Orthogonal Genetic Algorithm for Planar Thinned Array Designs

Abstract

An orthogonal genetic algorithm (OGA) is applied to optimize the planar thinned array with a minimum peak sidelobe level. The method is a genetic algorithm based on orthogonal design. A crossover operator formed by the orthogonal array and the factor analysis is employed to enhance the genetic algorithm for optimization. In order to evaluate the performance of the OGA, 20×10-element planar thinned arrays have been designed to minimize peak sidelobe level. The optimization results by the OGA are better than the previously published results.

1. Introduction

Aperiodic thinned array is of great importance in practical applications because of the reduction of the array costs, weight, and power consumption so that it is very active topic in electromagnetics (EM) community [1–7]. However, such advantages usually come at the cost of a loss of sidelobe level and gain compared to the filled array. In the past, several methods [1, 2] had been employed to overcome these drawbacks. However, the synthesis problem is complex and cannot be well solved with these methods. In the 1990’s, cyclic difference set (CDS) was applied to design massively thinned arrays [3]. In recently years, many global optimization algorithms have been introduced to design thinned arrays, such as genetic algorithm (GA) [4, 5], a hybrid approach based on genetic algorithm (GA) and difference sets (CDS) [6], and ant colony optimization (ACO) [7]. These evolutionary algorithms have been used to arrive at thinned arrays featuring a specified peak sidelobe level (PSLL).

Orthogonal design, an experimental design method, has been applied to enhance evolutionary algorithm for optimization [8–11]. Zhang and Leung [8] proposed an orthogonal genetic algorithm by applying orthogonal design to enhance the crossover operator in genetic algorithm for multimedia multicast routing problems. To solve multidimensional knapsack problems, Li et al. [11] present a new genetic algorithm, in which the orthogonal design with factor analysis is incorporated into the genetic algorithm. Numerical results in [8, 11] demonstrate that these methods are an efficient and effective technique for solving 0-1 integer programming problems.

In this paper, an orthogonal genetic algorithm (OGA) is utilized to optimize the planar thinned array with a minimum PSLL. The method is a genetic algorithm based on orthogonal design. A crossover operator formed by the orthogonal array and the factor analysis is employed to enhance the genetic algorithm for optimization. The optimization of a thinned array can be formulated as a 0-1 integer optimization problem. The OGA adopts binary code so that it is very suitable for solving 0-1 integer programming problems. 20×10-element planar thinned arrays are designed to demonstrate the effectiveness of the OGA. Numerical results show that the OGA yields better results than previously published results.

2. Orthogonal Experiment Design with Factor Analysis

In this section, we briefly introduce the concept of experimental design methods. For more details, the readers can refer to [8, 9]. To aid explanation, we consider a simple example. The yield of a chemical product depends on three variables: temperature, time, and amount of catalyst at the values shown in Table 1. The temperature, time, and amount of catalyst are called the factors of the experiment. Each factor has two possible values, and we say that each factor has two levels.

Table 1 

An experiment design problem with three factors and two levels per factor.

To find the best combination of levels for maximum yield, we can do one experiment for every combination and then select the best one. In the above example, there are 2×2×2=8 combinations, and hence there are 8 experiments. In general, when there are 𝑁 factor at 𝑄 levels, the number of combinations is 𝑄^𝑁. When 𝑁 and 𝑄 are small, it may be feasible to test all the combinations. When 𝑁 and 𝑄 are large, it may not be possible to test all the combinations. Therefore, it is desirable to sample a small, but representative set of combinations for experimentation. The orthogonal design was developed for this purpose.

The orthogonal design provides a series of orthogonal arrays for different 𝑁 and 𝑄. We let 𝐿_𝑀(𝑄^𝑁) denote an orthogonal array for 𝑁 factors and 𝑄 levels, where 𝐿 stands for Latin square and 𝑀 represents a combination of levels. For convenience, we denote 𝐿_𝑀(𝑄^𝑁)=[𝑎_𝑖,𝑗]_𝑀×𝑁, where the 𝑗th factor in the 𝑖th combination has level 𝑎_𝑖,𝑗 and 𝑎_𝑖,𝑗∈{1,2,…,𝑄}. The following is an orthogonal array:𝐿₄2³=𝑎_{𝑖,𝑗4×3}=111122212221. (1)

In 𝐿₄(2³), there are three factors, two levels per factor, and four combinations of levels. In the first combination, the three factors have respective levels 1, 1, 1; in the second combination, the three factors have respective levels 1, 2, 2, and so forth. Based on 𝐿₄(2³), we get a sample of 4 combinations out of 2³ combinations. In this example, there are 8 combinations of levels. We apply the orthogonal array 𝐿₄(2³) to select four representative combinations to be tested, and these four combinations are shown in Table 2.

