to the next. Randomly distributing the population in a wide area helps to increase the diversity while guidance by the elite individuals often makes the search more efficient and hence promotes the evolution speed. Generally speaking, differential mutation operation in DE could be regarded as a local search. Differential mutation
22 International Journal of Communication Networks and Information Security (IJCNIS)
base vector acts as the local search center and the scaled vector difference determines the search range or search step around the center. The diversity of the population at the next generation is effected by the diversity of the local search centers (base vectors). The base vectors with high quality provide guidance making the search pay more attention to the promising regions, which often brings faster evolution speed. In DE/rand/1/*, the base vectors are randomly chosen from the population, thus a good diversity of base vectors is gained but no search guidance is provided. On the other hand, all the individuals in DE/best/1/* use the same base vector: the best vector so far. The guidance of the best individual generally leads to fast evolution speed but also increases the risk of stagnation due to the lack of enough diversity of base vectors. From the above analysis, we notice that differential mutation base providing both guidance and diversity should be a more promising choice to better balance the diversity and evolution of the population. Based on this idea, a new generator for differential mutation base, namely “best of random”, is proposed. Best of random differential mutation base uses the best individual among the three randomly chosen individuals as differential mutation base and the other two worse individuals as donors for vector difference. Thus both good quality and diversity of the base vectors can be obtained. Best of random mutation strategy can be expressed as:
x n+1,v ,i = x where x
n , p1 y
n, pb
, x
(
+ F x n , p1 − x n , p2
n, p2 y
and x
n, p b
)
(2)
are vectors of optimization
parameters of 3 randomly chosen individuals p n, p b
n , p1
, p
n , p2
,
, and f ( x ) ≤ min( f (x ), f (x )) . It can be easily found that DE/BoR/1/* is almost as simple as DE/rand/1/* and introduce no extra intrinsic control parameters.
p
n, pb
n , p1
n , p2
4. Array Synthesis Using DE with Best of Random Mutation Strategy 4.1 Description of Array Synthesis problem The array factor AF(θ) of a linear antenna array with N elements at angle θ can be expressed as [2]: N
AF (θ ) = ∑ I i e
j(
2π
λ
xi sin(θ ) +φi )
(3)
i =1
Where Ii and
φi
are the excitation amplitude and phase of
the element located at xi, respectively. λ is the wavelength. In this paper, the uniform amplitude excitation is considered as it is an effective way to reduce the system costs and hardware implementation complexity [2]. Thus Eq. (2) becomes: N
AF (θ ) = ∑ e i =1
j(
2π
λ
xi sin(θ ) +φi )
(4)
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4.2 Objective Function for Optimization of Sidelobe Level The task of the optimization in this paper is to minimize the sidelobe level (SLL) of the antenna array. The objective function to be minimized can be written as [2]:
f ( ρ ) = Maxθ∈S
AFρ (θ )
AFρ (θ 0 )
(5)
where S is the space spanned by the angle θ excluding the mainlobe and ρ represents the unknown parameter vector, such as element positions and phases. This objective function minimizes all the sidelobe levels and maximizes the power in the mainlobe located at θ = θ 0 . In this paper, θ 0 = 0◦.
4.3 Position-Phase Synthesis of Unequally Spaced Arrays In this subsection, position-phase synthesis of a 32element symmetric array with uniform amplitude excitation[2] is carried out by using DDE/BoR/1/bin. Due to the symmetry, the number of optimization parameters is 32, including 16 position parameters and 16 phase parameters. The other half position and phase parameters can be calculated by:
x−i = − xi , φ−i = φi
1 ≤ i ≤ 16
(6)
The distance between adjacent elements is restricted in [0.5λ, 1.0λ], that is:
0.25λ ≤ d1 = x1 ≤ 0.5λ 0.5λ ≤ d i = xi − xi −1 ≤ λ , 2 ≤ i ≤ 16
(7)
4.3.1 Coding And Decoding of optimization parameters As the element positions are subject to the above constraints, some special method is needed to tackle the constraints and ensure feasible solution if we directly optimize the element positions. For example, in [5] the coding resetting of gene variables and broad sense genetic operators are utilized to deal with the constraints, which makes the algorithm a little bit more complex. Instead of directly optimizing the element positions, we optimize the distance of the adjacent elements di. Thus the encoded optimization parameter vector ρ = [d1, d2,…, d16, Φ1, Φ2,…, Φ16], with –π ≤ Φi ≤ π and the search range for di being defined by (7). In this way the constrained optimization problem is converted into unconstrained optimization problem, and DE can be directly applied to antenna synthesis without any modification. After di have been obtained, xi can be calculated as follows:
x1 = d1 xi = xi −1 + d i , 2 ≤ i ≤ 16
(8)
23 International Journal of Communication Networks and Information Security (IJCNIS)
4.3.2 Parameter Settings for the Experiments The known best SLLs for this example reported in [2] and [12] are -23.34dB and -23.45dB, respectively. Fortuitously, it is noticed that the calculated SLL depends on the resolution or the sampling interval of angle θ due to ripples or spines. As the angle resolution becomes smaller, some ripples or spines in the sidelobe may become visible while they are invisible under the larger angle resolution. Under different angle resolutions ∆, the SLLs for this example computed with the optimization parameters given in [2] and [12] are shown in Table 1 in which inconsistence of SLLs under different angle resolutions are quite obvious.
