# Signal Source Localization with Long-Term Observations in Distributed Angle-Only Sensors

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## Abstract

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## 1. Introduction

## 2. Localization with Angle-Only Passive Sensors

#### 2.1. Signal Model of Passive Observations

#### 2.2. Estimation of Target Track

#### 2.3. The Gross LS Algorithm

#### 2.4. The Linear LS Algorithm

#### 2.5. The Truncated LS Algorithm

## 3. Numerical Results

#### 3.1. The Convergence Curves

#### 3.2. Computation Cost

#### 3.3. The Impact of Velocity Estimation Error

#### 3.4. Nonlinearity of the DOA Approximation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The convergence curves of the positioning errors with different numbers of observations. (

**a**) The RMSE of position estimation and its dB form (

**b**); (

**c**) The RMSE of velocity estimation and its dB form (

**d**); (

**e**) The gross RMSE and its dB form (

**f**).

**Figure 4.**The first (

**a**) and last (

**b**) 20 estimates of the position and velocity of the gross LS, the linear LS and the truncated LS algorithms.

**Figure 5.**The computation time in 10 random simulations of the gross LS, linear LS and truncated LS algorithms, in scale (

**a**) and dB (

**b**). The program is run on a computer with an Intel™i7-10700 CPU and 16 GB memory.

**Figure 6.**The RMSE of position with velocity estimated replaced by real velocity in scale (

**a**) and dB (

**b**) for ${\mathbf{C}}_{\mathrm{s}}(k,n)=\mathbf{I}$, and in scale (

**c**) and dB (

**d**) for ${\mathbf{C}}_{\mathrm{s}}(k,n)=10\mathbf{I}$.

**Figure 7.**The target velocity estimation results with real and estimated sensor velocity are shown in (

**a**) and (

**b**) for ${\mathbf{C}}_{\mathrm{s}}(k,n)=10\mathbf{I}$. The azimuth (

**c**) and elevation (

**d**) angles of the target in the four sensors.

Position (m) at t = 0 s | Velocity (m/s) | |
---|---|---|

Sensor #1 | ${\mathbf{p}}_{1}^{\mathrm{o}}\left(0\right)={[1000,1000,0]}^{\mathrm{T}}$ | ${[-100,0,0]}^{\mathrm{T}}$ |

Sensor #2 | ${\mathbf{p}}_{2}^{\mathrm{o}}\left(0\right)={[1000,2000,0]}^{\mathrm{T}}$ | ${[-100,-80,0]}^{\mathrm{T}}$ |

Sensor #3 | ${\mathbf{p}}_{3}^{\mathrm{o}}\left(0\right)={[2000,1000,0]}^{\mathrm{T}}$ | ${[-100,-50,0]}^{\mathrm{T}}$ |

Sensor #4 | ${\mathbf{p}}_{4}^{\mathrm{o}}\left(0\right)={[1500,1500,0]}^{\mathrm{T}}$ | ${[-100,-60,0]}^{\mathrm{T}}$ |

Target #1 | ${\mathbf{g}}_{1}^{\mathrm{o}}\left(0\right)={[0,100,1000]}^{\mathrm{T}}$ | ${[200,100,0]}^{\mathrm{T}}$ |

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## Share and Cite

**MDPI and ACS Style**

Zhou, S.; Wang, L.; Liu, R.; Chen, Y.; Peng, X.; Xie, X.; Yang, J.; Gao, S.; Shao, X.
Signal Source Localization with Long-Term Observations in Distributed Angle-Only Sensors. *Sensors* **2022**, *22*, 9655.
https://doi.org/10.3390/s22249655

**AMA Style**

Zhou S, Wang L, Liu R, Chen Y, Peng X, Xie X, Yang J, Gao S, Shao X.
Signal Source Localization with Long-Term Observations in Distributed Angle-Only Sensors. *Sensors*. 2022; 22(24):9655.
https://doi.org/10.3390/s22249655

**Chicago/Turabian Style**

Zhou, Shenghua, Linhai Wang, Ran Liu, Yidi Chen, Xiaojun Peng, Xiaoyang Xie, Jian Yang, Shibo Gao, and Xuehui Shao.
2022. "Signal Source Localization with Long-Term Observations in Distributed Angle-Only Sensors" *Sensors* 22, no. 24: 9655.
https://doi.org/10.3390/s22249655