Table 2 

Based on the 𝐿₄(2³), 4 combinations are selected form 8 combinations for experiments.

Two-level orthogonal arrays are used in this study. There are 𝑁 factors, where 𝑁 is the number of design factors (variables), and each factor has two levels. To establish an orthogonal array of 𝑁 factors with two levels, let 𝐿_𝑛(2^𝑛−1) denote 𝑛−1 columns and 𝑛 individual experiments corresponding to the 𝑛 rows, where 𝐿 stands for Latin square, 𝑛=2^𝑘, 𝑘 positive integer (𝑘>1), and 𝑁≤𝑛−1. More details about generating a two-level orthogonal array are available in [8, 9].

After evaluation of the 𝑛 combinations, the summarized data are analyzed using factor analysis. Factor analysis can evaluate the effects of individual factors on the objective (or fitness) function, rank the most effective factors, and determine the better level for each factor such that the function is optimized. In this study, the factor analysis in [10] is used. Let 𝑦_𝑡 denote a function value of the combination 𝑡, where 𝑡=1,…,𝑛. Define the main effect of factor 𝑗 with level 𝑘 as 𝐸_𝑗𝑘, where 𝑗=1,…,𝑁 and 𝑘=1,2:𝐸_𝑗𝑘=^𝑛_𝑡=1𝑦_𝑡⋅𝐹_𝑡, (2) where 𝐹_𝑡=1 if the level of factor 𝑗 of combination 𝑡 is 𝑘; otherwise, 𝐹_𝑡=0. Considering the case that the optimization objective is to be minimized, the level 1 of factor 𝑗 makes a better contribution to the function than level 2 of factor 𝑗 does when 𝐸_𝑗1<𝐸_𝑗2. If 𝐸_𝑗1>𝐸_𝑗2, level 2 is better. On the contrary, if the objective is to be maximized, level 1 (level 2) is better when 𝐸_𝑗1>𝐸_𝑗2(𝐸_𝑗1<𝐸_𝑗2). If 𝐸_𝑗1=𝐸_𝑗2, levels 1 and 2 have the same contribution. After the better one of two levels of each factor is determined, a reasoned combination consisting of 𝑁 factors with better levels can be easily derived.

3. Orthogonal Genetic Algorithm

In this section, the orthogonal genetic algorithm is introduced.

A crossover operator formed by the orthogonal array and the factor analysis is employed to enhance the performance of genetic algorithm. The application of the orthogonal experiment design and factor analysis in the crossover operator is firstly illustrated by a simple example. Here, we choose the test function 𝑓(𝑥)=∑⁷_𝑖=1𝑥²_𝑖, where 𝑥_𝑖∈{0,1}. The test function would be minimized.

First, two individuals 𝑥^′ and 𝑥^′′, each consists of seven factors corresponding to 𝑥_𝑖 of 𝑓(𝑥), are randomly chosen to execute various matrix experiments of an orthogonal array. Regarded as level 1, the values of seven factors in individual 𝑥^′ are 1, 1, 1, 1, 0, 0, and 0; those in individual 𝑥^′′ are 0, 0, 0, 0, 1, 1, and 1, which are regarded as level 2. Therefore, 𝑓(𝑥^′)=4 and 𝑓(𝑥^′′)=3. The orthogonal array 𝐿₈(2⁷) has been chosen, because each individual has seven variables (factors). Then, seven variables in individuals 𝑥^′ and 𝑥^′′ corresponding to the factors A, B, C, D, E, F, and G, respectively, are defined in Table 3.

Table 3 

Corresponding factors of 𝑥^′ and 𝑥^′′.

Next, as shown in Table 4, the values of level 1 and level 2 are assigned to the level cells in the orthogonal array 𝐿₈(2⁷). The function values of each row (offspring individual) are then calculated. Then, formulation (2) is applied in Table 4 to determine the optimum level for each factor. The function value of an optimum individual found by OGA is zeros, which is much better than its parent individuals 𝑥^′ and 𝑥^′′ with 𝑓(𝑥^′)=4 and 𝑓(𝑥^′′)=3. It is obvious that instead of executing all 2⁷ combinations of factor levels, the orthogonal design with factor analysis can offer an efficient approach toward finding the optimum individual by only executing eight experiments.

Table 4 

Generating a better individual from two individuals by orthogonal experiment design and factor analysis.

The pseudocode of OGA is given as shown in Algorithm 1.