Table 1 SLLs computed with the optimization parameters in [2] and [12] ∆ (◦)
SLL (dB) in [2]
SLL (dB) in [12]
0.1
-23.22
-23.13
0.2
-23.22
-23.16
0.5
-23.23
-23.37
1.0
-23.25
-23.37
2.0
-23.25
-23.46
It is observed from Table 1 that the results from ∆=0.2◦ downwards are trustable while those from ∆=0.5◦ onwards are not. However, to study the effect of angle resolution on the performance of the algorithm, two different angle resolutions, i.e., ∆=0.2◦ and ∆=0.5◦, are used in our simulation. According to our experiment and the results reported in [2] and [12], it is found that SLL=-23.3dB is very close to the optimal SLL of the array even though neither of the two reported results actually reaches it. In order to see the capability of DDE/BoR/1/bin, SLL = -23.3dB is used as the value to reach (VTR) for the algorithm. The maximum number of objective function evaluations NEmax is 500,000. If the obtained SLL meets the VTR or the number of objective function evaluations (NFE) reaches NEmax, the algorithm is terminated. 100 independent runs are repeated for each angle resolution. A run is said to be successful if the algorithm can find the VTR within NEmax objective function evaluations, otherwise it is unsuccessful. The successful rate rs = NS/100, where NS is the number of successful runs. The control parameters of DDE/BoR/1/bin in this example are: Np=60, F=0.4, Cr=0.7.
4.3.3 Simulation Results The simulation results of the array synthesis with ∆=0.5◦ and ∆=0.2◦ are shown in Table 2. Where Emin, Eavg, Emax are the minimal, average and maximum number of objective function evaluations of the successful runs, respectively. SLLw is the obtained worst SLL in the 100 independent runs. The distribution of the NFE and SLLs in the 100 independent runs for ∆=0.5◦ and ∆=0.2◦ are shown in Figs. 2~5.
Table 2 Simulation results of position-phase array synthesis using DE/BoR/1/bin rs
∆=0.5◦ 0.92
∆=0.2◦ 0.74
Vol. 1, No. 1, April 2009
Emin
27411
35784
Eavg Emax SLLw (dB)
73808 266545 -21.74
111493 455870 -22.47
SLLdavg(dB)
0.290
0.066
In order to see the effect of angle resolution, the returned optimization parameters from 100 independent runs with ∆=0.5◦ are reused to calculate the SLLs with the angle resolution ∆=0.2◦. The SLL differences between the new SLLs with ∆=0.2◦ and the original SLLs with ∆=0.5◦ are shown in Fig. 6. Similarly, the returned optimization parameters with ∆=0.2◦ are reused to calculate the SLLs with ∆=0.1◦. The SLL differences between the new SLLs with ∆=0.1◦ and the original SLLs with ∆=0.2◦ are shown in Fig. 7. The average sidelobe level difference SLLdavg when using the same optimization parameters but different angle resolutions over the 100 runs in the two cases are shown in the last row of Table 2. From the simulation results, it has been observed that the returned SLL with angle resolution ∆=0.5◦ is not very much trustable as the SLL will be increased by 0.290dB in average when it is recalculated using the same optimization parameters with ∆=0.2◦. If ∆=0.2◦ is used in the evaluation of the objective function, the returned SLL is more trustable as the SLL is only increased by 0.066dB when it is recalculated using the same optimization parameters with ∆=0.1◦. However, by decreasing the angle resolution, the success rate of the algorithm decreases and the number of objective function evaluations increases. The computation time of the function evaluation is also increased by using smaller angle resolution. In the unsuccessful runs, most of the SLLs with ∆=0.5◦ and all the SLLs with ∆=0.2◦ are lower than -22 dB. This means although the algorithm may not converge to the optimal SLL sometimes, it can still reach the suboptimal SLL. For verification purpose, one example of returned positions and phases with SLL=-23.3dB is shown in Table 3. The corresponding radiation pattern is shown in Fig. 8.
