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In this paper, an improved genetic algorithm with dynamic weight vector (IGA-DWV) is proposed for the pattern synthesis of a linear array. To maintain the diversity of the selected solution in each generation, the objective function space is divided by the dynamic weight vector, which is uniformly distributed on the Pareto front (PF). The individuals closer to the dynamic weight vector can be chosen to the new population. Binary- and real-coded genetic algorithms (GAs) with a mapping method are implemented for different optimization problems. To reduce the computation complexity, the repeat calculation of the fitness function in each generation is replaced by a precomputed discrete cosine transform matrix. By transforming the array pattern synthesis into a multiobjective optimization problem, the conflict among the side lobe level (SLL), directivity, and nulls can be efficiently addressed. The proposed method is compared with real number particle swarm optimization (RNPSO) and quantized particle swarm optimization (QPSO) as applied in the pattern synthesis of a linear thinned array and a digital phased array. The numerical examples show that IGA-DWV can achieve a high performance with a lower SLL and more accurate nulls.

Linear array pattern synthesis with multiobjective optimization has been a hot research topic in recent years [

Many optimization methods construct objective functions into one function, which is defined as aggregating function [

In [

Thinned arrays have been widely used in array pattern synthesis due to the excellent performance [

Phased arrays are considered to be the best solution in beam scanning and widely used in engineering [

A genetic algorithm (GA) is generally believed to be suitable for discrete optimization problems because it uses a discrete coding method and deals directly with discrete variables. In the present study, an improved genetic algorithm with dynamic weight vector (IGA-DWV) is proposed for array pattern synthesis. The objectives of the linear array are synchronously optimized for different fitness functions. The innovations and the effectiveness obtained in this work are described below:

The remainder of this paper is organized as follows. In Section

The multiobjective optimization problem can be expressed as

Very often, because the objectives in (

Considering a minimization problem for each objective and

In this study, IGA-DWV is proposed for array pattern synthesis. Figure

The main flowchart of IGA-DWV.

The main step in the process is the nondominated sorting approach which is the same as that in [

The uniform crossover and bitwise mutation is used for binary-coded GAs. The crossover operator for binary-coded GAs can be expressed as

For real-coded GAs, the simulated binary crossover (SBX) operator and polynomial mutation [

To select the individuals with high performance, nondominated ranks are effective for the convergence of population. The solutions in the last front should be carefully considered. Here, a group of weight vector uniformly distributed on the PF is expected to be obtained. The weight vector can divide the space to

For most MOPs, the PFs are not flat, which is shown in Figure

The weight vector uniformly distributed on convex and nonconvex PFs.

To get the uniformly distributed points on the PF, one group of excellent weight vector can achieve this goal, which is calculated according to the slope of the PF. The weight vector will be updated in each generation. In the

The main step is to select

The length of a certain period of curve is related to the slope here. In the initial weight vector, there are

For the nonconvex and convex PFs, the weight vectors have different origins. In order to unitize different MOPs,

For the convex PF, the normalized length of the

For the nonconvex and convex PFs, the normalized lengths are different from each other. For nonconvex PF, more weights should be selected on both sides, but in the middle for convex PF. The probability of choosing the

According to

1:

2:

3:

4: Choose the

5:

6:

After getting

The purpose of this paper is the same as that of [

In this paper, IGA-DWV calculates one group of excellent weight vector uniformly distributed in the PF instead of the uniformly distributed points. The first step of IGA-DWV is to calculate the slope in the PF associated with every weight vector. The normalized length of the

Considering the complexity of one iteration of IGA-DWV, the basic operations and their worst-case complexities can be divided into three classes. The computational complexity of nondominated sorting is

The fitness function evaluations are the main part of time costing for the methods used in array pattern synthesis. For every iteration, the function value of each individual should be calculated. This operator takes a lot of time, especially for the large array. Hence, the time costing is closely related to the population size and the number of iterations.

Thinned array means turning off some elements from the uniformly spaced array to create a desired radiation pattern [

There are many evolutionary methods for the pattern synthesis of a thinned array. These apply discrete coding to calculate this discrete problem. In [

Consider a linear array of

Geometry of the

For the thinned array, the objectives of optimization are set as the minimum SLL and the maximum directivity by adjusting the switch of the elements. The fitness function can be defined as

For the directivity, the minus coefficient can make it a minimization problem. In the fitness function,

Array directivity is defined as the radiant intensity in a specific direction divided by the isotropic radiant intensity. The isotropic radiant intensity is the total radiated power of the array divided by

1: Find the position of the elements turned on:

2: Calculate the number of 1 in

3: Calculate the number of positions in which the value should be reversed:

4: Generate a random position distribution according to

5:

6: Set the excess number as 0:

7:

8: Set the excess number as 1:

9:

Phased array can achieve beam scanning and interference suppression by adjusting the phase without changing the excitation amplitude. With the rapid development of digital phase shifters, phased arrays have been widely used in engineering applications. For the

In this paper, a linear array of

Geometry of the

The array pattern can be expressed as [

The first fitness function is the same as that in the pattern synthesis of a linear thinned array. In the second fitness function,

For the thinned array and the phased array, the switch and the phase of each element constitute the decision vector of the MOP, respectively. These variable parameters can adjust the relative relation of received or emitted electromagnetic waves. The different gains can be obtained for different steering directions in the space. The above fitness functions are to set the expected gain in the specified steering directions. After the optimal variable parameters are obtained by the proposed IGA-DWV, the expected array pattern can be formed with a high performance. As thus, the useful signals can be received or emitted in the expected directions, and the useless signals can also be effectively suppressed.

