A hybrid method that combines the iterative Fourier transform (IFT) and the algorithm of differential evolution (DE) is proposed to address the synthesis of low sidelobe sparse linear array (SLA) including many elements. Firstly, a thinned linear array (TLA) with the lattices spaced at half wavelength is obtained by the IFT. Then, for the elements of the TLA whose left or right spacing are greater than half wavelength, their placements are selected as the candidates which will be further optimized by the DE, as long as the interelement spacing is not less than half wavelength. Consequently, the convergence pressure of DE is greatly alleviated for the reason that the selected elements only accounts for a small part of the total. Therefore, the SLA with improved sidelobe performance can be obtained at relative low hardware cost. Several numerical instances confirmed the effectiveness of the proposed method.

In some antenna applications, we focus the antenna array with narrow beamwidth to increase the resolution while allow sacrificing some antenna gain, so a certain percentage of the elements can be removed from a periodic, fully populated array. This is commonly known as thinned array (TA). For the advantages in reducing weight, power, and cost, TA had been used in satellite communications, ground-based radars, radio astronomy, and so on [

Another kind of structure analogous to TA, which has the elements arbitrarily located within the antenna aperture and thereby provides more degree of freedom to achieve sidelobe suppression, is called sparse array (SA). Unfortunately, the synthesis for SA refers to a question of multiple constraints including the minimum element spacing, the antenna aperture, and the number of elements. Therefore, the use of stochastic optimization algorithms such as the simulated annealing [

In this manuscript, the synthesis of uniform-amplitude sparse linear array with many elements is considered. Naturally, the use of the stochastic method, typically as DE, always involves a large set of parameters and thereby requires high computational resources. To ease the impact, we propose a hybrid method that combines the IFT and DE (IFT-DE) to address the issue. The method is based on DE with the individual parameters constituted by the element locations. Differently, unlike the conventional DE in which all the individual parameters are randomly initialized, for the assumption that the element spacing is not less than

The manuscript is mainly organized as follows. Section

Consider a linear array with

The IFT in the proposed method is the same as that used in [

Accordingly, a single trial of the IFT-DE can be divided into three steps, as are in turn described below.

Perform the IFT to get a thinned linear array [

Randomly initialize the element excitations

Compute the array factor

Force the values in the sidelobe region of

Compute

Truncate the

For

Decrease

Repeat steps (2)–(7) until

Select the element locations that need to be optimized.

For the above thinned array, select the elements whose left or right spacing are greater than

The optimization for the selected element locations using DE.

Determine the lower and upper bounds for locations of the selected elements.

Assume that a total of

It depicts that the maximum range of the adjacent elements moving in the opposite direction is set equal, and the minimum distance after the movement is

Perform DE.

A conventional DE similar to that presented in [

A thinned array to illustrate the way how to select the element locations that need to be optimized.

The proposed method is applied to the design cases similar to that taken from [

We consider synthesizing a symmetrical/asymmetrical SLA that is originally based on 200 lattices, symmetrical/asymmetrical TLA with different filling factors. The first case refers to synthesizing a symmetrical SLA through a symmetrical TLA with filling factor of 66%. The fitness convergence curve indicates that the optimal fitness value among 100 trials of IFT-DE is obtained at trial number 94 (Figure

The sparse linear array obtained by the IFT-DE based on a symmetrical TLA with a filling factor of 66%. (a) The convergence curve of the fitness value. (b) The optimal far-field pattern among 100 trials.

The element locations of the symmetrical TLA with a filling factor of 66% obtained by the IFT.

The element locations of the sparse linear array obtained by the IFT-DE based on the TLA described in Figure

Similarly, as far as a symmetrical TLA with a filling factor of 77% is originally concerned, the far-field pattern of the best symmetrical SLA among 100 trials of IFT-DE shows 1 dB PSLL reduction compared with the report presented in [

The sparse linear array obtained by the IFT-DE based on an asymmetrical TLA with a filling factor of 39%. (a) The convergence curve of the fitness value. (b) The optimal far-field pattern among 100 trials.

The element locations of the sparse linear array obtained by the IFT-DE based on an asymmetrical TLA with a filling factor of 39%.

The fourth example assumes an asymmetrical TLA that is 69.5% filled. The obtained SLA shows about 1.16 dB PSLL decrement with no sacrifice of beamwidth, as is compared to the value presented in [

We first consider synthesizing a low sidelobe symmetrical SLA based on 100 lattices, symmetrical TLA with a filling factor of 80%. The PSLL of the sparse array obtained by IFT-DE is equal to −20.52 dB, which is 1.7 dB lower than the reports using ACO [

Comparative synthesizing results by using the IFT-DE and some published tools.

The structure of the SLA | Filling factor of the TLA originally obtained by the IFT (%) | PSLL (dB) | 3 dB beamwidth (degree) | ||
---|---|---|---|---|---|

IFT-DE | IFT [ |
IFT-DE | IFT [ | ||

Symmetrical | 77 ( |
−24.04 | −22.92 | 0.58 | 0.588 |

66 ( |
−24.59 | −22.86 | 0.68 | 0.685 | |

Asymmetrical | 69.5 ( |
−25.46 | −24.3 | 0.64 | 0.643 |

39 ( |
−20.38 | −17.33 | 0.8 | 0.546 | |

Symmetrical | 76 ( |
−23.56 | −20.53 (by GA in [ |
— | |

78 ( |
−22.97 | −20.56 (by GA in [ |
— | ||

80 ( |
−22.2 | −20.52 (by ACO in [ |
— |

All the aforementioned results are obtained by using a PC equipped with an 8 GB RAM as well as an Intel I7-6700 Processor that operates at 3.4 GHz, and the hardware cost is about an hour for each numerical instance.

The use of IFT could efficiently provide a thinned linear array with improved sidelobe performance. For the assumption that the element spacing is not less than half wavelength, most element locations of the thinned array are predetermined after performing the IFT. Therefore, the number of element locations needing to be optimized is largely reduced, which makes the solution space of DE greatly narrowed. Consequently, the low sidelobe SLA with many elements could be synthesized at relative low hardware cost. Some numerical results confirmed the effectiveness of the IFT-DE in synthesizing large SLA at a faster convergence speed. Furthermore, by simple modification, the proposed method can also be extended to 2D arrays and thereby provides a candidate way for pattern synthesis of planar sparse arrays.

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

The authors declare that they have no conflicts of interest.

This work is supported in part by the National Natural Science Foundation of China under Grant nos. 61601272 and 61772398, in part by the Natural Science Foundation of Shaanxi Province under Grant no. 2016JM6068, in part by the scientific research plan of the Education Department of Shaanxi Province under Grant no. 15JK1147, and in part by the scientific research plan of Shaanxi University of Technology under Grant no. SLGQD14-05.