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A Linear Technique for Artifacts Correction and Compensation in Phase Interferometric Angle of Arrival Estimation^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### Contributions and Paper Organization

- We mathematically introduce and experimentally prove, with a measurement campaign, the effectiveness of an artifacts linear compensation technique that can be employed when adopting the phase interferometric approach to AoA estimation.
- The technique embeds in one single computation all the possible mismatches due to systematic errors and a first-order (linear) approximation of the mutual coupling effects acting on the antenna array that can damage the integrity of the phase information.
- The matrix can be computed once and its values remain valid for the employed hardware and setup. Moreover, given the linearity of the approach, this is simple and fast to be implemented.
- Given the generality of the assumptions and the description, the technique can be implemented either in digital form by complex-number signal processing, or in the analog domain by means of Variable Gain Amplifiers (VGA) and networks of phase shifters.

## 2. Problem Description

#### 2.1. Phase Errors Due to Different Length of Signal Paths

#### 2.2. Phase Artifacts Due to the Mutual Coupling of the Antenna Array Elements

- Space waves, with ${\rho}^{-1}$ asymptotic radial decay;
- High-order waves, with ${\rho}^{-2}$ decay;
- Surface waves, with ${\rho}^{-0.5}$ decay;
- Leaky waves, with ${e}^{-\lambda \rho}\xb7{\rho}^{-0.5}$ decay;

## 3. Theory of the $\mathbf{\alpha}$-Matrix

## 4. Experimental Results

#### 4.1. Experiment Setup

#### 4.2. AoA Estimation without Compensation

#### 4.3. Computation and Analysis of the $\alpha $-Matrix

#### 4.4. AoA Estimation with Compensation

#### 4.5. Comparison

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. RF/IF α-Matrix Equivalence

**Theorem**

**A1.**

**Proof.**

## References

- Bounini, F.; Gingras, D.; Pollart, H.; Gruyer, D. From Simultaneous Localization and Mapping to Collaborative Localization for Intelligent Vehicles. IEEE Intell. Transp. Syst. Mag.
**2021**, 13, 196–216. [Google Scholar] [CrossRef] - Zekavat, S.A.R.; Kansal, S.; Levesque, A.H. Wireless Positioning Systems: Operation, Application, and Comparison. In Handbook of Position Location; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2011; Chapter 1; pp. 3–23. [Google Scholar] [CrossRef]
- Zekavat, S.; Buehrer, R.; Durgin, G.; Lovisolo, L.; Wang, Z.; Goh, S.T.; Ghasemi, A. An Overview on Position Location: Past, Present, Future. Int. J. Wirel. Inf. Netw.
**2021**, 28, 45–76. [Google Scholar] [CrossRef] - Björnson, E.; Sanguinetti, L.; Wymeersch, H.; Hoydis, J.; Marzetta, T.L. Massive MIMO is a reality—What is next?: Five promising research directions for antenna arrays. Digit. Signal Process.
**2019**, 94, 3–20. [Google Scholar] [CrossRef] - Larsson, E.G.; Edfors, O.; Tufvesson, F.; Marzetta, T.L. Massive MIMO for next generation wireless systems. IEEE Commun. Mag.
**2014**, 52, 186–195. [Google Scholar] [CrossRef][Green Version] - Piccinni, G.; Avitabile, G.; Coviello, G.; Talarico, C. Real-Time Distance Evaluation System for Wireless Localization. IEEE Trans. Circuits Syst. I Regul. Pap.
**2020**, 67, 3320–3330. [Google Scholar] [CrossRef] - Zafari, F.; Gkelias, A.; Leung, K.K. A Survey of Indoor Localization Systems and Technologies. IEEE Commun. Surv. Tutor.
**2019**, 21, 2568–2599. [Google Scholar] [CrossRef][Green Version] - Povalac, A.; Sebesta, J. Phase of arrival ranging method for UHF RFID tags using instantaneous frequency measurement. In Proceedings of the 2010 Conference Proceedings ICECom, 20th International Conference on Applied Electromagnetics and Communications, Dubrovnik, Croatia, 20–23 September 2010; pp. 1–4. [Google Scholar]
- Scherhäufl, M.; Pichler, M.; Müller, D.; Ziroff, A.; Stelzer, A. Phase-of-arrival-based localization of passive UHF RFID tags. In Proceedings of the 2013 IEEE MTT-S International Microwave Symposium Digest (MTT), Seattle, WA, USA, 2–7 June 2013; pp. 1–3. [Google Scholar] [CrossRef]
- International Telecommunications Union (I.T.U.) Spectrum Monitoring Handbook. Available online: http://handle.itu.int/11.1002/pub/80399e8b-en (accessed on 1 April 2021).
- Avitabile, G.; Florio, A.; Coviello, G. Angle of Arrival Estimation through a Full-Hardware Approach for Adaptive Beamforming. IEEE Trans. Circuits Syst. II Express Briefs
**2020**, 67, 3033–3037. [Google Scholar] [CrossRef] - Balanis, C.A. Antenna Theory: Analysis and Design; John Wiley & Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Pozar, D. Considerations for millimeter wave printed antennas. IEEE Trans. Antennas Propag.
**1983**, 31, 740–747. [Google Scholar] [CrossRef][Green Version] - Mailloux, R.J. Phased Array Antenna Handbook; Artech House: London, UK, 2017. [Google Scholar]
- Florio, A.; Avitabile, G.; Coviello, G. Digital Phase Estimation through an I/Q Approach for Angle of Arrival Full-Hardware Localization. In Proceedings of the 2020 IEEE Asia Pacific Conference on Circuits and Systems (APCCAS), Ha Long, Vietnam, 8–10 December 2020; pp. 106–109. [Google Scholar] [CrossRef]
- Florio, A.; Avitabile, G.; Coviello, G.; Ma, J.; Man, K.L. The Impact of Coherent Signal Reception on Interferometric Angle of Arrival Estimation. In Proceedings of the 2020 International SoC Design Conference (ISOCC), Yeosu, Korea, 21–24 October 2020; pp. 167–168. [Google Scholar] [CrossRef]

**Figure 1.**Block diagram depicting a 4-element ULA connected to a 4-channel RF front end with its basic sub-blocks for the amplification, down-conversion and band-pass filtering of the received signal, prior to the AoA estimation.

