# Low Discrepancy Sparse Phased Array Antennas

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}[21], The Atacama Large Millimeter/submillimeter Array (ALMA) that has 10 parabolic reflectors that are 12 m in diameter forming a total collecting area of 6600 m

^{2}[22], and the Very Large Array (VLA) that has 27 parabolic reflectors that are 25 m that extend in a “Y” shape with each arm over 15 km long [23].

- Sufficient elements to achieve a desired gain.
- Aperture size is large enough to achieve the desired beamwidth.
- Elements have a minimum separation distance, so they can physically fit into the aperture and mutual coupling is not a problem.
- No GLs present at maximum scan angles.
- The average element spacing is greater than $\lambda $.

## 2. Sampling Points on a Planar Aperture

_{s}and v

_{s}are the main beam location in the sine space.

#### 2.1. Random Sampling Approaches

#### 2.1.1. Random Sampling

#### 2.1.2. Random Sampling with Jitter

#### 2.1.3. Random Hyperuniform Spatial Arrangements

^{d}, i.e.,

#### 2.2. Low Discrepancy Sampling Approaches

#### 2.2.1. Hammersley Sampling

#### 2.2.2. Halton Sampling

_{1}and b

_{2}are two different prime bases. Note that Halton sampling is hierarchical.

#### 2.2.3. Sobol Sampling

#### 2.2.4. Poisson Disk Sampling

_{i}}, from a given domain, D, in N-dimensional space, that are tightly packed, but no closer than a specified minimum distance r. Here N is the number of elements in the array. The samples are at least a minimum distance apart, satisfying an empty disk criterion, i.e.,

## 3. Sparse Planar Phased Array Antennas

#### 3.1. Element Distributions on the Aperture

#### 3.2. Radiation Patterns of the Sparse Phased Array Antennas

_{n}is the element weight, and the other terms are as defined in (1). Here we consider elements that have unity amplitude and zero phase, so a

_{n}= 1. The elements are also isotropic, so E(u, v) = 1.

#### 3.3. Quantitative Analysis of Sparse Array Performances

_{SLL}), to the pattern value of the main lobe (F

_{max}). We note that for a boresight beam F

_{max}is the value of the far-field radiation pattern at F(0, 0). The directivity is defined as

## 4. Aperture Shape Effects on the Performance of Sparse Phased Array Antennas

^{2}as the square aperture studied in Section 3. The rectangular aperture has an aspect ratio of 9/4, i.e., the ratio of its longer side to its shorter side, corresponding to longer and shorter side lengths of 48λ and 64λ/3, respectively. The circular aperture has a radius of 18λ. To maintain the same aspect ratio as the rectangle and the same aperture size as the other arrays, the elliptical array major and minor axis are 27λ and 12λ, respectively. Here we compare the performance of two LDS sparse arrays that showed the best performance, namely Halton (with bases of 2 and 7) and Poisson distributions, with uniform distribution. The element distributions for these arrays are given in Figure 6.

## 5. Beam-Scanning Performance of Sparse Phased Array Antennas

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**A generic representation of the position of points on a unit-square with different distributions: (

**a**) uniform, (

**b**) random, (

**c**) LDS Poisson disk. The total number of elements is equal in all three cases but note that the LDS method fills the space more uniformly while avoiding too close spacings.

**Figure 3.**Position of elements on the aperture with different distributions: (

**a**) uniform, (

**b**) random, (

**c**) random with jitter, (

**d**) Hammersley (base 2), (

**e**) Hammersley (base 3), (

**f**) Hammersley (base 5), (

**g**) Hammersley (base 7), (

**h**) Halton (bases 2, 3), (

**i**) Halton (bases 2, 5), (

**j**) Halton (bases 2, 7), (

**k**) Halton (bases 3, 5), (

**l**) Halton (bases 3, 7), (

**m**) Halton (bases 5, 7), (

**n**) Sobol, (

**o**) Poisson disk.

**Figure 4.**Bar plot of minimum element spacings with different distributions: (

**a**) uniform, (

**b**) random, (

**c**) random with jitter, (

**d**) Hammersley (base 2), (

**e**) Hammersley (base 3), (

**f**) Hammersley (base 5), (

**g**) Hammersley (base 7), (

**h**) Halton (bases 2, 3), (

**i**) Halton (bases 2, 5), (

**j**) Halton (bases 2, 7), (

**k**) Halton (bases 3, 5), (

**l**) Halton (bases 3, 7), (

**m**) Halton (bases 5, 7), (

**n**) Sobol, (

**o**) Poisson disk.

**Figure 5.**Normalized power patterns of the antenna arrays with different element distributions in the u-v space: (

**a**) uniform, (

**b**) random, (

**c**) random with jitter, (

**d**) Hammersley (base 2), (

**e**) Hammersley (base 3), (

**f**) Hammersley (base 5), (

**g**) Hammersley (base 7), (

**h**) Halton (bases 2, 3), (

**i**) Halton (bases 2, 5), (

**j**) Halton (bases 2, 7), (

**k**) Halton (bases 3, 5), (

**l**) Halton (bases 3, 7), (

**m**) Halton (bases 5, 7), (

**n**) Sobol, (

**o**) Poisson disk.

**Figure 6.**Position of elements on rectangular, circular, and elliptical apertures with different distributions: (

**a**) uniform (rectangular), (

**b**) Halton (rectangular), (

**c**) Poisson disk (rectangular), (

**d**) uniform (circular), (

**e**) Halton (circular), (

**f**) Poisson disk (circular), (

**g**) uniform (elliptical), (

**h**) Halton (elliptical), (

**i**) Poisson disk (elliptical).

**Figure 7.**Bar plot of minimum element spacings on rectangular, circular, and elliptical apertures with different distributions: (

