Beampattern synthesis for large-scale antenna array via accurate array response control

doi:10.1016/j.dsp.2021.103152

Digital Signal Processing

Volume 117, October 2021, 103152

https://doi.org/10.1016/j.dsp.2021.103152 Get rights and content

Abstract

In this paper, two low computational complexity accurate array response control algorithms for large-scale antenna array are presented. The first proposed algorithm is an Improved Weight vector ORthogonal Decomposition (I-WORD) approach. We extend the WORD method by selecting the non-negative coefficient as the ultimate solution, and thus the ultimate weight vector will be in an exact form. Therefore, the proposed I-WORD algorithm has no longer a selection procedure and has low computational complexity compared with the WORD algorithm. Moreover, to achieve multi-point accurate array response control, we further develop a Multi-point control based on I-WORD (MI-WORD) algorithm. The MI-WORD algorithm is able to control multi-point response simultaneously by finding the weight vector from the intersection of weight vector sets with a new matrix constructing manner, which is different from the multi-point accurate array response control (MA²RC) method. Both the proposed algorithms can be applied to adjust the response accurately to synthesize the beampattern. Furthermore, the proposed MI-WORD method has the advantage of low computational complexity, especially when the number of array antennas is large. Simulation results show the effectiveness of the two algorithms and the property of low complexity of the MI-WORD algorithm for beampattern synthesis.

Introduction

Array signal processing has attracted significant attention from researchers for decades. An important branch of array signal processing is beampattern synthesis [1], [2], [3]. The problem of beampattern synthesis is to design a weight vector for antenna array, which can make the array beampattern meet some specific requirements. For example, the low uniform sidelobe is designed to against the interferences in the radar system. Moreover, antenna arrays with a narrow main beam and high gain are urgently demanded in modern applications. Large phased antenna arrays have been effectively used in realizing highly directional beamforming [4]. Thus, beampattern synthesis for large-scale antenna arrays of which the number of elements is large also has been researched [5].

Plenty of works have been devoted to beampattern synthesis in recent years. A weight vector calculated with the theory of Chebyshev polynomials was presented in [6], which can achieve the same level for all the sidelobes when the beam width is minimum. However, its applications are limited to uniform antenna arrays. As a result, several algorithms have been proposed to solve the beampattern synthesis problem for nonuniform antenna arrays. For example, a simple iterative solution of linearly constrained least squares method was developed in [7]. In [8], the problem of beampattern synthesis was formed as a quadratic program problem, which could synthesize the pattern of arbitrary array to any appropriate beampattern. The element beampattern in an unequally spaced array was considered as the beampattern radiated by a subarray of some equally spaced virtual elements in [9]. The synthesis of arbitrary sparse planar arrays based on the two-dimensional unitary matrix pencil was studied in [10].

Besides, a number of approaches based on global optimization have been developed for nonuniform arrays [11], [12], [13]. A genetic algorithm was applied to the beampattern synthesis for thinned arrays in [11]. In [12], the particle swarm optimization was implemented to handle arbitrary nonlinear cost functions. To synthesize beampattern for an unequally spaced array, simulated annealing was applied in [13]. However, all the global search algorithms bring heavy computation loads to the problem of beampattern synthesis.

Different from the global search approaches, a class of methods based on convex optimization techniques [14], [15], [16] have been presented in the past several years. In [17], Lebret and Boyd have expressed some antenna array beampattern synthesis problems as convex optimization problems, which can be solved by interior-point method. A method based on the second-order cone programming (SOCP) was developed in [18], which was applicable to general beampattern synthesis problem for arbitrary geometry array. In [19], a semidefinite programming has been presented to design the robust array beampattern, which can synthesize beampattern with the uncertainties, for example, array gain uncertainties. In addition, a novel beampattern synthesis method based on the semidefinite relaxation was developed to focus energy in the desired range-angle domain in [20].

Recently, some quite comfortable and effective methods have been researched. The fast Fourier transform (FFT) technique has been applied to synthesize unequally spaced arrays in [9], [21]. Then, an extend FFT method as iterative spatiotemporal Fourier transform has been used to design filter coefficients for generating frequency-invariant beam pattern in [22]. The modified iterative FFT technique was also used to synthesize thinned massive array for 5G communications in [23]. Besides, synthesizing the beampattern utilizing the differential evolution method also was researched in [24], [25]. Lately, [26] has presented a differential evolution algorithm to achieve shaped power pattern of a linear dipole array. With new encoding mechanism and Cauchy mutation, the differential evolution algorithm was applied to synthesize large unequally spaced planar arrays in [27].

