# Probability of Conjunction Estimation for Analyzing the Electromagnetic Environment Based on a Space Object Conjunction Methodology

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Conjunction Prediction

## 3. Analytical Expression of the Probability of EM Conjunction

## 4. Simulation Results

^{2}. In all simulations, we take the variance from this range. We further assumed the positioning errors in $x$ and $y$ are uncorrelated, i.e., ${\rho}_{xy}$ = 0, having a mean value equal to zero. Both antennas are assumed to be omnidirectional with unity gains. Since navigational radars are operating at high frequency, mainly in $\mathrm{GHz},$ we assume the frequency of operation is 8 $\mathrm{GHz}$. Due to the high power of the radars, transmitted power is taken from the range 60–66 dBm, while the threshold of $Rx$ (${\eta}_{r})$ is taken from 0–10 dBm, which is the upper limit assumed for impeccable and linear functionality of the $Rx$system. This range threshold is taken as microware amplifiers in EM receivers usually get saturated at these values, however, this threshold limit can be adjusted depending upon the type of EMI evaluation to be performed, e.g., EM interference from a jamming perspective, saturation, or burnout, etc. To compute the probability of EM conjunction, we neglect the effect of the physical size of the platforms. We further assumed that both platforms are moving with the same velocity. These initial conditions are applied in all simulations. Simulations are performed for three different trajectories. All three scenarios are shown in Figure 7, Figure 8 and Figure 9. First is the straight-line scenario where systems are moving close and crossing each other, second case is of a circular trajectory (tangent) scenario in which they make a tangent at the time of closest approach. The third case is also circular trajectory (crossing) scenario, but moving close and crossing each other at two different instances. ${P}_{cEM}$ is computed for all simulation scenarios by using Equation (17), written ${P}_{cE{M}_{analy}}$, and the results are validated with Monte Carlo simulations, and written as ${P}_{cE{M}_{MC}}$.

## 5. Discussions

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Matsumoto, Y.; Wiklundh, K. Evaluation of impact on digital radio systems by measuring amplitude probability distribution of interfering noise. IEICE Trans. Commun.
**2015**, 98, 1143–1155. [Google Scholar] [CrossRef] - Rahman, M.J.; Wang, X. Probabilistic analysis of mutual interference in cognitive radio communications. In Proceedings of the IEEE Global Telecommunications Conference, Kathmandu, Nepal, 5–9 December 2011; pp. 1–5. [Google Scholar]
- Violette, N.J.L.; White, D.R.J. Electromagnetic Compatibility Handbook, 1st ed.; Springer: New York, NY, USA, 1987; pp. 106–146. [Google Scholar]
- Morgan, D. A Handbook for EMC Testing and Measurement, 2nd ed.; The Institution of Engineering and Technology: London, UK, 2007; pp. 1–12, 14–36. [Google Scholar]
- Department of Defense USA. MIL-STD-464C, USA, 1 December 2010. Available online: http://everyspec.com/MIL-STD/MIL-STD-0300-0499/MIL-STD-464C_28312/ (accessed on 1 December 2010).
- Baqar, A.H.; Xu, P.; Tao, J.; Zhang, Y.C. Application of space object conjunction method in the system level EMC evaluation. In Proceedings of the Asia-Pacific International Symposium on Electromagnetic Compatibility, Seoul, Korea, 20–23 June 2017; pp. 253–255. [Google Scholar]
- Lei, C.; Xian, Z.B.; Yan, G.L.; Ke, B.L. Orbital Data Applications for Space Objects, 1st ed.; Springer: Singapore, 2017. [Google Scholar]
- Anselmo, L.; Pardini, C. The end-of-life disposal of the Italian geostationary satellites. Adv. Space Res.
**2014**, 34, 1203–1208. [Google Scholar] [CrossRef] - Chan, K. International Space Station Collision Probability; The Aerospace Corporation: Chantilly, VA, USA, 1997; Available online: http://aero.tamu.edu/sites/default/files/faculty/alfriend/S4.2%20Chan.pdf (accessed on 1 June 2018).
- Xiao, L.X.; Xiong, Y.Q. A method for calculating probability of collision between space objects. Res. Astron. Astrophys.
**2014**, 14, 601–609. [Google Scholar] [Green Version] - Alfano, S.; Oltrogge, D. Probability of Collision: Valuation, variability, visualization, and validity. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Long Beach, CA, USA, 13–16 September 2016; p. 5654. [Google Scholar]
- Patera, R.P. A general method for calculating satellite collision probability. J. Guidance Control Dyn.
**2001**, 24, 716–722. [Google Scholar] [CrossRef] - Papoulis, A. Probability, Random Variables, and Stochastic Processes, 3rd ed.; McGraw-Hill: New York, NY, USA, 1991. [Google Scholar]
- Alfano, S. Review of conjunction probability methods for short-term encounters. Adv. Astronaut. Sci.
**2007**, 127, 719–746. [Google Scholar] - Bai, X.Z.; Chen, L. A Rapid Algorithm of Space Debris Collision Probability Based on Space Compression and Infinite Series. Acta Math. Appl. Sin.
**2009**, 32, 336–353. [Google Scholar] - Rice, S.O. Mathematical Analysis of Random Noise. Bell Syst. Tech. J.
**1944**, 23, 282–332. [Google Scholar] [CrossRef] - Shaw, J.A. Radiometry and the Friis transmission equation. Am. J. Phys.
**2013**, 81, 33–37. [Google Scholar] [CrossRef] - Romeu, J.L. Understanding Series and Parallel Systems Reliability; Department of Defense, Reliability Analysis Center: Rome, NY, USA, 2004; Volume 11, 8p. [Google Scholar]
- Willmott, C.J.; Matsuura, K. On the use of dimensioned measured of error to evaluate the performance of spatial interpolators. Int. J. Geogr. Inf. Sci.
**2006**, 20, 89–102. [Google Scholar] [CrossRef] - Hughes, W.J. Global Positioning System (GPS) Standard Positioning Service (SPS) Performance Analysis Report; Federal Aviation Administration, GPS Product Team: Washington, DC, UK, 2014. Available online: http://www.nstb.tc.faa.gov/reports/PAN96_0117.pdf (accessed on 1 June 2010).

