1. Introduction

The objective of array pattern synthesis (APS) is to find the excitation of an array to produce a radiation beampattern which is close to the desired one. Dolph-Chebyshev method [1, 2] can be used to design an optimal pattern with the minimum sidelobe level and desired mainlobe width for a uniform linear array (ULA) with isotropic elements. While it is more difficult to solve the APS problem for an array of arbitrary geometric structures.

For nonuniformly spaced arbitrary arrays, there are several algorithms [3–6] that have been proposed to synthesize beampatterns. The design of thinned narrowbeam arrays has been well proposed in [3], which first fix element locations by eliminating the elements pair by pair according to the smallest possible sidelobe on the given interval and then optimize the weights via linear programming. For APS problem, which can also be formulated as a quadratic programming problem [4, 5], the objective function is to minimize the squared errors between the synthesized pattern and the desired pattern. Besides, additional linear constraints [4] or weighting functions [7] are also added to the quadratic objective function to minimize the peaks of the synthesis error. The challenge to weighting functions in the

quadratic programming is that it has to be adjusted in an ad hoc manner. Besides, an inverse matrix has to be computed at each iteration for updating the weighting functions, which will result in high computation requirements, especially for large size of the array. The author of [8] proposed a recursive least squares method to solve the problem. Another kind of evolutionary algorithm, such as simulated annealing [9], particle swarm optimization [10], and genetic algorithm [11–13], has also been used for APS problem optimization.

<sup>G</sup>ð Þ¼ <sup>θ</sup> <sup>X</sup> M

DOI: http://dx.doi.org/10.5772/intechopen.88881

vector w ¼ ½ � w1; …; wM

where <sup>Θ</sup> <sup>∈</sup> �90<sup>∘</sup>

2.2 The proposed algorithm

2.2.1 Creating a virtual array

subsequent processing.

37

2.2.2 Finding the sparse weight vector

using the peak error across θ, i.e.,

i¼1

min <sup>w</sup> max

We try to find w in Eq. (3) such that Eq. (2) is satisfied. The new solution of Eq. (3) can be summarized as follows:

where ϕ<sup>i</sup> ¼ 2πxi sin θ=λ is the phase delay due to propagation, complex weight

Convex Optimization and Array Orientation Diversity-Based Sparse Array Beampattern Synthesis

<sup>T</sup> <sup>∈</sup>CM, and the steering vector <sup>a</sup>ð Þ<sup>θ</sup> . Let Gdð Þθ be the desired array response at the direction θ. The APS problem is to

the array described above, how well Gð Þθ approximates Gdð Þθ can be measured by

; <sup>90</sup><sup>∘</sup> ½ � is a dense set of arrival angles that we are of interest. The

find the complex weight vector <sup>w</sup> such that <sup>G</sup>ð Þ¼ <sup>θ</sup> Gdð Þ<sup>θ</sup> for all <sup>θ</sup> <sup>∈</sup> �90<sup>∘</sup>

goal of the proposed algorithm is to find both optimal antenna locations and corresponding weights that approach the desired array pattern as well as possible.

The APS problem can be formulated as a following estimation problem:

For a given array size, to obtain more elements than those of a conventional array with λ=2 inter-element spacing, we first create a dense uniformly spaced linear array with much smaller inter-element spacing than conventional array and initialize a weight matrix Q as an identity matrix to create a more sparse array in

The specified synthesized pattern Gð Þθ is produced by a weight vector. The weight vector can be obtained by solving the following weighted l1-norm minimization convex problem Eq. (4), which is subject to minimizing the peak of the error

Subject to k k <sup>G</sup>ð Þ� <sup>θ</sup> Gdð Þ<sup>θ</sup> <sup>∞</sup> <sup>≤</sup> <sup>ε</sup>, <sup>∀</sup><sup>θ</sup> <sup>∈</sup> �90<sup>∘</sup>

Minimizing k k Qw <sup>1</sup> makes the vector Qw sparse, which is useful to create a nonuniformly spaced array. According to the situation that some weights of the original weight vector <sup>w</sup> <sup>¼</sup> ½ � <sup>w</sup>1; <sup>w</sup>2; … <sup>T</sup> from Eq. (4) are very small, they can be deleted without significantly decreasing the array performance. So a sparse weight vector can be obtained by retuning the small value elements of the original weight vector, that is, the wi will be retained if wi j j=k k w <sup>∞</sup> >η ð Þ i ¼ 1; 2; … , otherwise wi ¼ 0. The η is a designed threshold whose value should make a trade-off between

where ε is the fitting error between the synthesized pattern and desired pattern.

between the synthesized pattern Gð Þθ and the desired pattern Gdð Þθ :

Minimize k k Qw <sup>1</sup>

wi exp ð Þ¼ <sup>j</sup>2πxi sin <sup>θ</sup>=<sup>λ</sup> <sup>w</sup><sup>T</sup>að Þ<sup>θ</sup> (1)

<sup>θ</sup> <sup>∈</sup> <sup>Θ</sup> j j <sup>G</sup>ð Þ� <sup>θ</sup> Gdð Þ<sup>θ</sup> (2)

<sup>w</sup><sup>T</sup>að Þ¼ <sup>θ</sup> <sup>G</sup>ð Þ<sup>θ</sup> , <sup>∀</sup><sup>θ</sup> <sup>∈</sup> <sup>Θ</sup> (3)

; <sup>90</sup><sup>∘</sup> ½ � (4)

; <sup>90</sup><sup>∘</sup> ½ �. For

Recently, second-order cone programming (SOCP) and semi-definite programming (SDP), as convex optimization techniques [14, 15], have been proposed to solve the APS problem readily by using SOCP solver and SDP solver, respectively. While a general nonuniform APS problem cannot be directly formulated as a convex problem. An iterative procedure [15] was proposed to optimize the array pattern by solving an SDP problem at each iteration. All the abovementioned approaches to design an optimal nonuniform array are to construct an objective function of minimizing the synthesis error or peak error. When the positions of elements are given, the nonuniformly spaced arrays can be optimized using convex programming like that for uniformly spaced arrays. While it is impossible to solve the APS problem by complex programming if the positions of the array elements are unknown. In addition, to solve the problem of occupying more elements to obtain the desired beampattern, the authors in [16] proposed a matrix pencil-based noniterative synthesis algorithm, which can efficiently save the number of elements in a very short computation time. Zhang et al. [17] formulated the APS problem as a sparseness constrained optimization problem and solved the problem by using Bayesian compressive sensing (BCS) inversion algorithm; the authors in [18] proposed an approach for APS of linear sparse arrays, and then the multitask BCS has been used to design 2D sparse synthesis problem [19], sparse conformal array synthesis problem [20–22], and another CS-based sparse array synthesis problem [23–26].

In this chapter, we proposed an array pattern synthesis algorithm [27] by using reweighted l1-norm minimization [28] and convex optimization [29]. Then we extended our work to a new version [30] by using reweighted l1-norm minimization and array orientation diversity. Merits of the algorithm include the following: (1) it does not need a thorough search in the multidimensional parameter space, and (2) it can achieve the same array performance with fewer antenna elements when the array size is given and thus reduces the array cost significantly. Regarding the notation of this chapter, ð Þ� <sup>T</sup> represents the transpose operation of a vector or matrix, j j� denotes the absolute value operator, and k k� <sup>1</sup> and k k� <sup>∞</sup> represent the l1-norm and l∞-norm of a vector or matrix, respectively. And d ex denotes the smallest integer not less than x, and diag xð Þ means the diagonal matrix with the main diagonal elements equaled to the vector x.