Table 3 Element positions and phases for a symmetric 32element antenna array xi/λ (1≤i≤8) Φi◦(1≤i≤8) xi/λ (9≤i≤16) Φi◦ (9≤i≤16) 0.2535 0.7825 1.2992 1.8333 2.3342 2.9305 3.4813 4.0867
52.68 56.59 53.35 53.54 55.63 53.81 52.12 56.67
4.7599 5.2821 6.0467 6.8861 7.7974 8.6678 9.6504 10.5287
55.18 53.04 55.09 41.82 56.08 83.13 23.62 55.39
24 International Journal of Communication Networks and Information Security (IJCNIS)
0.5
SLL difference (dB)
400000
300000
NFE
Vol. 1, No. 1, April 2009
0.6
500000
200000
100000
0.4
0.3
0.2
0.1
0.0
0 20
40
60
80
20
100
40
60
80
100
Index of runs
Index of runs
Figure 6 Differences between the SLLs with ∆=0.2◦ and that with ∆=0.5◦
Figure 2 Distribution of NFE when ∆=0.5◦ -21.6
0.12 -21.8
0.10
-22.0
Difference of SLL (dB)
SLL (dB)
-22.2 -22.4 -22.6 -22.8 -23.0 -23.2 -23.4 20
40
60
80
0.08
0.06
0.04
0.02
0.00
100
20
Index of runs
40
60
80
100
Index of runs
Figure 7 Differences between the SLLs with ∆=0.1◦ and that with ∆=0.2◦
Figure 3 Distribution of SLLs when ∆=0.5◦ 500000
0 400000
-10 -20 |AF(θ)/AF(0)|
NFE
300000
200000
100000
-30 -40 -50
0 20
40
60
80
100
Index of runs
-60
Figure 4 Distribution of NFE when ∆=0.2◦
-80
-60
-40
-20
0
20
40
60
80
θ (deg)
Figure 8 Radiation pattern for a symmetric 32-element array -22.4
5. Conclusion -22.6
SLL (dB)
-22.8
-23.0
-23.2
-23.4 20
40
60
80
Index of runs
Figure 5 Distribution of SLLs when ∆=0.2◦
100
The new DE algorithm with best of random mutation base strategy is applied to the position-phase synthesis of the unequally spaced antenna array. With an indirect coding method, the constrained optimization problem is converted into an unconstrained problem. In order to study the effect of the angle resolution on the performance of the algorithm, two different angle resolutions are used for the evaluation of the objective function respectively. By comparing the SLLs obtained with the same returned optimization parameters but different resolutions, it is observed that the results obtained with angle resolution ∆=0.2◦ is more trustable compared with those with ∆=0.5◦, though smaller angle resolution may
25 International Journal of Communication Networks and Information Security (IJCNIS)
also lead to slower convergence rate and lower success rate. From the simulation results, it is demonstrated that the new DE algorithm is promising in the array synthesis problem.
References [1] B. P. Kumar and G. R. Branner, “Design of unequally spaced arrays for performance improvement”, IEEE Trans. Antennas and Propagation, vol. 47, no. 3, pp. 511-523, Mar. 1999 [2] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna arrays using the differential evolution algorithm”, IEEE Trans. Antennas and Propagation, vol. 51, no. 9, pp. 22102217, 2003 [3] N. Jin and Y. Rahmat-Samii, “Advances in Particle Swarm Optimization for Antenna Designs: RealNumber, Binary, Single-Objective and Multiobjective Implementations”, IEEE Trans. Antennas and Propagation, vol. 55, no. 3, pp. 556-567, Mar. 2007 [4] B. P. Kumar and G. R. Branner, “Generalized analytical technique for the synthesis of unequally spaced arrays with linear, planar, cylindrical or spherical geometry”, IEEE Trans. Antennas and Propagation, vol. 53, no. 2, pp. 621-634, Feb. 2005 [5] K. Chen, Z. He, and C. Han, “A modified real GA for the sparse linear array synthesis with multiple constraints”, IEEE Trans. Antennas and Propapgation, vol. 54, no. 7, pp. 2169-2173, 2006 [6] S. DeLuccia and D. H. Werner, “Nature-based design of aperiodic linear arrays with broadband elements using a combination of rapid neural-network estimation techniques and genetic algorithms”, IEEE Magazine Antennas and Propagation, vol. 49, no. 5, pp. 13-23, Oct. 2007 [7] L. Cen, W. Ser, Z. L. Yu, and S. Rahardja, “An improved genetic algorithm for aperiodic array synthesis”, IEEE Int. Conf. Acoustics, Speech and Signal Processing, Las Vegas, Nevada, 31 Mar. - 4 Apr. 2008, pp. 2465-2468 [8] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing”, IEEE Trans. Signal Processing, vol. 44, no. 1, pp. 119-122, Jan. 1996 [9] R. Storn and K. Price, “Differential evolution- a simple and efficient adaptive scheme for global optimization over continuous spaces”, Technical Report TR-95-012, International Computer Science Institute, Berkley, CA, 1995 [10] K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution: a Practical Approach to Global Optimization, Springer, Berlin, 2005 [11] A. Qing, Differential Evolution: Fundamentals and Applications in Electrical Engineering, John Wiley & Sons, New York, 2009 [12] Y. Chen, S. Yang, and Z. Nie, “The application of a modified differential evolution strategy to some array pattern synthesis problems”, IEEE Trans. Antennas and Propagation, vol. 56, no. 7, pp. 1919-1927, July 2008 [13] A. Qing, “Electromagnetic inverse scattering of multiple two-dimensional perfectly conducting objects by the
Vol. 1, No. 1, April 2009
differential evolution strategy”, IEEE Trans. Antennas Propagat., vol. 51, no. 6, pp. 1251-1262, 2003 [14] A. Qing, “Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems”, IEEE Trans. Geoscience Remote Sensing, vol. 44, no. 1, pp. 116-125, 2006 [15] A. Qing, “A parametric study on differential evolution based on benchmark electromagnetic inverse scattering problem”, 2007 IEEE Congress Evolutionary Computation, Singapore, Sept. 25-28, 2007, pp. 19041909 [16] A. Qing, “A study on base vector for differential evolution”, 2008 IEEE World Congress Computational Intelligence/2008 IEEE Congress Evolutionary Computation, Hong Kong, June 1-6, 2008, pp. 550-556