**Figure 1.**Communication link and electromagnetic interference sources (courtesy image: Navy continues electronic warfare upgrades for ships, by Kevin Mccaney, 2016).

**Figure 2.**Simulation of debris pattern orbiting around Earth (courtesy image: created by the Institute of Aerospace Systems at the Technische Universität Braunschweig Germany, 2015).

**Figure 5.**Parameters of ${P}_{c}$: probability density function and radius of conjunctional cross-section area.

**Figure 7.**Straight line trajectory: ${P}_{t}=60\mathrm{dBm}$; ${\eta}_{r}=10\mathrm{dBm}$; error parameters: mean:${\mu}_{x},{\mu}_{y}=0$, variance: ${\sigma}_{x}^{2},{\sigma}_{y}^{2}=$ 10${\mathrm{m}}^{2}$, cov: ${\rho}_{xy}=0$.

**Figure 8.**Circular trajectory (tangent): ${P}_{t}=60\mathrm{dBm}$; ${\eta}_{r}=10\mathrm{dBm}$; error parameters: mean:${\mu}_{x},{\mu}_{y}=0$, variance: ${\sigma}_{x}^{2},{\sigma}_{y}^{2}=$ 10${\text{}\mathrm{m}}^{2}$, cov: ${\rho}_{xy}=0$.

**Figure 9.**Circular trajectory (Crossing): ${P}_{t}=60\mathrm{dBm}$; ${\eta}_{r}=10\mathrm{dBm}$; error parameters: mean:${\mu}_{x},{\mu}_{y}=0$, variance: ${\sigma}_{x}^{2},{\sigma}_{y}^{2}=$ 10${\text{}\mathrm{m}}^{2}$, cov: ${\rho}_{xy}=0$.

**Figure 10.**${P}_{cEM}$ and error in ${P}_{cEM}$ (Monte Carlo and Analytical) for straight line trajectory; ${P}_{t}=60\mathrm{dBm}$; ${\eta}_{r}=10\mathrm{dBm}$; error parameters: mean:${\mu}_{x},{\mu}_{y}=0$, variance: ${\sigma}_{x}^{2},{\sigma}_{y}^{2}=$ 10 ${\mathrm{m}}^{2}$, cov: ${\rho}_{xy}=0$.

**Figure 11.**${P}_{cEM}$ and error in ${P}_{cEM}$ (Monte Carlo and Analytical) for circular trajectory (tangent); ${P}_{t}=60\text{}\mathrm{dBm}$; ${\eta}_{r}=10\mathrm{dBm}$; error parameters: mean:${\mu}_{x},{\mu}_{y}=0$, variance: ${\sigma}_{x}^{2},{\sigma}_{y}^{2}=$ 10 ${\mathrm{m}}^{2}$, cov: ${\rho}_{xy}=0$.

**Figure 12.**${P}_{cEM}$ and error in ${P}_{cEM}$ (Monte Carlo and Analytical) for circular trajectory (crossing); ${P}_{t}=60\mathrm{dBm}$; ${\eta}_{r}=10\mathrm{dBm}$; error parameters: mean:${\mu}_{x},{\mu}_{y}=0$, variance: ${\sigma}_{x}^{2},{\sigma}_{y}^{2}=$ 10 ${\mathrm{m}}^{2}$, cov: ${\rho}_{xy}=0$.

