^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

A multiobjective approach based on the third evolution step of generalized differential evolution (GDE3) algorithm is proposed for optimizing the time-modulated array (TMA) in this paper. Different from the single-objective optimization, which optimizes a weighted sum of the peak sidelobe level (PSLL) and the peak sideband level (PSBL) of the array, the multiobjective algorithm treats the PSLL and the PSBL as two distinct objectives that are to be optimized simultaneously. Furthermore, not only one outstanding optimization result can be acquired but also a set of solutions known as Pareto front is obtained by using the GDE3 algorithm, which will guide the design of time-modulated array more effectively. Users can choose one appropriate outcome which has a suitable tradeoff between the PSLL and the PSBL. This approach is illustrated through a time-modulated concentric circular ring array (CCRA). The optimal parameters and the corresponding radiation patterns are presented at last. Experimental results reveal that the multiobjective optimization can be an effective approach for the TMA synthesis problems.

The time-modulated array (TMA) was proposed in 1959 by Shanks and Bickmore [

As mentioned above, the TMA has been optimized by different kinds of optimization algorithms. In these algorithms, single objective is optimized, which means only one best result can be concluded after optimization. However, the electromagnetic optimization objectives are often in conflict with each other, and there may not exist a solution that is the global best one. In fact, there are a set of solutions known as Pareto front or nondominated solutions [

In this paper, an approach based on the third evolution step of generalized differential evolution (GDE3) [

The rest of this paper is organized as follows. Section

The configuration of _{r}_{n}_{n}

The configuration of the CCRA.

By decomposing into Fourier series with different frequency components (

At the center frequency (

It can be seen from the functions that the far-field array factor of the time-modulated CCRA can be adjusted through controlling the normalized switch-on time sequence

To optimize the PSLL of the time-modulated CCRA, and to get the relationship between the PSLL and the PSBL, the multiobjective evolutionary algorithm GDE3 [_{max} = 2000. The first design objective is the PSLL, and the second design objective is the PSBL. These two objects are incompatible objective functions. The algorithm is run for 10 times independently for each problem, and the best results are presented in the next section.

The flowchart of the GDE3 algorithm.

Consider a uniformly excited nine-ring CCRA presented in [

Ring spacing and numbers of elements for case 1.

Ring number | Ring spacing ( |
Number of elements |
---|---|---|

1 | 0.5 | 6 |

2 | 1.0 | 12 |

3 | 1.5 | 18 |

4 | 2.0 | 25 |

5 | 2.5 | 31 |

6 | 3.0 | 37 |

7 | 3.5 | 43 |

8 | 4.0 | 50 |

9 | 4.5 | 56 |

To realize low PSLL, three examples are presented in this paper. Firstly, the normalized switch-on time sequence is optimized in case 1, with the ring spacing and numbers of elements unchanged.

To reduce the PSLL, a thin nine-ring CCRA is considered as case 2. The ring spacing remains unchanged like case 1. The spacing between adjacent elements within the same ring and the normalized switch-on time sequence are optimized simultaneously.

In case 3, to realize much lower PSLL with acceptable PSBL, the ring spacing, the number of elements, and the normalized switch-on time sequence for each ring are all optimized. For comparison, the six-ring CCRA presented in [

As is known, the Pareto front of biobjective problem offers a set of solutions that fulfill the aforementioned constrain. So in each case, three representative individuals, denoted by A, B, and C, are selected for the design. Individual A has the lowest PSBL, and individual B has the lowest PSLL. In consideration of practical using and comparison, the PSLL of individual C is as low as possible if the PSBL is close to −20 dB. The numerical results of three examples are shown in the following.

The Pareto front of case 1 is shown in Figure

The Pareto front for case 1.