Initialize: population size NP, crossover rate 𝑃crossover, mutation rate 𝑃mutation, maximal number of generations 𝐺max and current generation counter 𝑔 Initialize randomly the initial population 𝑃0 of size NP binary strings Evaluate each individual 𝑥∈𝑃0 WHILE stopping rule is not invoked DO    Select a suitable two-level orthogonal array 𝐿𝑛(2𝑛−1) for matrix experiments    // Orthogonal Crossover Operation    FOR 𝑖=1 to NP      IF random(0,1)<𝑃crossover       Select two random parents 𝑥𝑎,𝑥𝑏∈𝑃𝑔       Execute the matrix experiment for 𝑥𝑎 and 𝑥𝑏       Calculate the function values of individuals generated by the matrix experiment       Calculate the effects of the various factors (𝐸𝑗1,𝐸𝑗2) by using factor analysis,       Generate an optimal individual 𝑥†𝑐      END IF     𝑋𝑐←𝑥†𝑐    END FOR     NP_temp=size(𝑃𝑔∪𝑋𝑐)     // Mutation Operation     FOR 𝑖=1 to NP_temp      IF random(0,1)<𝑃mutation       Select one random individual 𝑥𝑚∈𝑃𝑔∪𝑋𝑐, and choose randomly two gene bits in this individual       Change these two gens from 0 to 1 or vice versa, and generate a new individual 𝑥†𝑚      END IF        𝑋𝑚←𝑥†𝑚     END FOR    // Select Operation      Sort 𝑃𝑔∪𝑋𝑐∪𝑋𝑚      Copy the NP first solutions of 𝑃𝑔∪𝑋𝑐∪𝑋𝑚, and retain the best individual END WHILE

Algorithm 1 

The pseudocode of OGA.

4. Problem Formulation

Consider a planar thinned array consisting of 2𝑀×2𝑁 identical elements with half-wavelength spaced, which is symmetric about the 𝑥-axis and 𝑦-axis. Assume that the array elements are all isotropic. Radiation pattern of the array is calculated by the standard expression of the array factor:𝐹(𝐖,𝑢,𝑣)=4^𝑀𝑁_𝑚=1𝑛=1𝑤_𝑚𝑛cos[𝜋(2𝑚−1)𝑢]⋅cos[𝜋(2𝑛−1)𝑣], (3) where 𝑢=sin𝜃cos𝜑 and 𝑣=sin𝜃sin𝜑 are the direction parameters, and 𝐖={𝑤_𝑚𝑛,𝑚=1,2,…,𝑀;𝑛=1,2,…,𝑁}, 𝑤_𝑚𝑛=1or0 if an element is “ON” or “OFF” at the location (𝑚,𝑛).

The number of radiating elements in a thinned array depends on the array fill factor that represents the fraction of the turned ON elements in relation to the total number of elements and its value ranges between zero and one. The array fill factor is expressed as ∑^𝑀_𝑚=1∑^𝑁_𝑛=1𝐖_𝑚𝑛/𝑀×𝑁. Therefore, the array thinning factor (i.e., the fraction of the turned OFF elements in relation to the total number of elements) can be written as follows:1Γ=1−𝑀×𝑁^𝑀𝑁_𝑚=1𝑛=1𝐖_𝑚𝑛. (4)

The peak sidelobe level of the planar thinned array can be formulated as follows:𝑓_planar(𝐖)=max_{()𝑢,𝑣∈𝑆}𝐹(𝐖,𝑢,𝑣)𝐹𝐹_max, (5) where 𝑆 denotes the sidelobe region and 𝐹𝐹_max is the peak of the main beam.

To suppress PSLL in all 𝜑 planes, the fitness function can be defined as𝑓(𝐖)=𝑓_planar(𝐖). (6)

In [4], the fitness function is the sum of the maximum PSLL in the 𝜑=0^∘ and 𝜑=90^∘ planes. That is,𝑓(𝐖)=𝑓_planar(𝐖)_𝜑=0∘+𝑓_planar(𝐖)_𝜑=90∘. (7)

According to [7], the fitness function is written by𝑓(𝐖)=max𝑓_planar(𝐖)_𝜑=0∘,𝑓_planar(𝐖)_𝜑=90∘. (8)

Hence, a planar thinned array synthesis can be modeled into the following optimization problem: min𝑓(𝐖) (9) is solved by means of OGA to find the optimal array geometry, that is, the optimal amplitude weights 𝐖={𝑤_𝑚𝑛,𝑚=1,2,…,𝑀;𝑛=1,2,…,𝑁}.

5. Numerical Results and Comparisons

In order to assess the capabilities of the OGA, comparisons with the GA [4], MGA [5], and ACO [7] in thinning 20×10-element planar arrays have been performed in this section. Parameter settings of the OGA are as follows. Throughout this paper, population size 𝑁𝑃=50; crossover probability 𝑃_crossover=0.3; mutation probability 𝑃_mutation=0.1; and the maximum number of generations is set to 𝐺_max=50. Due to the randomness nature, the algorithm is run 20 times independently for each problem and the best results are presented in this section.