**a**) uniform (rectangular), (

**b**) Halton (rectangular), (

**c**) Poisson disk (rectangular), (

**d**) uniform (circular), (

**e**) Halton (circular), (

**f**) Poisson disk (circular), (

**g**) uniform (elliptical), (

**h**) Halton (elliptical), (

**i**) Poisson disk (elliptical).

**Figure 8.**Normalized power patterns of sparse antenna arrays with rectangular, circular, and elliptical apertures and different element distributions in the u-v space: (

**a**) uniform (rectangular), (

**b**) Halton (rectangular), (

**c**) Poisson disk (rectangular), (

**d**) uniform (circular), (

**e**) Halton (circular), (

**f**) Poisson disk (circular), (

**g**) uniform (elliptical), (

**h**) Halton (elliptical), (

**i**) Poisson disk (elliptical).

**Figure 9.**Normalized power patterns of the antenna arrays in the u-v space scanned along the elevation plane in ϕ = 0° direction with different element distributions: (

**a**) uniform (20° scan), (

**b**) uniform (40° scan), (

**c**) uniform (60° scan), (

**d**) random (20° scan), (

**e**) random (40° scan), (

**f**) random (60° scan), (

**g**) Halton (20° scan), (

**h**) Halton (40° scan), (

**i**) Halton (60° scan), (

**j**) Poisson disk (20° scan), (

**k**) Poisson disk (40° scan), (

**l**) Poisson disk (60° scan). The Halton sampling uses prime bases of 2 and 7.

**Figure 10.**Normalized power patterns of the antenna arrays in the u-v space scanned along the elevation plane in ϕ = 45° direction with different element distributions: (

**a**) uniform (20° scan), (

**b**) uniform (40° scan), (

**c**) uniform (60° scan), (

**d**) random (20° scan), (

**e**) random (40° scan), (

**f**) random (60° scan), (

**g**) Halton (20° scan), (

**h**) Halton (40° scan), (

**i**) Halton (60° scan), (

**j**) Poisson disk (20° scan), (

**k**) Poisson disk (40° scan), (

**l**) Poisson disk (60° scan). The Halton sampling uses prime bases of 2 and 7.

Method | Average Minimum Element Spacing (λ) | Peak SLL (dB) | Directivity (dB) | Aperture Efficiency (%) |
---|---|---|---|---|

Uniform | 1.3333 | 0 | 31.5478 | 11.168 |

Random | 0.6667 | −10.90 | 31.9619 | 12.209 |

Random with Jitter | 0.9325 | −9.36 | 32.5775 | 14.068 |

Hammersley (base 2) | 1.0037 | −7.0 | 32.7075 | 14.166 |

Hammersley (base 3) | 1.1688 | −2.69 | 32.6226 | 14.215 |

Hammersley (base 5) | 1.2624 | −0.55 | 31.6892 | 11.466 |

Hammersley (base 7) | 0.7538 | −0.25 | 33.8949 | 19.054 |

Halton (bases 2, 3) | 0.8436 | −10.10 | 32.4609 | 13.696 |

Halton (bases 2, 5) | 0.8115 | −4.33 | 32.3534 | 13.361 |

Halton (bases 2, 7) | 0.9172 | −12.35 | 32.8188 | 14.872 |

Halton (bases 3, 5) | 0.8430 | −8.10 | 32.8611 | 15.018 |

Halton (bases 3, 7) | 0.7663 | −5.90 | 32.2561 | 13.065 |

Halton (bases 5, 7) | 0.8633 | −5.60 | 32.9511 | 15.332 |

Sobol | 0.9307 | −6.58 | 32.7127 | 14.513 |

Poisson Disk | 0.9031 | −12.28 | 32.8212 | 14.880 |

**Table 2.**Performance metrics for sparse antenna arrays with rectangular, circular, and elliptical apertures and different element distributions.

Method/Aperture Type | Average Minimum Element Spacing (λ) | Peak SLL (dB) | Directivity (dB) | Aperture Efficiency (%) |
---|---|---|---|---|

Uniform/Rectangular | 1.3333 | 0 | 31.5778 | 11.175 |

Uniform/Circular | 1.3333 | 0 | 31.5848 | 11.261 |

Uniform/Elliptical | 1.3333 | 0 | 31.5181 | 11.228 |

Halton/Rectangular | 0.8481 | −6.74 | 32.4857 | 13.774 |

Halton/Circular | 0.8967 | −9.27 | 32.6474 | 14.383 |

Halton/Elliptical | 0.7724 | −8.94 | 32.5883 | 14.365 |

Poisson Disc/Rectangular | 0.9094 | −12.18 | 32.8062 | 14.829 |

Poisson Disc/Circular | 0.9016 | −15.30 | 32.8643 | 15.119 |

Poisson Disc/Elliptical | 0.9245 | −14.34 | 32.6634 | 14.616 |

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**MDPI and ACS Style**

Torres, T.; Anselmi, N.; Nayeri, P.; Rocca, P.; Haupt, R.
Low Discrepancy Sparse Phased Array Antennas. *Sensors* **2021**, *21*, 7816.
https://doi.org/10.3390/s21237816

**AMA Style**

Torres T, Anselmi N, Nayeri P, Rocca P, Haupt R.
Low Discrepancy Sparse Phased Array Antennas. *Sensors*. 2021; 21(23):7816.
https://doi.org/10.3390/s21237816

**Chicago/Turabian Style**

Torres, Travis, Nicola Anselmi, Payam Nayeri, Paolo Rocca, and Randy Haupt.
2021. "Low Discrepancy Sparse Phased Array Antennas" *Sensors* 21, no. 23: 7816.
https://doi.org/10.3390/s21237816