Our element positions and weights in a 19-element antenna array.

## Abstract

The sparse array pattern synthesis (APS) has many important implications in some special situations where the weights, size, and cost of antennas are limited. In this chapter, the APS with a minimum number of elements problem is investigated from the perspective of sparseness constrained optimization. Firstly, to reduce the number of antenna elements in the array, the APS problem is formulated as sparseness constrained optimization problem under compressive sensing (CS) framework and solved by using the reweighted L1-norm minimization algorithm. Besides, to address left-right radiation pattern ambiguity problem, the proposed algorithm exploits the array orientation diversity in the sparsity constraint framework. Simulation results demonstrate the proposed method’s validity of achieving the desired radiation beampattern with the minimum number of antenna elements.

### Keywords

- array beampattern synthesis
- compressive sensing
- array orientation diversity
- convex optimization

## 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, 4, 5, 6] that have been proposed to synthesize beampatterns. The design of thinned narrow-beam 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, 12, 13], has also been used for APS problem optimization.

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 non-iterative 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, 21, 22], and another CS-based sparse array synthesis problem [23, 24, 25, 26].

In this chapter, we proposed an array pattern synthesis algorithm [27] by using reweighted *l*_{1}-norm minimization [28] and convex optimization [29]. Then we extended our work to a new version [30] by using reweighted

## 2. Nonuniform array pattern synthesis using reweighted *l*_{1}-norm minimization

### 2.1 Problem formulation

Consider a narrowband linear array with

where

Let

where

### 2.2 The proposed algorithm

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

We try to find

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

#### 2.2.2 Finding the sparse weight vector

The specified synthesized pattern

where

#### 2.2.3 Updating the weight matrix

After obtaining the original weight vector

#### 2.2.4 Forming the nonuniform array

The sparse weight vector

The above steps (A, B, C) are repeated until the final synthesized array performance is satisfactory or the specified maximum number of iterations is attained.

#### 2.2.5 Optimizing the sparse weight vector

After obtaining the array antenna positions by steps (B, C, D), the optimal weight vector

### 2.3 Computer simulation and discussions

Given the array aperture, the objective is to design an array with the beampattern as shown in Figure 1, where region

The beampattern of Figure 2 is obtained by using our approach for a 19-element array, and the optimal beampattern exhibits the maximum sidelobe of −15.46 dB. The optimal antenna positions and the corresponding weights are displayed in Table 1. The designs proposed in [3, 17] describe a 25-element and a 29-element non-ULA with the approximate desired array pattern shown in Figure 2, respectively. The 25-element array beampattern described in [3] by eliminating the elements pair by pair has a maximum sidelobe −13.75 dB, while the maximum sidelobe of 29-element array beampattern obtained by the BCS algorithm [17] is −13.165 dB. The antenna positions and the corresponding weights of the two methods in [3, 17] are listed in Tables 2 and 3, respectively. Compared with the method in [3], we can see from Table 1 and Figure 2 that our proposed algorithm saves six elements without reducing the array performance and the our minimum inter-element spacing of the non-ULA is 0.375*λ* larger than that of the method of eliminating the elements pair by pair [3]. Compared with the reference method in [17], our proposed method offers an economization of 10 elements as well as 2.3 dB performance improvement, and the minimum inter-element spacing of the sparse array designed by our approach is 0.75*λ* larger than that of the reference array [17]. We also emphasize that the reference array [17] has 4.5*λ* larger array aperture than that of our array.

Element indices | Position (λ) | Weight value | Element indices | Position (λ) | Weight value |
---|---|---|---|---|---|

1,19 | ±10.500 | 0.2289 | 6,14 | ±3.875 | 0.2677 |

2,18 | ±8.625 | 0.2583 | 7,13 | ±2.875 | 0.2195 |

3,17 | ±7.6250 | 0.2207 | 8,12 | ±1.875 | 0.1813 |

4,16 | ±6.750 | 0.2904 | 9,11 | ±1.000 | 0.2347 |

5,15 | ±4.750 | 0.1567 | 10 | 0 | 0.2427 |

Element indices | Position (λ) | Weight value | Element indices | Position (λ) | Weight value |
---|---|---|---|---|---|