**Figure 13.**(

**a**) Effect of ${P}_{t}$ on ${P}_{cEM}$; (

**b**) Effect of ${\eta}_{r}$ on ${P}_{cEM}$; (

**c**) $\mathrm{Effect}\text{}\mathrm{of}$ variance $\left({\sigma}_{x}^{2}\text{},\text{}{\sigma}_{y}^{2}\text{}\right)$ on${P}_{cEM}$.

**Table 1.**Effect of transmitted power ${P}_{t}$ on ${P}_{cEM}$; position uncertainty (mean:${\mu}_{x},{\mu}_{y}$ = 0, variance: ${\sigma}_{x}^{2},{\sigma}_{y}^{2}$ = 10 ${\mathrm{m}}^{2}$, cov: ${\rho}_{xy}$ = 0).

Trajectory Type | Transmitter Power ${\mathit{P}}_{\mathit{t}}$ $\left(\mathbf{d}\mathbf{B}\mathbf{m}\right)$ | Receiver Threshold ${\mathit{\eta}}_{\mathit{r}}$ $\text{}\left(\mathbf{d}\mathbf{B}\mathbf{m}\right)$ | Max. Probability of Interference Monte Carlo ${\mathit{P}}_{\mathit{c}\mathit{E}{\mathit{M}}_{\mathit{M}\mathit{C}}}$at $\mathit{\tau}$ | Max. Probability of Interference Analytical ${\mathit{P}}_{\mathit{c}\mathit{E}{\mathit{M}}_{\mathit{a}\mathit{n}\mathit{a}\mathit{l}\mathit{y}}}$ at $\mathit{\tau}$ | % Error in ${\mathit{P}}_{\mathit{c}\mathit{E}\mathit{M}}$at $\mathit{\tau}$ |
---|---|---|---|---|---|

Straight line Figure 7 | 60 | 10 | 0.0221 | 0.0200 | 0.5362 |

63 | 0.0436 | 0.0435 | 0.1813 | ||

66 | 0.0846 | 0.0847 | 0.1412 | ||

Circular Tangent Figure 8 | 60 | 10 | 0.0221 | 0.0220 | 0.2773 |

63 | 0.0436 | 0.0436 | 0.0765 | ||

66 | 0.0851 | 0.0848 | 0.3691 | ||

Circular Crossing Figure 9 | 60 | 10 | 0.0220, 0.0222 | 0.0219, 0.0220 | 0.2140, 0.7585 |

63 | 0.0437, 0.0435 | 0.0434, 0.0434 | 0.5892, 0.3119 | ||

66 | 0.0850, 0.0848 | 0.0848, 0.0848 | 0.2381, −0.0012 |

**Table 2.**Effect of Receiver’s threshold ${\eta}_{r}$ on ${P}_{cEM}$; position uncertainty (mean:${\mu}_{x},{\mu}_{y}$ = 0, variance: ${\sigma}_{x}^{2},{\sigma}_{y}^{2}$ = 10 ${\mathrm{m}}^{2}$, cov: ${\rho}_{xy}$ = 0).

Trajectory Type | Receiver Threshold ${\mathit{\eta}}_{\mathit{r}}$ $\left(\mathbf{d}\mathbf{B}\mathbf{m}\right)$ | Transmitter Power ${\mathit{P}}_{\mathit{t}}$ $\left(\mathbf{d}\mathbf{B}\mathbf{m}\right)$ | Max. Probability of Interference— Monte Carlo ${\mathit{P}}_{\mathit{c}\mathit{E}{\mathit{M}}_{\mathit{M}\mathit{C}}}$at $\mathit{\tau}$ | Max. Probability of Interference— Analytical ${\mathit{P}}_{\mathit{c}\mathit{E}{\mathit{M}}_{\mathit{a}\mathit{n}\mathit{a}\mathit{l}\mathit{y}}}$ at $\mathit{\tau}$ | % Error in ${\mathit{P}}_{\mathit{c}\mathit{E}\mathit{M}}$at $\mathit{\tau}$ |
---|---|---|---|---|---|

