快速笔形波束合成：大型非均匀天线阵列的新方法

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"大型不等距天线阵列的快速笔形波束方向图合成" 在无线通信、雷达系统和卫星通信等领域，天线阵列的设计和优化是至关重要的。天线阵列能够通过控制其辐射模式来实现特定的性能需求，如增益、方向性和波束扫描范围。在大型天线阵列中，尤其是不等距分布的天线阵列，图案合成的计算成本通常很高，这限制了其在实际应用中的效率。针对这一问题，本文提出了一种新的快速笔形波束方向图合成方法，专门适用于大型非均匀天线阵列。该方法的核心在于结合了最小二乘意义下的插值和迭代快速傅里叶变换（FFT）。首先，通过对非均匀分布的天线元素进行插值，将其转换为均匀虚拟元素。这一过程旨在保持原有的阵列特性，同时简化后续的计算。然后，利用FFT对这些均匀虚拟元素进行处理，合成出所需的波束方向图。由于FFT算法的高效性，这种方法在计算速度上显著优于传统方法，特别是在处理大规模阵列时。文章详细探讨了插值参数的选择，这是保证插值精度的关键。合适的插值参数可以确保从非均匀分布到均匀分布的转换过程中，阵列的原始特性得到尽可能精确的保留。此外，作者还提供了线性和平面两种类型的非均匀阵列实例，通过实例验证了该方法的有效性和优势。 "Array pattern synthesis"（阵列图案合成）是天线阵列设计中的核心步骤，目的是通过调整各个天线单元的相位或幅度来形成特定形状的辐射模式。在这个过程中，非均匀分布的天线阵列因其灵活的设计空间和可能的性能优势而受到关注，但其计算复杂度也相应增加。 "Nonuniform fast Fourier transform"（非均匀快速傅里叶变换）是一种处理非均匀数据的FFT变体，它允许对非均匀采样数据进行快速频谱分析。在本文中，非均匀FFT被用来快速合成笔形波束方向图，降低了计算负担。 "Unequally spaced array"（不等距天线阵列）是指阵列中的天线元素不是等间距排列的，这种排列方式可以实现更复杂和灵活的波束形状，但同时也增加了图案合成的复杂度。本文提出的快速笔形波束方向图合成方法为大型非均匀天线阵列的优化提供了一个高效的新途径，有助于在保持性能的同时，减少计算成本，这对于现代通信系统中大规模天线阵列的应用具有重要意义。

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 2, FEBRUARY 2013 627

Fast Pencil Beam Pattern Synthesis of Large

Unequally Spaced Antenna Arrays

Kai Yang, Student Member, IEEE, Zhiqin Zhao, Senior Member, IEEE, and Qing Huo Liu, Fellow, IEEE

Abstract—Reducing the computational cost of large array pat-

tern synthesis is attractive in many applications. In this paper, a

fast pencil beam pattern synthesis method for large nonu niform

antenna arrays is proposed. This method is based on an interpo-

lation in a least square sense and iterative fast Fourier transform

(FFT), i.e., interpolate the nonuniform distributed elements int

uniform virtual elements , and th en apply FFT to synthesis the uni-

form array. Thanks to the efﬁciency of FFT, the proposed method

is much faster and can handle much larger arrays than the existing

methods. To guarantee the interpolation accuracy, the choice of

the interpolation parameters is discussed. Both linear and planar

nonuniform array examples are shown to validate the adv

antages

of the proposed method.

Index Terms—Array pattern synthesis, nonuniform fast Fourier

transform, unequally spaced array.

I. INTRODUCTION

ONUNIFORM or unequally spaced arrays have a

wide range of applications in radar, sonar and wireless

communication, and othe

r areas. One example of using a

nonuniform array is to prevent grating lobes or obtain a better

pattern performance than a uni form array. Another example is

the use of a nonunifo

rm array may be more suitable when the

antennas are restricted into a limited space on the body of ﬂight

vehicles. In recent literature, several novel algorithms were

developed to op

timize the element positions and excitations of

nonuniform array s to improve pattern performance [1]–[3] and

reduce the computational cost [4]–[6] or obtain a given pattern

[7]–[10].

