Pattern synthesis with minimum mainlobe width via sparse optimization

Highlights

• Achieving minimum mainlobe width is formulated as sparse optimization problem with $l_0$ norm.

• A log-sum-exp penalty function is proposed to replace $l_0$ norm for the sparse optimization problem.

• The proposed method is suitable for not only focused beam pattern but also shaped beam pattern.

• The proposed method is suitable for arbitrary arrays.

• There is no need to accurately determine the mainlobe region.

Abstract

This paper presents a pattern synthesis algorithm achieving minimum mainlobe width via sparse optimization. The problem of synthesizing a pattern with minimum mainlobe width is formulated as a sparse optimization problem with $l_0$ norm by introducing a slack variable. To solve the sparse optimization problem, three existing relaxations for $l_0$ norm are presented. Moreover, a novel log-sum-exp penalty function is proposed to replace $l_0$ norm to be minimized in this paper, leading to a convex problem which can be solved directly and do not need to solve a sequence of sparse optimization problems compared with the iterative reweighted $l_1$ norm. Both the focused beam pattern and shaped beam pattern for arbitrary arrays can be synthesized utilizing the proposed algorithm. Besides, the mainlobe width of pattern synthesized by the proposed algorithm is minimum. Simultaneously, it brings an extra advantage that it is no longer necessary to accurately determine the mainlobe region without synthesis performance degradation. Numerical examples are presented to verify the effectiveness and superiority of the proposed algorithm.

Introduction

The pattern synthesis of sensor array is an issue of increasing relevance and has been extensively applied in many fields, such as, radar, wireless communication and remote sensing [1]. For instance, synthesizing a pattern with nulls is an effective way to suppress interference in the radar system. For some communication scenarios, multiple-beams pattern is designed to achieve multi-user reception. In addition, shaping a pattern with flat-top mainlobe is attractive in remote sensing field [2], [3].

In the past decades, many approaches have been developed for pattern synthesis. For example, uniform sidelobes were achieved by the classical Chebyshev (Cheb) method with the beam width to the first null that is the minimum for given sidelobe level in [4]. However, the Cheb method is limited to the uniformly spaced and isotropic element arrays. To synthesis nonuniform arrays, global search algorithms, such as, genetic algorithm [5], simulated annealing method [6], and particle swarm optimization method [7], [8], have been discussed in the literature. Thanks to the powerful mathematical tool, many effective convex optimization approaches have been applied to pattern synthesis. In [9], it has shown how to formulate the problem of synthesizing pattern as an optimization problem. Furthermore, semidefinite relaxation methods [10], [11], [12] and optimization methods [13], [14], [15] have been investigated. Besides, the alternative direction method of multipliers (ADMM) framework has been also utilized to improve the array pattern synthesis performance [16], [17].

Recently, some fast methods based on fast Fourier transform (FFT) have been presented. For instance, the iterative FFT was applied to synthesize unequally spaced arrays in [18]. Then, it has been extended as iterative spatiotemporal Fourier transform to design filter coefficients for generating frequency-invariant beam pattern in [19]. Moreover, a modified iterative FFT technique synthesized thinned massive array for 5G communications in [20]. Additionally, to control the array response level at some angles, a numerical pattern synthesis (NPS) method by imposing artificial interferences was discussed in [21]. What's more, to flexibly and precisely control the array response, an accurate array response control (A²RC) approach [22], a weight vector orthogonal decomposition (WORD) approach [23] and an optimal and precise array response control (OPARC) [24] were presented. Besides, a multi-point accurate array response control (MA²RC) [25], a multi-point based on improved WORD (MI-WORD) algorithm [26] and a flexible array response control via oblique projection (FARCOP) [27] were developed to adjust multi-point responses.

It is noticed that some of the afore-mentioned methods have referred to the mainlobe width. But they lack further researching on minimizing the mainlobe width or some of them exist restrictions. For example, the Cheb method in [4] is limited to not only the uniform geometry but also the uniform sidelobe level. The method mentioned in [8] needs to redesign the element spacing to minimize the beam width after the desired sidelobe level is achieved. In addition, it only focuses on determining the mainlobe region in [21]. To the best of our knowledge, a more practical synthesis method to achieve optimal minimum mainlobe width has not been discussed yet. Besides, to synthesize perfectly, the sidelobe region is needed to be determined precisely in most of existing methods, while it is hard to know the exact sidelobe region for some patterns. To overcome the afore-mentioned weaknesses, we present a new algorithm which synthesizes pattern for arbitrary arrays with minimum mainlobe width in this paper. By introducing a slack variable, the problem of synthesizing pattern with minimum mainlobe width can be formulated as a sparse optimization problem with $l_0$ norm.

Then, the works are focused on finding a solution for sparse optimization problem with $l_0$ norm. A general method is replacing the $l_0$ norm with $l_1$ norm as penalty function, leading to a solvable convex optimization problem [28]. However, the sparse performance of the $l_1$ minimization is limited. The iterative reweighted $l_1$ minimization algorithm is more efficient compared with $l_1$ minimization [29], [30], [31]. Nevertheless, it needs to solve a sequence of weighted $l_1$ norm minimizations which may lead to other problems, for example, determining the number of iterations. Lately, a log-sum penalty function which can replace $l_0$ norm was shown in [32]. Considering that the initial problem with log-sum penalty function is non-convex, it is hard to find a global solution although approximate solution ($l_1$ reweighting and $l_2$ reweighting) was provided in [33]. Inspired by the log-sum sparse encourage function, a new penalty function, i.e., log-sum-exp function, is firstly proposed to replace $l_0$ norm in this paper. On this basis, the sparse optimization problem of synthesizing pattern with minimum mainlobe width can be converted to a log-sum-exp sparsity encourage function minimization problem, which can be solved efficiently.

The rest of the paper is organized as follows. In Section 2, the array model and the problem of pattern synthesis is briefly introduced. The proposed algorithm is presented in Section 3. Numerical examples are shown in Section 4 and conclusions are drawn in Section 5.