# Efficient Large Sparse Arrays Synthesis by Means of Smooth Re-Weighted L1 Minimization

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

## 3. Smooth Re-Weighted L1 Minimization

## 4. Ring Population from the Calculated Excitation

- Choose the values of $\delta $, $\mathbf{d}$, and $\eta $ to use.
- we extract the ring radii and excitations from the sparse vector $\tilde{\mathbf{e}}$;

`cvx_begin`

`cvx_solver sedumi`

`variable e(nu)`

`minimize (norm (G*e, 1))`

`subject to`

`abs (A*e) <= Mup`

`A0*e == 1`

`cvx_end`

`e(nu)`” is the $\nu $ length column vector ${\tilde{\mathbf{e}}}^{p}$, “

`G`” is a diagonal matrix implementing the smooth weight (15), “

`A`" and “

`Mup`” are, respectively, the matrix and the column vector implementing (13), and “

`A0`” is a row vector implementing (14).

## 5. Numerical Examples

#### 5.1. Large Array with Variable Excitation

#### 5.2. Small Isophoric Sparse Array

#### 5.3. Very Large Isophoric Sparse Array

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Layout of the 597-element sparse array. Each radiator is represented by a circle of half a wavelength diameter.

**Figure 4.**Imagemap of the array factor in the $u,v$ plane for the 597-element array. The dual tone (pink/cyan) colormap does not show any violation of the SLL constraint out of the main beam.

**Figure 5.**Layout of the 167-element sparse array. Each radiator is represented by a circle of half a wavelength diameter.

**Figure 7.**Imagemap of the array factor in the $u,v$ plane for the 167-element array. The dual tone (pink/cyan) colormap does not show any violation of the SLL constraint out of the main beam.

**Figure 8.**Layout of the 3516-element sparse array. Each radiator is represented by a circle of half a wavelength diameter.

**Figure 10.**Imagemap of the array factor for the broadside beam in the $u,v$ plane for the 3516-element array. Only the inner part of the $u,v$ plane, relative to Earth’s surface as seen from GEO orbit is displayed.

**Figure 11.**Imagemap of the array factor for the beam scanned to 8° deg in the $u,v$ plane for the 3516 element array. Only the inner part of the $u,v$ plane, relative to the Earth’s surface as seen from GEO orbit, is displayed. The dual tone (pink/cyan) colormap does not show any violation of the SLL constraint out of the main beam.

# | ${\mathit{R}}_{\mathit{p}}$ ($\mathit{\lambda}$) | ${\mathit{N}}_{\mathit{p}}$ | ${\mathit{A}}_{\mathit{p}}$ |
---|---|---|---|

1 | 0.9 | 7 | 0.693 |

2 | 1.805 | 15 | 1 |

3 | 2.665 | 22 | 0.943 |

4 | 3.55 | 29 | 0.794 |

5 | 4.476 | 37 | 0.72 |

6 | 5.403 | 45 | 0.631 |

7 | 6.352 | 53 | 0.52 |

8 | 7.309 | 61 | 0.464 |

9 | 8.222 | 68 | 0.368 |

10 | 9.161 | 76 | 0.269 |

11 | 10.15 | 85 | 0.277 |

12 | 11.85 | 99 | 0.15 |

# | ${\mathit{R}}_{\mathit{p}}$ ($\mathit{\lambda}$) | ${\mathit{N}}_{\mathit{p}}$ |
---|---|---|

1 | 1.127 | 14 |

2 | 1.85 | 22 |

3 | 2.7 | 27 |

4 | 3.5 | 27 |

5 | 4.317 | 29 |

6 | 5.45 | 48 |

# | ${\mathit{R}}_{\mathit{p}}$ ($\mathit{\lambda}$) | ${\mathit{N}}_{\mathit{p}}$ |
---|---|---|

1 | 19.602 | 144 |

2 | 30.594 | 192 |

3 | 38.25 | 139 |

4 | 43.45 | 167 |

5 | 48.647 | 103 |

6 | 52 | 117 |

7 | 60.1 | 119 |

8 | 63.45 | 130 |

9 | 69.048 | 162 |

10 | 75.5 | 238 |

11 | 80.84 | 232 |

12 | 90.75 | 301 |

13 | 103.45 | 272 |

14 | 110.95 | 264 |

15 | 118.721 | 223 |

16 | 126.9 | 215 |

17 | 144.459 | 498 |

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**MDPI and ACS Style**

Pinchera, D.; Migliore, M.D.; Lucido, M.; Schettino, F.; Panariello, G. Efficient Large Sparse Arrays Synthesis by Means of Smooth Re-Weighted L1 Minimization. *Electronics* **2019**, *8*, 83.
https://doi.org/10.3390/electronics8010083

**AMA Style**

Pinchera D, Migliore MD, Lucido M, Schettino F, Panariello G. Efficient Large Sparse Arrays Synthesis by Means of Smooth Re-Weighted L1 Minimization. *Electronics*. 2019; 8(1):83.
https://doi.org/10.3390/electronics8010083

**Chicago/Turabian Style**

Pinchera, Daniele, Marco Donald Migliore, Mario Lucido, Fulvio Schettino, and Gaetano Panariello. 2019. "Efficient Large Sparse Arrays Synthesis by Means of Smooth Re-Weighted L1 Minimization" *Electronics* 8, no. 1: 83.
https://doi.org/10.3390/electronics8010083