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An Efficient Multi-Path Multitarget Tracking Algorithm for Over-The-Horizon Radar^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Assumptions and Models

#### 2.1. Target Motion Model

#### 2.2. Measurement Generation Model

## 3. Multi-Path Linear Multitarget Integrated Probabilistic Data Association

#### 3.1. Track State Expression

#### 3.2. Measurement Utilization

- Assuming that only one of the selected measurements is generated by the target (${\phi}_{\tau}=1$, i.e., each measurement cell consists of one of the selected measurements), and the corresponding measurement cells are$$\begin{array}{c}\hfill \begin{array}{c}\hfill {z}_{1,1}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{1}\left(k\right),{z}_{1,2}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{2}\left(k\right)& ,{z}_{1,3}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{3}\left(k\right)\\ ,{z}_{1,4}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{4}\left(k\right)\end{array}.\end{array}$$
- Assuming that two of the selected measurements are generated by the target (${\phi}_{\tau}=2$, i.e., each measurement cell consists of two of the selected measurements), and the corresponding measurement cells are$$\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}{z}_{2,1}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{1}\left(k\right),{z}_{2}\left(k\right),{z}_{2,2}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{1}\left(k\right),{z}_{3}\left(k\right)\hfill & ,{z}_{2,3}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{1}\left(k\right),{z}_{4}\left(k\right)\\ ,\end{array}\\ {z}_{2,4}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{2}\left(k\right),{z}_{3}\left(k\right),{z}_{2,5}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{2}\left(k\right),{z}_{4}\left(k\right)\hfill & ,{z}_{2,6}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{3}\left(k\right),{z}_{4}\left(k\right)\\ .\end{array}\end{array}$$
- Assuming that three of the selected measurements are generated by the target (${\phi}_{\tau}=3$, i.e., each measurement cell consists of three of the selected measurements), and the corresponding measurement cells are$$\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}{z}_{3,1}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{1}\left(k\right),{z}_{2}\left(k\right),{z}_{3}\left(k\right),{z}_{3,2}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{1}\left(k\right),{z}_{2}\left(k\right),{z}_{4}\left(k\right)\hfill & ,\end{array}& {z}_{3,3}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{1}\left(k\right),{z}_{3}\left(k\right),{z}_{4}\left(k\right),{z}_{3,4}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{2}\left(k\right),{z}_{3}\left(k\right),{z}_{4}\left(k\right)\hfill & .\end{array}\end{array}$$
- Assuming that all of the selected measurements are generated by the target (${\phi}_{\tau}=4$, i.e., a measurement cell consists of all of the selected measurements), and the corresponding measurement cell is$$\begin{array}{c}\hfill \begin{array}{c}\hfill {z}_{4,1}\left(k\right)=\left(\right)open="\{"\; close="\}">{z}_{1}\left(k\right),{z}_{2}\left(k\right),{z}_{3}\left(k\right),{z}_{4}\left(k\right).\end{array}\end{array}$$

#### 3.3. MP-IPDA

#### 3.4. Modulated Clutter Measurement Density for the Path Pattern Combined Measurement Cell

#### 3.5. MP-LM-IPDA

Algorithm 1: MD-LM-IPDA Track Update Process |

1: for each track $\tau $ find2: The track state prediction $p\left(\right)open="("\; close=")">{x}_{k}^{\tau}|{\chi}_{k}^{\tau},{Z}^{k-1}$ and the probability of target existence prediction $P\left(\right)open="("\; close=")">{\chi}_{k}^{\tau}|{Z}^{k-1}$ 3: The measurement selection (the gating method), the measurement cell generation (the measurement partition) and the path pattern combination 4: The modulated clutter measurement density ${\tilde{\rho}}_{\left(\right)open="\langle "\; close="\rangle ">{z}_{{\phi}_{\tau},{n}_{{\phi}_{\tau}}}\left(k\right)|{A}_{j}\left(\right)open="("\; close=")">{z}_{{\phi}_{\tau},{n}_{{\phi}_{\tau}}}\left(k\right)}^{}\tau $ for each path pattern combined measurement cell 5: A posteriori data association probabilities ${\beta}_{\left(\right)open="\langle "\; close="\rangle ">{z}_{{\phi}_{\tau},{n}_{{\phi}_{\tau}}}\left(k\right)|{A}_{j}\left(\right)open="("\; close=")">{z}_{{\phi}_{\tau},{n}_{{\phi}_{\tau}}}\left(k\right)}^{}\tau $ in (44) and ${\beta}_{k,0}^{\tau}$ in (45) 6: The updated probability of target existence $P\left(\right)open="("\; close=")">{\chi}_{k}^{\tau}|{Z}^{k}$ in (47) 7: The updated track state $p\left(\right)open="("\; close=")">{x}_{k}^{\tau}|{\chi}_{k}^{\tau},{Z}^{k}$ generated by a Gaussian mixture based on all the data association events 8: end for |

## 4. Complexity Analyses

## 5. Simulation

#### 5.1. Simulation Scenario 1

#### 5.2. Simulation Scenario 2

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PDA | Probabilistic data association. |

