2.2 The proposed algorithm

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

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 pat-

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

2. Nonuniform array pattern synthesis using reweighted l1-norm

Consider a narrowband linear array with M isotropic antennas located at

across the array with incident direction θ. The M signal outputs si are converted to the baseband, weighted by the weights wi, and summed. Then the array response

. Assume that a harmonic plane wave with wavelength λ propagates

tern 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

also been used for APS problem optimization.

Advances in Array Optimization

the main diagonal elements equaled to the vector x.

[23–26].

minimization

x1, …, xM ∈R<sup>2</sup>

36

can represented as

2.1 Problem formulation

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

$$\mathfrak{w}^T \mathfrak{a}(\theta) = G(\theta), \forall \theta \in \Theta \tag{3}$$

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:

#### 2.2.1 Creating a virtual array

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 subsequent processing.

#### 2.2.2 Finding the sparse weight vector

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

$$\begin{aligned} \text{Minimize } & \|\mathsf{Q}w\|\_1 \\ \text{Subject to } & \|G(\theta) - G\_d(\theta)\|\_\infty \le \varepsilon, \forall \theta \in [-\mathsf{90}^\*, \mathsf{90}^\*] \end{aligned} \tag{4}$$

where ε is the fitting error between the synthesized pattern and desired pattern. 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 APS performance and convergence rate. Because more elements of the original weight vector will be pruned if the threshold value η is increased, which make us probably cannot find the optimal array element positions of the array, correspondingly the array synthesis performance is not optimal for a given array element number. Conversely, if the threshold value is decreased, less elements of the original weight vector will be pruned in each iteration, which increases the algorithm complexity. So we should make a good balance between APS performance and convergence rate when setting the value of η.