Unfortunately, all algorithms mentioned above cannot flexibly control the array response. It has to be completely redesigned the weight vector even if only a slight change of the desired pattern is needed. When the response at a given direction is needed to be adjusted accurately, the weight vector needs to be resolved from the beginning. To solve this problem, a scheme called optimal and precise array response control (OPARC) which assigns a virtual interference to a direction to control the response level precisely has been investigated in [28]. It can only control the response at one angle in one step. In addition, a beampattern synthesis method based on weight vector orthogonal decomposition (WORD) was proposed in [29], which can control the array response accurately at a given direction. However, there are two weight vectors obtained by the WORD algorithm, which need a criterion to choose one of them as the ultimate weight vector. What's more, both the WORD method and the OPARC method work as point-by-point manner when beampattern is synthesized, which adjust the response at only one given direction in each iteration. Although a multi-point accurate array response control (MA²RC) method based on the accurate array response control (A²RC) method has been presented in [30] and a multi-point method based on oblique projection (OBPJ) has been presented in [31], the computational complexity of them is high when the number of elements and the number of the directions needed to be controlled are large.

Considering the drawbacks afore-mentioned, this paper is dedicated to two low computational complexity and accurate array response control algorithms. Firstly, the two weight vectors obtained by the WORD algorithm will be analyzed and the nonpositive one of the solutions would be discarded. Thus, the ultimate weight vector will be in an exact form and we present an Improved WORD (I-WORD) algorithm which can omit the selection procedure compared with the WORD algorithm. Then, combining the MA²RC method, a multi-point response control at one step method based on the I-WORD algorithm is under consideration. To achieve low computational complexity for large-scale antenna array, a new matrix constructing manner leading to low computational complexity is applied, which is different from the MA²RC method in [30]. Therefore, we obtain a low complexity Multi-point accurate array response control based on I-WORD (MI-WORD) algorithm. Finally, the proposed MI-WORD algorithm is applied to synthesize beampattern for antenna arrays.

The rest of this paper is organized as follows. In Section 2, the beampattern synthesis problem is formulated and the WORD algorithm is introduced. The improved WORD algorithm is proposed in Section 3. Then, the proposed MI-WORD method is developed in Section 4. In Section 5, the application of the MI-WORD method for beampattern synthesis is described. Numerical examples are presented in Section 6. Finally, conclusions are drawn in Section 7.

Section snippets

Formulation of beampattern synthesis problem

Considering an arbitrary geometry array with N elements, the steering vector in direction θ can be expressed as $a (θ) = {[f_{1} (θ) e^{- j ϕ_{1} (θ)}, \dots, f_{N} (θ) e^{- j ϕ_{N} (θ)}]}^{T}$ where $f_{n} (θ)$ denotes the element pattern, $j = \sqrt{- 1}$ is the imaginary unit, ${(\cdot)}^{T}$ is the transpose operation and $ϕ_{n} (θ)$ represents the phase delay between the nth element and the reference element. Generally, we define the complex weight vector for the array as $w = {[w_{1}, w_{2}, \dots, w_{N}]}^{T}$ . Thus, the far-field array response can be given as $P (θ) = | \sum_{n = 1}^{N} w_{n}^{⁎} f_{n} (θ) e^{- j ϕ_{n} (θ)} | = | w$

The proposed improved WORD algorithm

In this section, we dedicate to present an improved WORD algorithm which does not need to select ultimate β from $β_{a}$ and $β_{b}$ but calculates the weight vector with an exact form.

For clarity, two solutions of the $β_{k + 1}$ are written below as $β_{k + 1, a} = \frac{- Re (B (1, 2)) + d}{B (2, 2)}, β_{k + 1, b} = \frac{- Re (B (1, 2)) - d}{B (2, 2)}$ where $d = \sqrt{[Re (B (1, {2))]}^{2} - B (1, 1) B (2, 2)}$ , $Re (\cdot)$ returns the real part of a complex number, and B is given by $B = [\begin{matrix} - ρ_{k + 1} | w_{k, ⊥}^{H} a (θ_{0}) |^{2} & - ρ_{k + 1} w_{k, ⊥}^{H} a (θ_{0}) a^{H} (θ_{0}) w_{k, ∥} \\ - ρ_{k + 1} w_{k, ∥}^{H} a (θ_{0}) a^{H} (θ_{0}) w_{k, ⊥} & | w_{k, ∥}^{H} a (θ_{k + 1}) |^{2} - ρ_{k + 1} | w_{k, ∥}^{H} a (θ_{0}) |^{2} \end{matrix}],$

The proposed low complexity multi-point control method

In the preceding section, we have presented the I-WORD algorithm, which adjusts the array response at a single direction in each step. In this section, a low complexity Multi-point accurate array response control based on I-WORD (MI-WORD) method will be developed.

Beampattern synthesis using the proposed MI-WORD algorithm

In the above sections, the I-WORD and MI-WORD algorithms have been presented. Following, the application of the proposed MI-WORD algorithm to beampattern synthesis will be introduced. To achieve synthesis, the MI-WORD algorithm is iterative to control multi-point response, where the I-WORD algorithm obtains the weight vector for each point by a single step.