Vol. 1, No. 1, April 2009
Synthesis of unequally Spaced Antenna Arrays by a new Differential Evolutionary Algorithm Chuan Lin1, 2, Anyong Qing2, Quanyuan Feng1 1
School of Information Science and Technology, Jiaotong University, Chengdu, China 610031 2 Temasek Laboratories, National University of Singapore, 5 Sports Drive 2, Singapore 117508 {tslqay, tslv6}@nus.edu.sg
Abstract: The differential evolution (DE) algorithm with a new differential mutation base strategy, namely best of random, is applied to the synthesis of unequally spaced antenna arrays. In the best of random mutation strategy, the best individual among three randomly chosen individuals is used as the mutation base while the other two are for the vector difference. Hence a good balance of diversity and evolution speed can be obtained. An indirect coding method is utilized to convert the constrained synthesis problem to an unconstrained one. The effect of angle resolution of the radiation pattern is also discussed. Simulation results demonstrate that the new DE algorithm is promising. Keywords: Antenna arrays, pattern synthesis, differential evolution, best of random, angle resolution
1. Introduction In array pattern synthesis, the main objective is to find an appropriate weighting vector and layout of the elements to yield the desired radiation pattern. Over the past few decades, the synthesis of uniformly spaced antenna arrays has been extensively studied. In recent years, the unequally spaced array, also termed as aperiodic array, has attracted increasing attention. Compared with the equally spaced array, the unequally spaced array has more degree of freedom and is able to lower the peak sidelobe level with smaller number of elements. The difficulties in the design of unequally spaced arrays are mainly caused by the nonlinear and non-convex dependency of the array factor to the weights and the sensor positions [1][2][3]. Various analytical and numerical array pattern synthesis techniques have been developed to meet the challenge[1]-[8]. Recently, evolutionary algorithms (EAs) including genetic algorithms (GAs)[5][6][7], simulated annealing (SA)[8], particle swarm optimization (PSO)[3], and so on, have become more popular as they are applicable not only to regular arrays (such as linear arrays and circular arrays) but also to arrays with complicated geometry layout and radiation pattern. Moreover, they can simultaneously optimize the weight vector and sensor positions. Differential evolution (DE) algorithm, introduced by Price and Storn, is a simple, efficient and robust evolutionary algorithm [9][10][11]. It has also been successfully applied to many electromagnetic fields, including array synthesis problems [2][12] and electromagnetic inverse problems[13][14]. A DE variant is usually marked as DE/x/y/z, where x denotes how the differential mutation base is chosen, y denotes the number of vector differences added to the base vector and z indicates the crossover method. Best and
random are the two most often implemented differential mutation bases. Although DE performs well as compared to many other methods, it still remains a challenge in DE to properly balance the exploration and exploitation. Through a comprehensive parametric study on DE, it has been observed that the balance of exploration and exploitation is greatly affected by differential mutation base strategies [11][15][16]. For example, guidance provided by the best differential mutation base often leads to higher efficiency at the cost of diversity while blind search by using random differential mutation bases gains diversity and sacrifices efficiency. Hence, using differential mutation base providing both guidance and diversity simultaneously might be a more promising solution to effectively balance exploration and exploitation. A new differential mutation base strategy, named “best of random” differential mutation base, is accordingly proposed. The new differential mutation strategy is very simple and introduces no additional intrinsic control parameter. Like random differential mutation, best of random differential mutation also randomly chooses three mutually different individuals. However, unlike random differential mutation, best of random differential mutation uses the best individual among the three individuals as differential mutation base while the other two worse individuals are donors for vector difference. DE algorithm with best of random mutation base is denoted as DE/BoR/1/*. Its advantages have been verified by our comprehensive simulations over a set of benchmark functions. In this paper, DE/BoR/1/* is applied to the unequally spaced array synthesis. Its efficiency in the array synthesis is also demonstrated by the numerical results. The rest of this paper proceeds as follows. Section II briefly describes differential evolution. Details of the new differential mutation base strategy are given in Section III. In Section IV, DE with best of random mutation base is applied to the array synthesis problem. Conclusion is given in Section V.