**Figure 2.**The 3-element subarray under test (

**a**) and the interaction between the antenna array elements considered in the model (

**b**).

**Figure 3.**AoA data concerning Exp.#1. In detail, the AoA were computed thanks to the couples {21},{32},{43} and their average.

**Figure 4.**AoA data concerning Exp.#2. In detail, the AoA were computed thanks to the couples {21},{32},{43} and their average.

**Figure 6.**Phase values for the $\alpha $-matrix coefficient during an experimental snapshot. In particular, in (

**a**) phase values for the ${\alpha}_{23}$ coefficient and in (

**b**) phase values for the ${\alpha}_{13}$ coefficient.

**Figure 7.**Analysis of the repeatability of the phase adjustment for the center position on 50 experimental samples, with $er{r}_{ij}$ representing the absolute error committed for the antenna elements $\left\{ij\right\}$ without compensation and $er{r}_{ij}$-C the absolute error committed with the compensation procedure.

**Figure 8.**AoA estimation values for Exp#1 before and after compensation obtained for (

**a**) the antenna couple $\left\{23\right\}$ and (

**b**) the antenna couple $\left\{43\right\}$.

**Figure 9.**AoA estimation values for Exp#2 before and after compensation obtained for (

**a**) the antenna couple $\left\{23\right\}$ and (

**b**) the antenna couple $\left\{43\right\}$.

**Figure 12.**Time- domain comparison of the (

**a**) acquired waveforms versus the (

**b**) compensated waveforms in the Exp #2 calibration point. The initial waveforms were preprocessed through a digital FIR filtering and downconversion stage to the IF of 40 MHz for better view the results.

Exp# | Average SNR [dB] | ||||
---|---|---|---|---|---|

CH1 | CH2 | CH3 | CH4 | AVG | |

1 | 29.55 | 30.61 | 28.56 | 29.54 | 29.56 |

2 | 28.87 | 31.20 | 27.26 | 28.17 | 28.88 |

${\mathit{err}}_{21}^{\mathit{k}}$ [deg] | ${\mathit{err}}_{32}^{\mathit{k}}$ [deg] | ${\mathit{err}}_{43}^{\mathit{k}}$ [deg] | ${\mathit{err}}_{\mathit{avg}}^{\mathit{k}}$ [deg] | |||||
---|---|---|---|---|---|---|---|---|

avg | std | avg | std | avg | std | avg | std | |

Exp#1 | 2.06 | 1.67 | 3.68 | 1.76 | 3.64 | 1.75 | 1.81 | 0.72 |

Exp#2 | 2.25 | 1.69 | 3.32 | 1.74 | 4.36 | 3.07 | 1.93 | 0.89 |

**Table 3.**Statistical indexes of the amplitude and phase of the $\alpha $-matrix coefficients on the 50 snapshots of the single experimental calibration point.

avg($|\xb7|$) [dB] | std($|\xb7|$) [dB] | avg($\mathit{\angle}\xb7$) [rad] | std($\mathit{\angle}\xb7$) [rad] | |
---|---|---|---|---|

${\alpha}_{23}$ | 1.17 | 0.06 | −3.13 | 0.007 |

${\alpha}_{13}$ | 2.13 | 0.07 | −0.13 | 0.008 |

**Table 4.**Statistical indexes of the absolute AoA estimation errors after compensation for antenna couples {23} and {43}.

${\mathit{err}}_{23}^{\mathit{k}}$ [deg] | ${\mathit{err}}_{43}^{\mathit{k}}$ [deg] | |||||||
---|---|---|---|---|---|---|---|---|

avg | std | avg | std | |||||

Exp#1 | 1.78 | −51.6% | 1.47 | −16.4% | 1.66 | −54.4% | 1.49 | −14.9% |

Exp#2 | 2.51 | −24.4% | 2.25 | −22.7% | 2.57 | −41.0% | 2.16 | −29.6% |

**Table 5.**Statistical indexes of the absolute AoA estimation errors after averaging the estimated AoAs after compensation for antenna couples {23} and {43}.

${\mathit{err}}_{\mathbf{avg}}^{\mathit{k}}$ [deg] | ||||
---|---|---|---|---|

avg | std | |||

Exp#1 | 1.11 | −38.7% | 0.88 | +18.1% |

Exp#2 | 1.54 | −54.0% | 1.47 | +39.4% |

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**MDPI and ACS Style**

Florio, A.; Avitabile, G.; Coviello, G.
A Linear Technique for Artifacts Correction and Compensation in Phase Interferometric Angle of Arrival Estimation. *Sensors* **2022**, *22*, 1427.
https://doi.org/10.3390/s22041427

**AMA Style**

Florio A, Avitabile G, Coviello G.
A Linear Technique for Artifacts Correction and Compensation in Phase Interferometric Angle of Arrival Estimation. *Sensors*. 2022; 22(4):1427.
https://doi.org/10.3390/s22041427

**Chicago/Turabian Style**

Florio, Antonello, Gianfranco Avitabile, and Giuseppe Coviello.
2022. "A Linear Technique for Artifacts Correction and Compensation in Phase Interferometric Angle of Arrival Estimation" *Sensors* 22, no. 4: 1427.
https://doi.org/10.3390/s22041427