To be specific, we set $a (θ_{0})$ as the initial weight vector ${\overline{w}}_{0}$ and obtain the beampattern $L_{0} (θ)$ . Let $L_{d} (θ)$ denote the desired beampattern.

Simulations

In this section, the proposed I-WORD algorithm will be firstly simulated. Then, the effectiveness and the property of low computational complexity of the proposed MI-WORD algorithm will be illustrated with several examples. For comparison, the convex-optimization-based method (labelled as “convex method” following) in [17] and the MA²RC method are simulated if available. Note that all the examples about the running time of the method are simulated on a computer with Intel Core CPU i7-10700 at

Conclusions

In this paper, we have proposed a low computational complexity multi-point accurate array response control algorithm for beampattern synthesis. After the analysis of the two solutions in WORD algorithm, we can discard the weight vector calculated with the nonpositive coefficient and obtain the I-WORD algorithm. The proposed I-WORD algorithm can omit the selection procedure in the WORD algorithm. Then, based on the I-WORD algorithm, the proposed MI-WORD method is developed by finding the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants 61771095 and 62031007.

Weilai Peng was born in Jiangxi, China. He received the B.Eng. degree in electronic engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 2015, where he is currently pursuing the Ph.D. degree in electronic engineering. His current research interests include array signal processing , digital beamforming, MIMO radar, and optimization theory.

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Xuejing Zhang was born in Hebei, China. He received the B.S. degree in electrical engineering from Huaqiao University, Xiamen, China, and the M.S. degree in signal and information processing from Xidian University, Xi'an, China, and the Ph.D. degree in signal and information processing from University of Electronic Science and Technology of China, Chengdu, China, in 2011, 2014, and 2019, respectively. From 2017 to 2019, he was a visiting student with the University of Delaware, Newark, DE, USA. His research interests include array signal processing and wireless communications.

Zishu He was born in Sichuan, China, in 1962. He received the B.S., M.S., and Ph.D. degrees in signal and information processing from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1984, 1988, and 2000, respectively. He is currently a Professor with the School of Information and Communication Engineering, UESTC. His current research interests include array signal processing, digital beam forming, the theory on multiple-input multiple-output (MIMO) communication, and MIMO radar, adaptive signal processing and interference cancellation.

Julan Xie was born in Leiyang, China. She received the B.S. and Ph.D. degrees in signal and information processing from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2005 and 2012, respectively. She is currently an Associate Professor of signal and information processing with the School of Information and Communication Engineering, UESTC. Her current research interests include array signal processing, digital beamforming, interference cancellation, and array optimization.

Chunlin Han was born in Hebei, China, in 1962. He received the B.S., M.S., and Ph.D. degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1984, 1989, and 2004, respectively. He is currently a Professor with the School of Information and Communication Engineering, UESTC. His current research interests include array signal processing and digital beamforming.

View full text

Beampattern synthesis for large-scale antenna array via accurate array response control

Abstract

Introduction

Section snippets

Formulation of beampattern synthesis problem

The proposed improved WORD algorithm

The proposed low complexity multi-point control method

Beampattern synthesis using the proposed MI-WORD algorithm

Simulations

Conclusions

Declaration of Competing Interest

Acknowledgements

Digit. Signal Process.

Digit. Signal Process.

Digit. Signal Process.

Linearly polarized shaped power pattern synthesis with sidelobe and cross-polarization control by using semidefinite relaxation

IEEE Trans. Antennas Propag.

Synthesizing shaped power patterns for linear and planar antenna arrays including mutual coupling by refined joint rotation/phase optimization

IEEE Trans. Antennas Propag.

Position aided beam alignment for millimeter wave backhaul systems with large phased arrays

Low-dimensional sdp formulation for large antenna array synthesis

IEEE Trans. Antennas Propag.

A current distribution for broadside arrays which optimizes the relationship between beam width and side-lobe level

Proc. IRE

A simple algorithm to achieve desired patterns for arbitrary arrays

IEEE Trans. Signal Process.

A flexible array synthesis method using quadratic programming

IEEE Trans. Antennas Propag.

Pattern synthesis of unequally spaced linear arrays including mutual coupling using iterative fft via virtual active element pattern expansion

IEEE Trans. Antennas Propag.

Modified compact genetic algorithm for thinned array synthesis

IEEE Antennas Wirel. Propag. Lett.

Particle swarm optimization versus genetic algorithms for phased array synthesis

IEEE Trans. Antennas Propag.

Synthesis of unequally spaced arrays by simulated annealing

IEEE Trans. Signal Process.

Application of convex relaxation to array synthesis problems

IEEE Trans. Antennas Propag.

Optimum Array Processing