MD-JPDA | Multiple detection joint probabilistic data association. |

MHT | Multiple hypothesis tracker. |

LM-IPDA | Linear multitarget integrated probabilistic data association. |

PTE | Probability of target existence. |

OTHR | Over-the-horizon-radar. |

MP-JIPDA | Multi-path joint integrated probabilistic data association. |

MP-LM-IPDA | Multi-path LM-IPDA. |

NCV | Nearly constant velocity. |

CFT | Confirmed false track. |

CTTs | Number of the confirmed true tracks. |

RMSE | Root mean square error. |

Nomenclature | |

$\tau $ | A track as well as the potential target being tracked by this track. |

${m}_{k}$ | The number of selected measurements at scan k. |

L | The number of signal propagation paths in the OTHR system. |

${\phi}_{\tau ,max}$ | The maximum number of target-originated measurements, which satisfies ${\phi}_{\tau ,max}=min(L,{m}_{k})$. |

${\phi}_{\tau}$ | The number of target originated measurements ${\phi}_{\tau}\in \left(\right)open="\{"\; close="\}">1,2,\dots ,{\phi}_{\tau ,max}$. |

${n}_{{\phi}_{\tau}}$ | A variable that enumerates the measurement cells under the condition that there are ${\phi}_{\tau}$ measurements generated by target $\tau $, ${n}_{{\phi}_{\tau}}\in \left(\right)open="\{"\; close="\}">1,2,\dots ,{c}_{{\phi}_{\tau}}$ and ${c}_{{\phi}_{\tau}}={C}_{{\phi}_{\tau}}^{{m}_{k}}=\frac{{m}_{k}!}{{\phi}_{\tau}!\left(\right)open="("\; close=")">{m}_{k}-{\phi}_{\tau}}$. |

${z}_{{\phi}_{\tau},{n}_{{\phi}_{\tau}}}\left(k\right)$ | A measurement cell specified by ${\phi}_{\tau}$ and ${n}_{{\phi}_{\tau}}$ at scan k. |

${\chi}_{k}^{\tau}$ | The event that target $\tau $ exists at scan k. |

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$\mathbf{Path}$ | Model | Transmit Layer ${\mathit{h}}_{\mathit{t}}$ | Receive Layer ${\mathit{h}}_{\mathit{r}}$ |
---|---|---|---|

1 | $EE$ | ${h}_{E}$ | ${h}_{E}$ |

2 | $EF$ | ${h}_{E}$ | ${h}_{F}$ |

3 | $FE$ | ${h}_{F}$ | ${h}_{E}$ |

4 | $FF$ | ${h}_{F}$ | ${h}_{F}$ |

Parameter | Value |
---|---|

Slant range size | 1000–1400 km |

Rate of slant range size | 0.013889–0.22222 km/s |

Apparent azimuth size | 0.069813–0.17453 rad |

Mean number of clutter per each scan | 25 |

Transmitter to receiver distance d | 100 km |

Hight of layer E, ${h}_{E}$ | 100 km |

Hight of layer F, ${h}_{F}$ | 260 km |

Target detection probability in each path | ${P}_{D}$ = 0.4 |

Gating probability | ${P}_{G}$ = 0.997 |

Measurement noise covariance R | diag (25 km${}^{2}$, 1 × 10${}^{-6}$ km${}^{2}$/s${}^{2}$, 9 × 10${}^{-6}$ rad${}^{2}$) |

SP-LM-IPDA | MP-LM-IPDA | MP-JIPDA | |
---|---|---|---|

Initial PTE | 0.000093 | 0.0009 | 0.0025 |

Confirmation PTE | 0.98 | 0.98 | 0.98 |

Termination PTE | 0.000093/5 | 0.0009/5 | 0.0025/5 |

Number of CFTs | 5 | 5 | 5 |

SP-LM-IPDA | MP-LM-IPDA | MP-JIPDA | |
---|---|---|---|

Initial PTE | 0.00008 | 0.0009 | 0.0027 |

Confirmation PTE | 0.98 | 0.98 | 0.98 |

Termination PTE | 0.00008/5 | 0.0009/5 | 0.0027/5 |

Number of CFTs | 7 | 7 | 7 |

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**MDPI and ACS Style**

Huang, Y.; Shi, Y.; Song, T.L.
An Efficient Multi-Path Multitarget Tracking Algorithm for Over-The-Horizon Radar. *Sensors* **2019**, *19*, 1384.
https://doi.org/10.3390/s19061384

**AMA Style**

Huang Y, Shi Y, Song TL.
An Efficient Multi-Path Multitarget Tracking Algorithm for Over-The-Horizon Radar. *Sensors*. 2019; 19(6):1384.
https://doi.org/10.3390/s19061384

**Chicago/Turabian Style**

Huang, Yuan, Yifang Shi, and Taek Lyul Song.
2019. "An Efficient Multi-Path Multitarget Tracking Algorithm for Over-The-Horizon Radar" *Sensors* 19, no. 6: 1384.
https://doi.org/10.3390/s19061384