2. Differential evolution 2.1 Evolution Mechanism Classic DE optimizes an objective function with a population of Np individuals. It involves two stages, namely, initialization and evolution. Initialization generates initial population P0. Then the population evolves from one generation (Pn) to the next (Pn+1) until the termination conditions are met. While evolving from Pn to Pn+1, the three
21 International Journal of Communication Networks and Information Security (IJCNIS)
evolutionary operations, namely, differential mutation, crossover and selection are executed in sequence. The Fortran-style pseudo-code of classic differential evolution is shown in Fig. 1 where xn, i is the vector of optimization parameters of the ith individual pn,i in population Pn, xn+1,v,i is the vector of optimization parameters of mutant vn+1,i, xn,b,i is the vector of optimization parameters of differential mutation base bn,i, p1y and p2y are integer numbers, usually random, Fy, a constant usually in [0,1], is the mutation intensity for the yth vector difference n , p1 y
n, p
n , p2 y
n, p
, x 1 y and x 2 y are vectors of optimization n, p n, p parameters of donors p 1 y and p 2 y from Pn, xn+1,c, i is the vector of optimization parameters of child (or trial individual) cn+1,i. αji is a real random number in [0,1], [biL,biU] is the search space of the ith optimization parameter, xi. Initialization n=0 do i = 1, Np do j = 1, N
x
−x
x
0, i j
(
= b +α b −b L j
i j
U j
L j
)
(
y ≥1
n , p1 y
−x
n , p2 y
),
1 ≤ i ≠ p1 y ≠ p 2 y ≤ N p regularize infeasible mutant vn+1,i do j = 1, N n +1, v , i
do while ( x j
< b Lj )
x nj +1, v ,i = x nj +1, v ,i + bUj − b Lj end do n +1, v , i
do while ( x j
2.2 Differential Mutation Base Differential mutation base can be chosen in a variety of ways. Best and random are the two most often implemented differential mutation bases. DE/best/1/* uses the best individual pn,best in Pn as the differential mutation base while DE/rand/1/* randomly chooses the differential mutation base from Pn. 2.3 Binomial Crossover Binomial crossover and exponential crossover are two most often implemented crossover methods in DE. In this paper, binomial crossover is used. In binomial crossover scheme, cn+1,i is generated as follows:
x
n +1,c ,i j
where
x nj+1,v ,i β jn+1,i ≤ C r = n ,i otherwise x j
β jn +1,i
(1)
is a real random number uniform in the range
[0,1] and Cr, a constant in [0,1], is the crossover probability. There is an extreme case. The child duplicates vn+1,i. In this n,i
end do evaluate f(x0,i) end do Evolution do while termination conditions are not satisfied n=n+1 do i = 1, Np differential mutation to obtain mutant vn+1,i choose differential mutation base bn,i, n, p n, p donors p 1 y and p 2 y
x n +1, v ,i = x n , b ,i + ∑ F y x
Vol. 1, No. 1, April 2009
> bUj )
x nj +1, v ,i = x nj +1, v ,i − bUj + b Lj end do end do crossover mutant vn+1,i with pn,i to deliver child cn+1,i evaluate child f(xn+1,c, i) selection to get individual pn + 1,i if f(xn+1,c, i) < f(xn, i) pn + 1,i = cn+1,i else pn + 1,i = pn,i end if end do end do Figure 1 Fortran-style Pseudo-code of Classic Differential Evolution
case, a randomly chosen parameter of pn,i, x j , will replace n +1, c , i
the corresponding parameter of the child cn+1,i, x j
.
n+1,i
There is another extreme case. c inherits no parameter from vn+1,i and hence no evolution happens. In this case, a n +1, c , i
randomly chosen parameter of the child cn+1,i, x j
, will
be replaced by the corresponding parameter of the mutant n +1, v , i
vn+1,i, x j
.
2.4 Dynamic Differential Evolution Classic differential evolution is static in nature from the point of view of population updating and is therefore inherently inefficient. Inspired by the advantage of GaussSeidel method against Jacobi method for linear equations, the dynamic differential evolution (DDE) [14] was developed. The dynamic updating of population in DDE leads to a larger virtual population size and quicker response to change of population status. Thus in general, dynamic differential evolution outperforms classic differential evolution: faster and more reliable, just as the Gauss Seidel method does over the Jacobi method. For more details of dynamic differential evolution, please refer to [14].
