# Geometrical Synthesis of Sparse Antenna Arrays Using Compressive Sensing for 5G IoT Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Synthesis Method

## 4. Results and Discussion

#### 4.1. First Example

#### 4.2. Second Example

#### 4.3. Third Example

#### 4.4. Fourth Example

#### 4.5. Fifth Example

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

IoT | Internet of Things |

CS | Compressive Sensing |

MBR | Main Beam Region |

SLR | SideLobe Region |

NR | Null Region |

SOCP | Second-Order Cone Program |

BS | Base Station |

AP | Access Point |

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**Figure 1.**First example: linear array. Flat-top pattern radiated by the optimized 19 elements (blue solid line), upper and lower bounds of the mask in the main beam region (MBR) (green solid line), upper bound of the mask in the sidelobe region (SLR) (red solid line), pattern synthesized by the 17 elements obtained after minimum inter-element distance control (red dashed line). The inset shows a zoom of the MBR. The final positions and excitations are listed in Table 2.

**Figure 2.**Second example: Linear array. Flat-top pattern radiated by the optimized 18 elements (blue solid line), desired pattern in the MBR (green solid line), upper bound of the mask in the SLR (red solid line). The inset shows a zoom of the MBR. The final positions and excitations are listed in Table 3.

**Figure 3.**Third example: linear array. Cosecant-like pattern radiated by the optimized 12 elements (blue solid line), desired pattern in the MBR (green solid line), upper bound of the mask in the SLR and null region (NR) (red solid line). The final positions and excitations are listed in Table 4.

**Figure 5.**Fourth example: positions of the candidate (red cross) and final (blue circles) array elements.

**Figure 7.**Fifth example: positions of the candidate (red cross) and final (blue circles) array elements.

Step 1 | Problem specifications(i) Initial set of candidate positions ${\overline{\mathbf{r}}}_{n}$ ($n=1,\dots ,N$); (ii) Upper ${M}^{\mathrm{up}}\left(\widehat{\mathbf{r}}\right)$ and lower ${M}^{\mathrm{low}}\left(\widehat{\mathbf{r}}\right)$ bounds of the mask; (iii) MBR, SLR, and NR; |

Step 2 | Initializations(i) Iteration $k=0$; (ii) Excitations ${w}_{n}^{0}=1$ ($n=1,\dots ,N$). |

Step 3 | Updates(i) Iteration $k\to k+1$; (ii) Current pattern $F({\mathbf{w}}^{k-1},\widehat{\mathbf{r}})$ using (1); |

Step 4 | EvaluationSolve the SOCP problem given by (8a)–(8c). |

Step 5 | Stop conditionIf $k\ge 3$ and $\parallel {\mathbf{w}}^{k}{\parallel}_{0}=\parallel {\mathbf{w}}^{k-1}{\parallel}_{0}={\parallel {\mathbf{w}}^{k-2}\parallel}_{0}$ ${\mathbf{w}}^{k}$ identifies the elements of the final sparse array and their excitations; Exit else Go to Step 3 end |

n | ${\mathit{z}}_{\mathit{n}}/\mathit{\lambda}$ | ${\mathit{w}}_{\mathit{n}}/{\mathit{w}}_{\mathbf{10}}$ | n | ${\mathit{z}}_{\mathit{n}}/\mathit{\lambda}$ | ${\mathit{w}}_{\mathit{n}}/{\mathit{w}}_{\mathbf{10}}$ | n | ${\mathit{z}}_{\mathit{n}}/\mathit{\lambda}$ | ${\mathit{w}}_{\mathit{n}}/{\mathit{w}}_{\mathbf{10}}$ |
---|---|---|---|---|---|---|---|---|