Straight line | 0 | 60 | 0.2014 | 0.1996 | 0.9233 |

1 | 0.1625 | 0.1621 | 0.2439 | ||

3 | 0.1054 | 0.1052 | 0.2259 | ||

5 | 0.0682 | 0.0679 | 0.3717 | ||

7 | 0.0435 | 0.0433 | 0.3933 | ||

9 | 0.0278 | 0.0276 | 0.5632 | ||

10 | 0.0221 | 0.0220 | 0.5362 | ||

Circular Tangent | 0 | 60 | 0.1996 | 0.1995 | 0.0711 |

1 | 0.1630 | 0.1619 | 0.6292 | ||

3 | 0.1060 | 0.1056 | 0.3603 | ||

5 | 0.0680 | 0.0680 | −0.0158 | ||

7 | 0.0435 | 0.0434 | 0.2033 | ||

9 | 0.0277 | 0.0276 | 0.2671 | ||

10 | 0.0221 | 0.0220 | 0.2773 | ||

Circular Crossing | 0 | 60 | 0.1989, 0.1996 | 0.1990, 0.1990 | −0.0313, 0.2789 |

1 | 0.1619, 0.1631 | 0.1620, 0.1620 | −0.0140, 0.7323 | ||

3 | 0.1056, 0.1055 | 0.1055, 0.1055 | 0.0724, −0.0336 | ||

5 | 0.0682, 0.0679 | 0.0679, 0.0679 | 0.4257, −0.0283 | ||

7 | 0.0435, 0.0434 | 0.0434, 0.0433 | 0.1440, 0.1134 | ||

9 | 0.0277, 0.0275 | 0.0276, 0.0275 | 0.4398, −0.1769 | ||

10 | 0.0220, 0.0222 | 0.0219, 0.0220 | 0.2141, 0.7585 |

**Table 3.**Effect of variance ${\sigma}_{x}^{2},{\sigma}_{y}^{2}$ on ${P}_{cEM}$; position uncertainty (mean:${\mu}_{x},{\mu}_{y}$ = 0, cov: ${\rho}_{xy}$ = 0).

Trajectory Type | Position Variance ${\mathit{\sigma}}_{\mathit{x}}^{2},{\mathit{\sigma}}_{\mathit{y}}^{2}$ $({\mathbf{m}}^{2}$) | Transmitter Power ${\mathit{P}}_{\mathit{t}}$ $\left(\mathbf{d}\mathbf{B}\mathbf{m}\right)$ | Receiver Threshold ${\mathit{\eta}}_{\mathit{r}}$ $\left(\mathbf{d}\mathbf{B}\mathbf{m}\right)$ | Max. Probability of Interference Monte Carlo ${\mathit{P}}_{\mathit{c}\mathit{E}{\mathit{M}}_{\mathit{M}\mathit{C}}}$at $\mathit{\tau}$ | Max. Probability of Interference Analytical ${\mathit{P}}_{\mathit{c}\mathit{E}{\mathit{M}}_{\mathit{a}\mathit{n}\mathit{a}\mathit{l}\mathit{y}}}$ at $\mathit{\tau}$ | % Error in ${\mathit{P}}_{\mathit{c}\mathit{E}\mathit{M}}$at $\mathit{\tau}$ |
---|---|---|---|---|---|---|

Straight line | 10 | 60 | 10 | 0.0221 | 0.0220 | 0.5362 |

35 | 0.0064 | 0.0063 | 0.8982 | |||

50 | 0.0045 | 0.0044 | 0.6667 | |||

64 | 0.0035 | 0.0035 | 0.9443 | |||

Circular Tangent | 10 | 60 | 10 | 0.0221 | 0.0220 | 0.2773 |

35 | 0.0064 | 0.0063 | 0.6218 | |||

50 | 0.0045 | 0.0044 | 0.7721 | |||

64 | 0.0035 | 0.0035 | 0.9169 | |||

Circular Crossing | 10 | 60 | 10 | 0.0220, 0.0222 | 0.0219, 0.0220 | 0.2141, 0.7585 |

35 | 0.0064, 0.0064 | 0.0063, 0.0063 | 1.1322, 1.3058 | |||

50 | 0.0045, 0.0045 | 0.0044, 0.0044 | 0.8144, 1.2290 | |||

64 | 0.0035, 0.0035 | 0.0035, 0.0035 | 0.5313, 0.9591 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Baqar, A.H.; Jiang, T.; Hussain, I.; Farid, G.
Probability of Conjunction Estimation for Analyzing the Electromagnetic Environment Based on a Space Object Conjunction Methodology. *Symmetry* **2018**, *10*, 255.
https://doi.org/10.3390/sym10070255

**AMA Style**

Baqar AH, Jiang T, Hussain I, Farid G.
Probability of Conjunction Estimation for Analyzing the Electromagnetic Environment Based on a Space Object Conjunction Methodology. *Symmetry*. 2018; 10(7):255.
https://doi.org/10.3390/sym10070255

**Chicago/Turabian Style**

Baqar, Asad Husnain, Tao Jiang, Ishfaq Hussain, and Ghulam Farid.
2018. "Probability of Conjunction Estimation for Analyzing the Electromagnetic Environment Based on a Space Object Conjunction Methodology" *Symmetry* 10, no. 7: 255.
https://doi.org/10.3390/sym10070255