3. Best of Random Differential Mutation Base How to balance exploration and exploitation is a crucial problem for most evolutionary algorithms including DE. In fact, this problem can also be understood from another point of view: the balance of diversity and evolution of the population. Diversity here means the dispersing of the individuals in the population, while evolution means constant improvement of the (average) performance of the population as they evolve from one generation to the next. Randomly distributing the population in a wide area helps to increase the diversity while guidance by the elite individuals often makes the search more efficient and hence promotes the evolution speed. Generally speaking, differential mutation operation in DE could be regarded as a local search. Differential mutation
22 International Journal of Communication Networks and Information Security (IJCNIS)
base vector acts as the local search center and the scaled vector difference determines the search range or search step around the center. The diversity of the population at the next generation is effected by the diversity of the local search centers (base vectors). The base vectors with high quality provide guidance making the search pay more attention to the promising regions, which often brings faster evolution speed. In DE/rand/1/*, the base vectors are randomly chosen from the population, thus a good diversity of base vectors is gained but no search guidance is provided. On the other hand, all the individuals in DE/best/1/* use the same base vector: the best vector so far. The guidance of the best individual generally leads to fast evolution speed but also increases the risk of stagnation due to the lack of enough diversity of base vectors. From the above analysis, we notice that differential mutation base providing both guidance and diversity should be a more promising choice to better balance the diversity and evolution of the population. Based on this idea, a new generator for differential mutation base, namely “best of random”, is proposed. Best of random differential mutation base uses the best individual among the three randomly chosen individuals as differential mutation base and the other two worse individuals as donors for vector difference. Thus both good quality and diversity of the base vectors can be obtained. Best of random mutation strategy can be expressed as:
x n+1,v ,i = x where x
n , p1 y
n, pb
, x
(
+ F x n , p1 − x n , p2
n, p2 y
and x
n, p b
)
(2)
are vectors of optimization
parameters of 3 randomly chosen individuals p n, p b
n , p1
, p
n , p2
,
, and f ( x ) ≤ min( f (x ), f (x )) . It can be easily found that DE/BoR/1/* is almost as simple as DE/rand/1/* and introduce no extra intrinsic control parameters.
p
n, pb
n , p1
n , p2
4. Array Synthesis Using DE with Best of Random Mutation Strategy 4.1 Description of Array Synthesis problem The array factor AF(θ) of a linear antenna array with N elements at angle θ can be expressed as [2]: N
AF (θ ) = ∑ I i e
j(
2π
λ
xi sin(θ ) +φi )
(3)
i =1
Where Ii and
φi
are the excitation amplitude and phase of
the element located at xi, respectively. λ is the wavelength. In this paper, the uniform amplitude excitation is considered as it is an effective way to reduce the system costs and hardware implementation complexity [2]. Thus Eq. (2) becomes: N
AF (θ ) = ∑ e i =1
j(
2π
λ
xi sin(θ ) +φi )
(4)
Vol. 1, No. 1, April 2009
4.2 Objective Function for Optimization of Sidelobe Level The task of the optimization in this paper is to minimize the sidelobe level (SLL) of the antenna array. The objective function to be minimized can be written as [2]:
f ( ρ ) = Maxθ∈S
AFρ (θ )
AFρ (θ 0 )
(5)
where S is the space spanned by the angle θ excluding the mainlobe and ρ represents the unknown parameter vector, such as element positions and phases. This objective function minimizes all the sidelobe levels and maximizes the power in the mainlobe located at θ = θ 0 . In this paper, θ 0 = 0◦.
4.3 Position-Phase Synthesis of Unequally Spaced Arrays In this subsection, position-phase synthesis of a 32element symmetric array with uniform amplitude excitation[2] is carried out by using DDE/BoR/1/bin. Due to the symmetry, the number of optimization parameters is 32, including 16 position parameters and 16 phase parameters. The other half position and phase parameters can be calculated by:
x−i = − xi , φ−i = φi
1 ≤ i ≤ 16
(6)
The distance between adjacent elements is restricted in [0.5λ, 1.0λ], that is:
0.25λ ≤ d1 = x1 ≤ 0.5λ 0.5λ ≤ d i = xi − xi −1 ≤ λ , 2 ≤ i ≤ 16
(7)
4.3.1 Coding And Decoding of optimization parameters As the element positions are subject to the above constraints, some special method is needed to tackle the constraints and ensure feasible solution if we directly optimize the element positions. For example, in [5] the coding resetting of gene variables and broad sense genetic operators are utilized to deal with the constraints, which makes the algorithm a little bit more complex. Instead of directly optimizing the element positions, we optimize the distance of the adjacent elements di. Thus the encoded optimization parameter vector ρ = [d1, d2,…, d16, Φ1, Φ2,…, Φ16], with –π ≤ Φi ≤ π and the search range for di being defined by (7). In this way the constrained optimization problem is converted into unconstrained optimization problem, and DE can be directly applied to antenna synthesis without any modification. After di have been obtained, xi can be calculated as follows:
x1 = d1 xi = xi −1 + d i , 2 ≤ i ≤ 16
(8)
23 International Journal of Communication Networks and Information Security (IJCNIS)
4.3.2 Parameter Settings for the Experiments The known best SLLs for this example reported in [2] and [12] are -23.34dB and -23.45dB, respectively. Fortuitously, it is noticed that the calculated SLL depends on the resolution or the sampling interval of angle θ due to ripples or spines. As the angle resolution becomes smaller, some ripples or spines in the sidelobe may become visible while they are invisible under the larger angle resolution. Under different angle resolutions ∆, the SLLs for this example computed with the optimization parameters given in [2] and [12] are shown in Table 1 in which inconsistence of SLLs under different angle resolutions are quite obvious.