The variable parameters in the array pattern synthesis can be optimized by using the proposed multiobjective optimization method of IGA-DWV. The objective functions are obtained by the array factor, which are supposed to be uniformly distributed on the PF. By using IGA-DWV, one group of excellent weight vectors uniformly distributed on the PF can ensure the diversity of the final solutions, and the weight vectors are calculated according to the slope of the PF. With the help of the efficient convergence of IGA-DWV, the smaller values of the objective functions can be optimized, which means that the array pattern has the lower SLL, lower nulls, and higher directivity.

To reduce the computation complexity, one linear transformation similar to a discrete cosine transform (DCT) is used for the pattern synthesis of a linear thinned array and a digital phased array. By discretizing the two array pattern functions with

For the 200-element (

Table

Comparison of results for the linear thinned array.

Design parameter | IGA-DWV | RNPSO | IGA-DWV | RNPSO |
---|---|---|---|---|

| 0.77 | 0.77 | 0.78 | 0.78 |

SLL (dB) | -23.26 | -23.05 | -23.05 | -22.94 |

Directivity (dB) | 30.44 | 28.82 | 30.72 | 29.33 |

Mean time costing (m) | 15.28 | 14.96 | 15.36 | 15.03 |

Simulation results for the switched-off elements.

Result | Number of switched-off elements |
---|---|

IGA-DWV with | ±54; ±55; ±60; ±64; ±65; ±68; ±69; ±72; ±75; ±76; ±77; ±78; ±81; ±83; ±84; ±85; ±87; ±89; ±90; ±91; ±92; ±94; ±96 |

IGA-DWV with | ±51; ±52; ±54; ±61; ±64; ±65; ±70; ±71; ±72; ±76; ±77;±79; ±80; ±82; ±84; ±85; ±87; ±88; ±89; ±91; ±93; ±97; ±99 |

IGA-DWV with | ±52; ±54; ±61; ±64; ±65; ±69; ±71; ±72; ±74; ±78; ±79; ±80; ±82; ±83; ±85; ±86; ±88; ±89; ±90; ±91; ±94; ±95 |

IGA-DWV with | ±53; ±57; ±60; ±64; ±65; ±69; ±70; ±71; ±73; ±75; ±79; ±81; ±82; ±83; ±84; ±86; ±87; ±89; ±92; ±94; ±95; ±96 |

Pattern results for the linear thinned array obtained by using IGA-DWV and RNPSO with

Distribution of the final solution obtained by IGA-DWV with

Contrasting distribution of the best SLL obtained in 50 trials by using IGA-DWV and RNPSO with

In the second case of

Pattern results for the linear thinned array obtained by using IGA-DWV and RNPSO with

Distribution of the final solution obtained by IGA-DWV with

Contrasting distribution of the best SLL obtained in 50 trials by using IGA-DWV with

In this experimental study, a 100-element linear array with a 4-bit digital phase shifter is considered. In this paper, two cases are implemented. The first case is SLL with nulls at

Figure

The digital phase sequences and results for different solutions.

Result | Digital Phase Sequence | Best SLL (dB) | Null depth (dB) | Mean time costing (m) |
---|---|---|---|---|

Nulls at 30° to 31° obtained by IGA-DWV | 7877787898779887768966686B761F8F0FF0F00E000BEF9858 | -18.16 | -61.72 | 8.76 |

Nulls at 30° to 31° obtained by QPSO | 8D6C8EF77049635D76668DDCAC410587A4847784DA79838A6C | -12.03 | -37.41 | 8.62 |

Nulls at 30° and 31° obtained by IGA-DWV | 887788778778888695786974A4C61F0D0F800FFFF00C006999 | -18.55 | -65.66 and -71.51 | 8.51 |

Nulls at 30° and 31° obtained by RNPSO | 9588AABA98AA4A68CEB53129A9D99BCEDEB1904ED3771D669A | -14.78 | -60.71 and 60.93 | 8.56 |

Pattern results for a 4-bit linear array obtained by using IGA-DWV and QPSO in the first case.

Comparison of patterns for the minimum SLL and nulls at

Distribution of the final solution obtained by IGA-DWV in the first case.

One more individual is extracted from Figure

Pattern results for a 4-bit linear array obtained by using IGA-DWV and QPSO just for minimum SLL.

Table

Pattern results for a 4-bit linear array obtained by using IGA-DWV and RNPSO in the second case.

Comparison of patterns for the minimum SLL and nulls at

Distribution of the final solution obtained by using IGA-DWV in the second case.

The objective of the proposed IGA-DWV is to obtain a set of solutions with good convergence and diversity for the pattern synthesis of a linear array. A multiobjective evolutionary algorithm, instead of a single objective obtained by aggregating functions, is applied to the linear array pattern synthesis. Contributing to uniformly selecting individuals based on the dynamic weight vector, the final solution can effectively deal with the conflict between the multiobjective functions. For a 200-element linear thinned array with

The data used to support the findings of this study are available from the corresponding author upon request.

The author declares that there are no competing interests.