In the first case, the planar thinned array was optimized by the GA [4] and MGA [5]. Equation (7) is selected as fitness function. The fitness value of the optimal solution by the GA [4] and MGA [5] was −39.83 dB and −45.456 dB, respectively. Here, the OGA is employed to optimize the planar thinned arrays. In order to maintain a fair comparison with published result [4, 5], the array with 54% filled is constrained and the aperture size of the array is 9.5𝜆×4.5𝜆. Figure 1 shows the radiation patterns and the corresponding geometry of the optimized array by OGA. The best fitness value is −51.18 dB (PSLL = −26.09 dB in 𝜑=0^∘ plane, and PSLL = −25.09 dB in 𝜑=90^∘ plane). As shown in Table 5, the optimal result by OGA is 11.35 dB and 5.724 dB lower than that of the best array in [4, 5], respectively. The optimal array by OGA has a directivity of 24.66 dB. In comparison, the array optimized by the GA has a directivity of 24.64 dB and the array optimized by the MGA has a directivity of 25.07 dB. Compared with the array by MGA, the directivity of the array by OGA is reduced by 0.59 dB. The convergence curve of the algorithm is plotted in Figure 2. Figure 3 reports the resultant fitness value of 20 runs. It can be seen from Figure 3 that, with the same maximum number of iterations, the fitness values obtained by OGA vary in the range from −47.93 dB to −51.18 dB. The worst-case fitness value found by OGA is also lower than that by the GA [4] and MGA [5].

Table 5 

Comparison with the GA [4] and MGA [5] for the planar array.

Figure 1 

Radiation patterns of the optimized 20×10-element planar array with the fitness function as (7).

Figure 2 

The convergence curve of the OGA for the planar array with the fitness function as (7).

Figure 3 

The fitness value of the objective function with (7) obtained by the OGA in 20 runs.

Another optimization problem is the reduction of PSLLs in all 𝜑 planes. To maintain a fair comparison with published result in [5], the array with 50% filled is optimized by OGA. Equation (6) is acted as the fitness function. Figure 4 shows the three-dimension radiation pattern of the optimal array by OGA. Figure 5 is the radiation patterns in 𝜑=0^∘, 𝜑=45^∘, and 𝜑=90^∘ planes and the corresponding geometry of the optimal array by OGA. The corresponding array pattern in Figure 4 has a directivity of 24.82 dB with a maximum PSLL 19.44 dB below the main beam. In comparison with the results in [5], the peak PSLL is reduced by 0.6 dB.

Figure 4 

Three dimension radiation pattern of the optimized array for reduction PSLL in all 𝜑 planes.

Figure 5 

Radiation patterns in 𝜑=0∘, 𝜑=45∘ and 𝜑=90∘ planes.

The second thinned planar array with published results is the same array geometry as before but with different fitness function. Equation (8) is selected as the fitness function. In order to maintain a fair comparison with published result [7], the thinning percentage of the array is not constrained. Figure 6 shows the radiation patterns and the corresponding geometry of the optimized array. The best OGA synthesized array yields PSLL = −28.34 dB in 𝜑=0^∘ plane, and PSLL = −26.59 dB in 𝜑=90^∘ plane, thus improving the PSLL of the best results (PSLL = −25.76 dB in 𝜑=0^∘ plane, and PSLL = −25.67 dB in 𝜑=90^∘ plane reported in [7]) in the literature of about 2.58 dB and 0.92 dB, respectively. The thinning percentage of the optimal array is 42%, which is better than 32% of the optimal array by the ACO [7]. The optimal array by OGA has a directivity of 25.44 dB. Compared with the array optimized by the ACO, the directivity of the optimal array is reduced by 0.55 dB. As shown in Table 6, it can be found that the results by OGA are better than that by the ACO [7] in terms of both the thinning percentage and PSLL. It can be seen from Figure 7 that the algorithm has very fast convergence speed (about 11th iteration). Moreover, Figure 8 reveals that out of 20 independent trials, there are 14 times for which OGA can find the best result, and even the worst result found by OGA is also equal to the best result of ACO [7].

Table 6 

Comparison with the ACO [7] for the planar array.

Figure 6 

Radiation patterns of the optimized 20×10-element planar array with the fitness function as (8).

Figure 7 

The convergence curve of the OGA for the planar array with the fitness function as (8).

Figure 8 

The fitness value of the objective function with (8) obtained by the OGA in 20 runs.

6. Conclusion

In this paper, the orthogonal genetic algorithm is employed to design thinned array with a minimum PSLL. The method is a genetic algorithm based on orthogonal design. The performance of the OGA has been evaluated by synthesis 20×10-element planar arrays. The experiment results show that the OGA yields lower PSLL than the results reported in literature.

Acknowledgment

This work was supported by the Fundamental Research Funds for the Central Universities (K50511020007).