Table 1 SLLs computed with the optimization parameters in [2] and [12] ∆ (◦)
SLL (dB) in [2]
SLL (dB) in [12]
0.1
-23.22
-23.13
0.2
-23.22
-23.16
0.5
-23.23
-23.37
1.0
-23.25
-23.37
2.0
-23.25
-23.46
It is observed from Table 1 that the results from ∆=0.2◦ downwards are trustable while those from ∆=0.5◦ onwards are not. However, to study the effect of angle resolution on the performance of the algorithm, two different angle resolutions, i.e., ∆=0.2◦ and ∆=0.5◦, are used in our simulation. According to our experiment and the results reported in [2] and [12], it is found that SLL=-23.3dB is very close to the optimal SLL of the array even though neither of the two reported results actually reaches it. In order to see the capability of DDE/BoR/1/bin, SLL = -23.3dB is used as the value to reach (VTR) for the algorithm. The maximum number of objective function evaluations NEmax is 500,000. If the obtained SLL meets the VTR or the number of objective function evaluations (NFE) reaches NEmax, the algorithm is terminated. 100 independent runs are repeated for each angle resolution. A run is said to be successful if the algorithm can find the VTR within NEmax objective function evaluations, otherwise it is unsuccessful. The successful rate rs = NS/100, where NS is the number of successful runs. The control parameters of DDE/BoR/1/bin in this example are: Np=60, F=0.4, Cr=0.7.
4.3.3 Simulation Results The simulation results of the array synthesis with ∆=0.5◦ and ∆=0.2◦ are shown in Table 2. Where Emin, Eavg, Emax are the minimal, average and maximum number of objective function evaluations of the successful runs, respectively. SLLw is the obtained worst SLL in the 100 independent runs. The distribution of the NFE and SLLs in the 100 independent runs for ∆=0.5◦ and ∆=0.2◦ are shown in Figs. 2~5.
Table 2 Simulation results of position-phase array synthesis using DE/BoR/1/bin rs
∆=0.5◦ 0.92
∆=0.2◦ 0.74
Vol. 1, No. 1, April 2009
Emin
27411
35784
Eavg Emax SLLw (dB)
73808 266545 -21.74
111493 455870 -22.47
SLLdavg(dB)
0.290
0.066
In order to see the effect of angle resolution, the returned optimization parameters from 100 independent runs with ∆=0.5◦ are reused to calculate the SLLs with the angle resolution ∆=0.2◦. The SLL differences between the new SLLs with ∆=0.2◦ and the original SLLs with ∆=0.5◦ are shown in Fig. 6. Similarly, the returned optimization parameters with ∆=0.2◦ are reused to calculate the SLLs with ∆=0.1◦. The SLL differences between the new SLLs with ∆=0.1◦ and the original SLLs with ∆=0.2◦ are shown in Fig. 7. The average sidelobe level difference SLLdavg when using the same optimization parameters but different angle resolutions over the 100 runs in the two cases are shown in the last row of Table 2. From the simulation results, it has been observed that the returned SLL with angle resolution ∆=0.5◦ is not very much trustable as the SLL will be increased by 0.290dB in average when it is recalculated using the same optimization parameters with ∆=0.2◦. If ∆=0.2◦ is used in the evaluation of the objective function, the returned SLL is more trustable as the SLL is only increased by 0.066dB when it is recalculated using the same optimization parameters with ∆=0.1◦. However, by decreasing the angle resolution, the success rate of the algorithm decreases and the number of objective function evaluations increases. The computation time of the function evaluation is also increased by using smaller angle resolution. In the unsuccessful runs, most of the SLLs with ∆=0.5◦ and all the SLLs with ∆=0.2◦ are lower than -22 dB. This means although the algorithm may not converge to the optimal SLL sometimes, it can still reach the suboptimal SLL. For verification purpose, one example of returned positions and phases with SLL=-23.3dB is shown in Table 3. The corresponding radiation pattern is shown in Fig. 8.