The performances and the optimized normalized switch-on time sequences of the three representative individuals are listed in Tables ^{1} has the lowest PSBL, but its PSLL is also the highest. For individual B^{1}, it has the lowest PSLL but with the highest PSBL. There is a tradeoff between the PSLL and the PSBL for individual C^{1} with PSLL = −25.4 dB and PSBL = −20.14 dB. Compared with the conventional nine-ring CCRA in [^{1} has much lower PSLL whilst it has acceptable PSBL, though the FNBW becomes little larger. As is shown in Table ^{1} and C^{1} show. It indicates that the tapered normalized switch-on time sequence is equal to the tapered excitation distribution. Meanwhile, to realize low PSBL, the normalized switch-on time sequence should be as large as possible just like A^{1}. The radiation patterns of individual C^{1} are presented in Figure

Performances of three individuals in case 1.

Individual | PSLL (dB) | PSBL (dB) | FNBW (deg) |
---|---|---|---|

A^{1} |
−20.11 | −40.42 | 16 |

B^{1} |
−40.92 | −8.82 | 25.2 |

C^{1} |
−25.40 | −20.14 | 20.6 |

The normalized switch-on time sequences for individuals A^{1}, B^{1}, and C^{1}.

Ring number | |||
---|---|---|---|

A^{1} |
B^{1} |
C^{1} | |

1 | 0.9268 | 0.9949 | 0.9956 |

2 | 0.9456 | 0.9988 | 0.9685 |

3 | 0.9814 | 0.9545 | 0.9412 |

4 | 0.9818 | 0.6999 | 0.9977 |

5 | 0.9673 | 0.5898 | 0.8362 |

6 | 0.9834 | 0.3792 | 0.1515 |

7 | 0.9799 | 0.2443 | 0.8213 |

8 | 0.1007 | 0.1486 | 0.1197 |

9 | 0.9781 | 0.1005 | 0.1011 |

Radiation patterns for individual C^{1}: (a) at the center frequency (

The Pareto front is shown in Figure ^{2} and B^{2} and the acceptable individual C^{2} are shown in Table ^{2} has a well balance between the PSLL and the PSBL. Compared with that in [^{2} has lower PSLL = −33.95 dB, which is 8 dB lower than the conventional CCRA, whilst it has acceptable PSBL = −20.01 dB. The number of elements and the normalized switch-on time sequences for individuals A^{2}, B^{2}, and C^{2} are listed in Table ^{2}, B^{2}, and C^{2}. The total number of elements decreases from 278 in case 1 to 206 in case 2 for individual C^{2}. Also it can be seen that the tapered excitation distribution transforms into the tapered switch-on time sequence in B^{2} and C^{2}, which reveals that, even with the uniform excitation, the time-modulated technology can realize much lower PSLL with acceptable PSBL than the conventional CCRA can. The multiobjective optimization can also offer the relationship between the PSLL and the PSBL, and users can select one appropriate mode from it. The radiation patterns for individual C^{2} are shown in Figure

The Pareto front for case 2.

Performances of three individuals in case 2.

Individual | PSLL (dB) | PSBL (dB) | FNBW (deg) |
---|---|---|---|

A^{2} |
−24.34 | −40.28 | 17.2 |

B^{2} |
−42.70 | −12.06 | 26.8 |

C^{2} |
−33.95 | −20.01 | 22.8 |

The numbers of elements and the normalized switch-on time sequences for individuals A^{2}, B^{2}, and C^{2}.

Ring number | Number of elements | |||||
---|---|---|---|---|---|---|

A^{2} |
B^{2} |
C^{2} |
A^{2} |
B^{2} |
C^{2} | |

1 | 6 | 6 | 6 | 0.9806 | 0.9945 | 0.9862 |

2 | 12 | 12 | 13 | 0.9837 | 0.9740 | 0.9985 |

3 | 19 | 17 | 18 | 0.9896 | 0.9093 | 0.9849 |

4 | 25 | 20 | 20 | 0.9917 | 0.8247 | 0.9592 |

5 | 20 | 21 | 19 | 0.9784 | 0.7642 | 0.8653 |

6 | 23 | 26 | 22 | 0.9884 | 0.4796 | 0.8162 |

7 | 25 | 31 | 25 | 0.9961 | 0.2931 | 0.7361 |

8 | 28 | 43 | 38 | 0.1062 | 0.1430 | 0.1345 |

9 | 55 | 31 | 45 | 0.9662 | 0.1063 | 0.1366 |

Radiation patterns for individual C^{2}: (a) at the center frequency (

Figure

The Pareto front for case 3.