1 | −9.74 | −0.0020 | 7 | −1.97 | −0.2027 | 14 | 3.25 | 0.1189 |

2 | −8.48 | 0.0283 | 8 | −0.65 | 0.1223 | 15 | 4.58 | −0.0862 |

3 | −7.18 | −0.0349 | 9 | −0.64 | 0.5128 | 16 | 5.87 | 0.0540 |

4 | −5.87 | 0.0570 | 11 | 0.64 | 0.4826 | 17 | 7.18 | −0.0399 |

5 | −4.58 | −0.0748 | 12 | 0.65 | 0.1515 | 18 | 8.48 | 0.0186 |

6 | −3.25 | 0.1252 | 13 | 1.97 | −0.2138 | 19 | 9.74 | −0.0172 |

n | ${\mathit{z}}_{\mathit{n}}/\mathit{\lambda}$ | $\left(|{\mathit{w}}_{\mathit{n}}|\right.,\text{}\left.\angle {\mathit{w}}_{\mathit{n}}\right)\phantom{\rule{1.em}{0ex}}$ | n | ${\mathit{z}}_{\mathit{n}}/\mathit{\lambda}$ | $\left(|{\mathit{w}}_{\mathit{n}}|\right.,\text{}\left.\angle {\mathit{w}}_{\mathit{n}}\right)\phantom{\rule{1.em}{0ex}}$ | n | ${\mathit{z}}_{\mathit{n}}/\mathit{\lambda}$ | $\left(|{\mathit{w}}_{\mathit{n}}|\right.,\text{}\left.\angle {\mathit{w}}_{\mathit{n}}\right)\phantom{\rule{1.em}{0ex}}$ |
---|---|---|---|---|---|---|---|---|

1 | −7.33 | (0.0510, −77.4970) | 7 | −2.25 | (0.1816, 110.4115) | 13 | 1.92 | (0.2459, −22.6474) |

2 | −6.01 | (0.0065, −34.0712) | 8 | −1.92 | (0.2638, 11.3857) | 14 | 2.25 | (0.1407, −120.9294) |

3 | −4.93 | (0.1053, 50.8963) | 9 | −0.73 | (0.5726, 101.9257) | 15 | 3.24 | (0.1420, 29.7340) |

4 | −4.58 | (0.1075, −46.5715) | 10 | −0.27 | (1.0000, 28.4632) | 16 | 3.56 | (0.1400, −78.2697) |

5 | −3.56 | (0.0257, 65.7662) | 11 | 0.27 | (0.9706, −24.9160) | 17 | 6.01 | (0.1090, 67.4450) |

6 | −3.24 | (0.0771, 7.0465) | 12 | 0.73 | (0.5570, −97.4171) | 18 | 6.32 | (0.1081, −58.2957) |

n | ${\mathit{z}}_{\mathit{n}}/\mathit{\lambda}$ | n | ${\mathit{z}}_{\mathit{n}}/\mathit{\lambda}$ | n | ${\mathit{z}}_{\mathit{n}}/\mathit{\lambda}$ | |||
---|---|---|---|---|---|---|---|---|

1 | −3.72 | (0.5171, 15.4295) | 5 | −1.50 | (0.7644, −149.0791) | 9 | 1.38 | (0.1902, −31.2221) |

2 | −3.01 | (1.0000, 100.9695) | 6 | −0.70 | (0.6749, −113.2490) | 10 | 2.12 | (0.2963, −8.6059) |

3 | −2.53 | (0.0721, 156.1117) | 7 | 0.14 | (0.4836, −77.9490) | 11 | 2.93 | (0.1309, 3.5823) |

4 | −2.29 | (0.9066, 171.0874) | 8 | 0.83 | (0.3659, −55.6914) | 12 | 3.75 | (0.2083, 72.6957) |

n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | −0.7703 | 11 | 3.9079 | 21 | 9.2605 | 31 | 4.3223 | 41 | 4.2946 | 51 | 3.1712 |

2 | −0.7830 | 12 | 4.1300 | 22 | 8.3720 | 32 | 1.1883 | 42 | 1.5319 | 52 | −0.6131 |

3 | −1.1008 | 13 | −1.1552 | 23 | 2.6786 | 33 | −0.7428 | 43 | 4.2036 | 53 | −0.5462 |