This paper focuses on the problem of optimization of the an-

tenna elem ent excitations with all the element positions ﬁxed to

obtain a d

esired array pattern performance, especially for large

linear

elements and planar elements arrays.

Many modern applications require arrays that are capable of

achie

ving very narrow main beam. Therefore, large array pat-

tern synthesis methods are requ ired [11], [12]. During the past

Manuscript received March 13, 2012; revised June 16, 2012; accepted Au-

gust 30, 2012. Date of publication September 21, 2012; date of current version

January 30, 2013. This work was supporte d in part by the National Natural Sci-

ence Foundation of China under Grants 60927002 and 61171044.

K. Yang is with the School of Electronic Engineering, University of Elec-

tronic Science and Technology of China (UESTC), Chengdu 611731, China,

and also with the Department of Electrical and Compute r Engineering, Duke

University, Durham, NC 27708 USA (e-mail: eekaiyang@gmail.com).

Z. Zhao is with the School of Electronic Engin eering, University of Electronic

Science and Technology of China (UESTC), Chengdu 611731, China (e-mail:

zqzhao@uestc.edu.cn).

Q. H. Liu is with the Department of Electronic Engineering, Duke University,

Durham, NC 27708 U SA (e-mail: qh liu@ee.duke.ed u).

gital Object Identiﬁer 10.1109/TAP.2012.2220319

50 years, sev eral array pattern synthesis techniques, such as

analytical methods, stochastic methods, adaptive array theory

method and convex optimization methods, have been presented

for obtaining low side lobe levels. The analytical methods [13]

use the matrix relationship between the elements and the far-

zone array pattern of the array. This kind of method is a non-iter-

ative process, thus time efﬁcient. However, this method requires

a speciﬁed array pattern and is difﬁcult to be applied to larg

arrays. The stochastic m e tho ds, such as g enetic algorithm (GA)

[14], [15], particle swarm optimization (PSO) method [16], [17]

and simulated annealing (SA) method [18]–[20], a

re global op-

timization methods which can escape from local optimal so-

lutions. T he stochastic methods can solve large optimization

problem, but the computation time is huge and

will grow rapidly

with the p roblem size. The adaptive array theory method [21]

is an it erat ive method to minimize the errors between synth e-

sized and desired patterns by recursi

vely adjust ing the inter-

ference-to-noise-r ati o. S imilar to the stochastic met hods, this

method is difﬁcult to be applied to large arrays due to the com-

putational cost. Element excita

tions optimization can be posed

in a convex form [22]– [24], and then it can be solved using a

convex problem solver (for example, SeDuMi [25] and CV X

[26]). However, most conve

x optim izat ion solvers are unlikely

to work well (or at all) for very large problems [26].

A promising iterative f ast F ou rier transform (FFT) method

[27], [28] was presented

to synthesize large planar arrays. This

method is based on the Fourier transform-pair relationship be-

tween the array excitations and the array factor. Thanks to the ef-

ﬁciency of FFT, this

method can solve large array synthesis with

much less time compared with the stochastic methods. How-

ever, this method requires all the elem e nts i n a uniform grid.

In this pa per, we e

xtend the iterative FFT method to nonuni-

form array through exponential in terpolation. Each element in

the nonuniform array is interpolated by several elements in a vir-

tual uniform a

rray, a nd then we optim ize the excitations of this

virtual array to arrive at a desired array pattern. Finally, calcu-

late the real element excitations by the optimized virtual element

excitatio

ns. To ensure the agreement between the virtual array

pattern and the real array pattern, t he exponential interpolation

error should be small. N umerical simulation results show that

the int

erpolation strategy can make the interpolation error very

small if we choose the interpolation parameters appropriately.

In addition, we should make sure th e virtual element excitations

are

always in the subspace of the real element excitations during

the iterative process; otherwise, there will be no exact solutions

for obtaining the real element excitations from virtual element

xcitations.

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