A^{3}, B^{3}, and C^{3} are chosen as three representative individuals acquired by the GDE3 algorithm. Performances of these three individuals are shown in Table ^{3} has the lowest PSBL with highest PSLL. Individual B^{3} has the lowest PSLL but with highest PSBL. For individual C^{3}, there is a tradeoff between the PSLL and the PSBL with PSLL = −33.7 dB and PSBL = −20.24 dB. Meanwhile the PSLL of individual C^{3} is nearly 6 dB lower than that of [^{3}, B^{3}, and C^{3} are listed in Tables ^{3} are presented in Figure

Performances of three individuals in case 3.

Individual | PSLL (dB) | PSBL (dB) | FNBW (deg) |
---|---|---|---|

A^{3} |
−16.61 | −46.97 | 8.8 |

B^{3} |
−43.40 | −14.96 | 28.4 |

C^{3} |
−34.58 | −20.55 | 24.4 |

The ring spacing for individuals A^{3}, B^{3}, and C^{3}.

Ring number | Ring spacing ( | ||
---|---|---|---|

A^{3} |
B^{3} |
C^{3} | |

1 | 0.8999 | 0.5937 | 0.5543 |

2 | 2.2715 | 1.2132 | 1.1770 |

3 | 3.4688 | 1.9433 | 1.8761 |

4 | 4.4764 | 2.7259 | 2.6182 |

5 | 5.8588 | 3.5232 | 3.3798 |

6 | 7.0763 | 4.3241 | 4.1684 |

The numbers of elements for individuals A^{3}, B^{3}, and C^{3}.

Ring number | Number of elements | ||
---|---|---|---|

A^{3} |
B^{3} |
C^{3} | |

1 | 10 | 7 | 7 |

2 | 18 | 15 | 15 |

3 | 43 | 21 | 22 |

4 | 42 | 22 | 21 |

5 | 66 | 23 | 25 |

6 | 88 | 46 | 47 |

The normalized switch-on time sequences for individuals A^{3}, B^{3}, and C^{3}.

Ring number | |||
---|---|---|---|

A^{3} |
B^{3} |
C^{3} | |

1 | 0.9878 | 0.8974 | 0.9755 |

2 | 0.9776 | 0.9737 | 0.9759 |

3 | 0.9960 | 0.9348 | 0.9617 |

4 | 0.9954 | 0.8002 | 0.9479 |

5 | 0.9965 | 0.3768 | 0.7036 |

6 | 0.9987 | 0.1255 | 0.1418 |

Radiation patterns for individual C^{3}: (a) at the center frequency (

In this paper, an attempt based on the multiobjective algorithm is made to solve the pattern synthesis problems of the time-modulated CCRA. By utilizing the GDE3 multiobjective algorithm, a set of solutions known as the Pareto front is obtained, which is beneficial to the design of time-modulated antenna array. In this design, the PSLL and the PSBL are set as two objectives. Three examples are studied in detail. Firstly, the normalized switch-on time sequence is optimized only. Then, both the number of elements and the normalized switch-on time sequence are optimized. The ring spacing, the number of elements within the same ring, and the normalized switch-on time sequence are all optimized at last. Moreover, the corresponding Pareto front, parameters of representative individuals, and radiation patterns are presented for all cases. Compared with the conventional CCRA, the selected three time-modulated CCRAs, C^{1}, C^{2}, and C^{3}, have lower PSLL and acceptable PSBL. In addition, users can choose the most appropriate individuals from the Pareto front to fulfill different antenna design requirements. The numerical results reveal that the approach based on the multiobjective optimization is an effective method for the TMA design.

The authors declare that there is no conflict of interest regarding the publication of this article.

This work is supported by the Doctoral Scientific Research Foundation of Yulin Normal University G2017002.