4 | −0.9977 | 14 | 4.2489 | 24 | −0.6591 | 34 | 3.9587 | 44 | −1.0944 | 54 | 0.2170 |

5 | −1.0158 | 15 | 4.1157 | 25 | 2.8896 | 35 | 5.2375 | 45 | 0.9100 | 55 | −1.0237 |

6 | −0.6747 | 16 | −1.0884 | 26 | 2.1762 | 36 | 6.5611 | 46 | 3.5755 | 56 | −1.1029 |

7 | −0.4380 | 17 | 0.7129 | 27 | 3.1437 | 37 | 3.9997 | 47 | 3.0698 | 57 | −0.3591 |

8 | −0.5442 | 18 | 1.4489 | 28 | −0.9362 | 38 | −0.2134 | 48 | 1.0584 | 58 | −1.0865 |

9 | −0.9632 | 19 | −0.7999 | 29 | 0.7927 | 39 | 9.3600 | 49 | −0.9486 | 59 | −1.5257 |

10 | −1.0186 | 20 | 3.6794 | 30 | 7.0626 | 40 | −0.1826 | 50 | 2.8547 | 60 | −1.3586 |

n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ | n | ${\mathit{w}}_{\mathit{n}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | −0.4287 | 12 | 2.5582 | 23 | 1.6233 | 34 | 2.8681 | 45 | 1.1385 | 56 | 1.6511 |

2 | −0.8324 | 13 | 2.3506 | 24 | 2.0073 | 35 | 0.8899 | 46 | 1.3016 | 57 | 1.7920 |

3 | −1.3767 | 14 | 1.6307 | 25 | 2.3949 | 36 | 2.1484 | 47 | 2.2395 | 58 | 2.1606 |

4 | 0.5247 | 15 | 1.0195 | 26 | 1.5068 | 37 | 1.4631 | 48 | 2.3995 | 59 | 1.5831 |

5 | 0.7506 | 16 | 2.4243 | 27 | 1.1951 | 38 | 1.5917 | 49 | 2.8561 | 60 | 2.6363 |

6 | 1.7065 | 17 | 1.4812 | 28 | 1.9673 | 39 | 0.7089 | 50 | 1.1195 | 61 | 1.3877 |

7 | 2.0100 | 18 | 1.5143 | 29 | 2.4747 | 40 | 0.4596 | 51 | 1.4155 | 62 | 1.1193 |

8 | 1.6065 | 19 | 2.4225 | 30 | 2.6175 | 41 | 2.3680 | 52 | 1.2087 | ||

9 | 2.1001 | 20 | 0.5591 | 31 | 1.9543 | 42 | 1.0167 | 53 | 1.7925 | ||

10 | 1.3948 | 21 | 1.1460 | 32 | 3.0988 | 43 | 1.7411 | 54 | 1.6238 | ||

11 | 2.0795 | 22 | 2.2760 | 33 | 2.6850 | 44 | 1.1867 | 55 | 1.6902 |

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**MDPI and ACS Style**

Buttazzoni, G.; Babich, F.; Vatta, F.; Comisso, M.
Geometrical Synthesis of Sparse Antenna Arrays Using Compressive Sensing for 5G IoT Applications. *Sensors* **2020**, *20*, 350.
https://doi.org/10.3390/s20020350

**AMA Style**

Buttazzoni G, Babich F, Vatta F, Comisso M.
Geometrical Synthesis of Sparse Antenna Arrays Using Compressive Sensing for 5G IoT Applications. *Sensors*. 2020; 20(2):350.
https://doi.org/10.3390/s20020350

**Chicago/Turabian Style**

Buttazzoni, Giulia, Fulvio Babich, Francesca Vatta, and Massimiliano Comisso.
2020. "Geometrical Synthesis of Sparse Antenna Arrays Using Compressive Sensing for 5G IoT Applications" *Sensors* 20, no. 2: 350.
https://doi.org/10.3390/s20020350