Table 3 Element positions and phases for a symmetric 32element antenna array xi/λ (1≤i≤8) Φi◦(1≤i≤8) xi/λ (9≤i≤16) Φi◦ (9≤i≤16) 0.2535 0.7825 1.2992 1.8333 2.3342 2.9305 3.4813 4.0867
52.68 56.59 53.35 53.54 55.63 53.81 52.12 56.67
4.7599 5.2821 6.0467 6.8861 7.7974 8.6678 9.6504 10.5287
55.18 53.04 55.09 41.82 56.08 83.13 23.62 55.39
24 International Journal of Communication Networks and Information Security (IJCNIS)
0.5
SLL difference (dB)
400000
300000
NFE
Vol. 1, No. 1, April 2009
0.6
500000
200000
100000
0.4
0.3
0.2
0.1
0.0
0 20
40
60
80
20
100
40
60
80
100
Index of runs
Index of runs
Figure 6 Differences between the SLLs with ∆=0.2◦ and that with ∆=0.5◦
Figure 2 Distribution of NFE when ∆=0.5◦ -21.6
0.12 -21.8
0.10
-22.0
Difference of SLL (dB)
SLL (dB)
-22.2 -22.4 -22.6 -22.8 -23.0 -23.2 -23.4 20
40
60
80
0.08
0.06
0.04
0.02
0.00
100
20
Index of runs
40
60
80
100
Index of runs
Figure 7 Differences between the SLLs with ∆=0.1◦ and that with ∆=0.2◦
Figure 3 Distribution of SLLs when ∆=0.5◦ 500000
0 400000
-10 -20 |AF(θ)/AF(0)|
NFE
300000
200000
100000
-30 -40 -50
0 20
40
60
80
100
Index of runs
-60
Figure 4 Distribution of NFE when ∆=0.2◦
-80
-60
-40
-20
0
20
40
60
80
θ (deg)
Figure 8 Radiation pattern for a symmetric 32-element array -22.4
5. Conclusion -22.6
SLL (dB)
-22.8
-23.0
-23.2
-23.4 20
40
60
80
Index of runs
Figure 5 Distribution of SLLs when ∆=0.2◦
100
The new DE algorithm with best of random mutation base strategy is applied to the position-phase synthesis of the unequally spaced antenna array. With an indirect coding method, the constrained optimization problem is converted into an unconstrained problem. In order to study the effect of the angle resolution on the performance of the algorithm, two different angle resolutions are used for the evaluation of the objective function respectively. By comparing the SLLs obtained with the same returned optimization parameters but different resolutions, it is observed that the results obtained with angle resolution ∆=0.2◦ is more trustable compared with those with ∆=0.5◦, though smaller angle resolution may
25 International Journal of Communication Networks and Information Security (IJCNIS)
also lead to slower convergence rate and lower success rate. From the simulation results, it is demonstrated that the new DE algorithm is promising in the array synthesis problem.
References [1] B. P. Kumar and G. R. Branner, “Design of unequally spaced arrays for performance improvement”, IEEE Trans. Antennas and Propagation, vol. 47, no. 3, pp. 511-523, Mar. 1999 [2] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna arrays using the differential evolution algorithm”, IEEE Trans. Antennas and Propagation, vol. 51, no. 9, pp. 22102217, 2003 [3] N. Jin and Y. Rahmat-Samii, “Advances in Particle Swarm Optimization for Antenna Designs: RealNumber, Binary, Single-Objective and Multiobjective Implementations”, IEEE Trans. Antennas and Propagation, vol. 55, no. 3, pp. 556-567, Mar. 2007 [4] B. P. Kumar and G. R. Branner, “Generalized analytical technique for the synthesis of unequally spaced arrays with linear, planar, cylindrical or spherical geometry”, IEEE Trans. Antennas and Propagation, vol. 53, no. 2, pp. 621-634, Feb. 2005 [5] K. Chen, Z. He, and C. Han, “A modified real GA for the sparse linear array synthesis with multiple constraints”, IEEE Trans. Antennas and Propapgation, vol. 54, no. 7, pp. 2169-2173, 2006 [6] S. DeLuccia and D. H. Werner, “Nature-based design of aperiodic linear arrays with broadband elements using a combination of rapid neural-network estimation techniques and genetic algorithms”, IEEE Magazine Antennas and Propagation, vol. 49, no. 5, pp. 13-23, Oct. 2007 [7] L. Cen, W. Ser, Z. L. Yu, and S. Rahardja, “An improved genetic algorithm for aperiodic array synthesis”, IEEE Int. Conf. Acoustics, Speech and Signal Processing, Las Vegas, Nevada, 31 Mar. - 4 Apr. 2008, pp. 2465-2468 [8] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing”, IEEE Trans. Signal Processing, vol. 44, no. 1, pp. 119-122, Jan. 1996 [9] R. Storn and K. Price, “Differential evolution- a simple and efficient adaptive scheme for global optimization over continuous spaces”, Technical Report TR-95-012, International Computer Science Institute, Berkley, CA, 1995 [10] K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution: a Practical Approach to Global Optimization, Springer, Berlin, 2005 [11] A. Qing, Differential Evolution: Fundamentals and Applications in Electrical Engineering, John Wiley & Sons, New York, 2009 [12] Y. Chen, S. Yang, and Z. Nie, “The application of a modified differential evolution strategy to some array pattern synthesis problems”, IEEE Trans. Antennas and Propagation, vol. 56, no. 7, pp. 1919-1927, July 2008 [13] A. Qing, “Electromagnetic inverse scattering of multiple two-dimensional perfectly conducting objects by the
Vol. 1, No. 1, April 2009
differential evolution strategy”, IEEE Trans. Antennas Propagat., vol. 51, no. 6, pp. 1251-1262, 2003 [14] A. Qing, “Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems”, IEEE Trans. Geoscience Remote Sensing, vol. 44, no. 1, pp. 116-125, 2006 [15] A. Qing, “A parametric study on differential evolution based on benchmark electromagnetic inverse scattering problem”, 2007 IEEE Congress Evolutionary Computation, Singapore, Sept. 25-28, 2007, pp. 19041909 [16] A. Qing, “A study on base vector for differential evolution”, 2008 IEEE World Congress Computational Intelligence/2008 IEEE Congress Evolutionary Computation, Hong Kong, June 1-6, 2008, pp. 550-556