1,25 | ±12.0 | 0.2100 | 8,18 | ±4.5 | 0.1924 |

2,24 | ±8.5 | 0.2605 | 9,17 | ±3.5 | 0.2296 |

3,23 | ±8.0 | 0.2276 | 10,16 | ±2.0 | 0.2282 |

4,22 | ±7.5 | 0.2554 | 11,15 | ±1.5 | 0.0876 |

5,21 | ±7.0 | 0.2103 | 12,14 | ±0.5 | 0.1143 |

6,20 | ±6.0 | 0.200 | 13 | 0 | 0.2084 |

7,19 | ±5.0 | 0.2037 |

Element indices | Position (λ) | Weight value |
---|---|---|

1,2 | −1.375, −0.500 | 0.0876, 0.1178 |

3,4 | 0.375, 0.500 | 0.0532, 0.1025 |

5,6 | 1.250, 1.375 | 0.0497, 0.1844 |

7,8 | 2.125, 4.375 | 0.1895, 0.1086 |

9,10 | 4.500, 5.250 | 0.0778, 0.2679 |

11,12 | 6.125, 7.000 | 0.2440, 0.1098 |

13,14 | 7.125, 8.125 | 0.0643, 0.2400 |

15,16 | 8.875, 10.125 | 0.1953, 0.2249 |

17,18 | 11.000, 11.750 | 0.2297, 0.0720 |

19,20 | 12.000, 12.750 | 0.1480, 0.1810 |

21,22 | 13.750, 13.875 | 0.0761, 0.0554 |

23,24 | 14.875, 15.750 | 0.0840, 0.1833 |

25,26 | 16.625, 16.750 | 0.1860, 0.0516 |

27,28 | 17.375, 19.625 | 0.1625, 0.0745 |

29 | 24.125 | 0.0317 |

The proposed APS algorithm based on convex optimization and reweighted *l*_{1}-norm minimization is proven to be effective in reducing array elements, suppressing the sidelobe, and reducing the aperture. This simple and effective design method can be extended to solving the 2D array synthesis problem.

## 3. Beampattern synthesis using reweighted *l*_{1}-norm minimization and array orientation diversity

To address left-right radiation pattern ambiguity problem, we allow exploitation of the array orientation diversity in the CS framework.

### 3.1 Problem formulation

We assume that transmit signals and the array are coplanar, so the antenna array synthesis problem can be described as follows:

where

Suppose that all the antenna elements in each array orientation

In order to solve Eqs. (7) and (8), we can assume that all the antenna elements are equally spaced from

where h is the number of sampled antenna radiation pattern, *l*th position of the

In Eq. (10) the smallest number of nonzero elements in the excitation vector

### 3.2 The proposed algorithm

In this subsection, the new solution of Eq. (6) can be summarized as follows:

#### 3.2.1 Initializing a virtual array and a weight matrix

To place more antenna elements than those of a conventional array with the same array size, we first create *l*_{1}-norm minimization in the following step, we set a

#### 3.2.2 Finding the sparse weight vector

Let *l*_{1}-norm minimization convex problem which is to try to minimize the peak value of the error between the synthesized pattern and the desired pattern:

where

#### 3.2.3 Updating the weight matrix

After obtaining the original weight vector

#### 3.2.4 Creating the nonuniform arrays

After obtaining the sparse weight vector

Repeat steps (2, 3, and 4) until the synthesized array beampattern performance is satisfactory or the specified maximum number of iterations or minimum antenna number is attained.

#### 3.2.5 Finding the optimal weight vector

After optimizing the antenna element positions by the above steps, we introduce convex optimization to obtain the optimal weight vector which can further improve the performance of the array beampattern synthesized by the sparse weight vector:

The optimal sparse weight vector

### 3.3 Computer simulations and discussion

The objective is to design an array with the desired beampattern for given the array physical size, as shown in Figure 1, where region

To show the performance of our beampattern synthesis, we will consider two cases, same element number array and same beampattern performance, since all formulated problems in Eqs. (6), (10), (11), and (12) are convex, so we adopt the optimization toolbox to solve the formulated problems.

#### 3.3.1 Same element number array with array orientation diversity

In this section, we analyzed the influence of the array orientation diversity on the beampattern synthesis by simulation results. We initialize four virtual ULAs (named Array 1, Array 2, Array 3, Array 4, with orientation

#### 3.3.2 Approximate beampattern performance with array orientation diversity

To demonstrate another advantage of array orientation diversity, we examine the beampattern synthesis of an 18-element array, 11-element array, 10-element array, and 10-element array correspondingly with one orientation, two orientations, three orientations, and four orientations using BCS algorithm and our method, respectively. The optimal beampatterns exhibit maximal sidelobes of −7.72, −8.01, −7.59, and −7.88 dB, respectively, which are shown in Figure 9. Figures 10–13 provide all the corresponding antenna positions and excitation amplitudes for all the four cases mentioned above. Obviously, given the array size, using orientation diversity can economize seven (or eight) elements without reducing the array performance. But more diversity is not always better enough, as shown in Figures 11–13. Besides, the excitation amplitudes in Figures 10–13 show that our proposed method needs less radiation energy for all four cases.

The proposed APS algorithm based on reweighted

## 4. Conclusions

This chapter focuses on the APS problem with sparse antenna array, which has practical applications, especially for massive antenna array. By using array orientation diversity and solving reweighted

## Acknowledgments

This work was supported in part by Sichuan Science and Technology Program (No. 18ZDYF2551) and in part by Fundamental Research Funds for the Central Universities (Program No. ZYGX2018J005).