# Multi-Target Tracking Algorithm Based on 2-D Velocity Measurements Using Dual-Frequency Interferometric Radar

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## Abstract

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## 1. Introduction

- We propose a multi-target tracking algorithm by establishing the relationship between 2-D velocity measurements and kinematic state of the target in terms of Cartesian coordinates. Based on 2-D velocity measurement function, the proposed MTT algorithm comprises the following steps: (i) data association using global nearest neighbor (GNN) based on auction method, (ii) target state estimation using interacting multiple model (IMM) estimator combined with square-root cubature Kalman filter (SCKF), and (iii) track management using rule-based M/N logic.
- We analyze the performance of the proposed algorithm in terms of tracking accuracy, computational complexity and IMM mean model probabilities for different scenarios with multiple (non-maneuvering and maneuvering) targets.

## 2. Mathematical Formulation of Problem

#### 2.1. The 2-D Velocity of a Point Source

#### 2.2. Process Model and Measurement Model

## 3. The Proposed Multi-Target Tracking Algorithm

- Detection and radial velocity measurement mode:For the purpose of detection and radial velocity measurement mode, the transmitting and receiving antennas operating at carrier frequency ${f}_{{c}_{2}}$ are connected to the processing unit. For detection, the radar scans the region of interest and range-radial velocity map is obtained by applying 2D fast Fourier transform (2D-FFT) to the beat frequency signal at the receiving antenna Rx1, providing the number of potential targets and their initial ranges. After detection, time-varying Doppler spectrogram is obtained by performing short-time Fourier transform (STFT) to the same beat frequency signal. According to Equation (1), the radial velocities of the targets are calculated by extracting their instantaneous frequencies from Doppler spectrogram.
- Angular velocity measurement mode:For angular velocity measurement, one transmitting and two receiving antennas operating at carrier frequency ${f}_{{c}_{1}}$ are connected to the processing unit. The two beat frequency signals at Rx2 and Rx3 are fed to the interferometric correlator to generate the output. Then, STFT is applied to the interferometric output to obtain the time-varying interferometric spectrogram. According to Equation (11), the angular velocities of targets are calculated by extracting their instantaneous frequencies from interferometric spectrogram.

## 4. Data Association Using Global Nearest Neighbor (GNN) Method

#### 4.1. Ellipsoid Gating

#### 4.2. Global Nearest Neighbor (GNN) Method

## 5. Target State Estimation Using IMM-SCKF Estimator

#### 5.1. Square-Root Cubature Kalman Filter (SCKF)

- Calculate the cubature points set ${\mathbf{X}}_{i}$ : (i = 1,2,…,m, where m = 2${n}_{x}$)$${\mathbf{X}}_{i,k-1|k-1}={\mathbf{S}}_{k-1|k-1}{\xi}_{i}+{\hat{\mathbf{x}}}_{k-1|k-1}$$$$\mathbf{P}=\mathbf{S}{\mathbf{S}}^{T}$$$${\xi}_{i}=\sqrt{\frac{m}{2}}{[1]}_{i}$$
- Time update: Propagate the cubature points and evaluate the predicted state and square-root error covariance.$${\mathbf{X}}_{i,k|k-1}^{*}=\mathbf{f}\left({\mathbf{X}}_{i,k-1|k-1}\right)$$$${\hat{\mathbf{x}}}_{k|k-1}=\frac{1}{m}\sum _{m=1}^{M}{\mathbf{X}}_{i,k|k-1}^{*}$$$${\mathbf{S}}_{k|k-1}=\mathbf{Tria}([{\mathcal{X}}_{k|k-1}^{*}\phantom{\rule{1.em}{0ex}}{\mathbf{S}}_{Q,k-1}])$$$${\mathbf{Q}}_{k-1}={\mathbf{S}}_{Q,k-1}{\mathbf{S}}_{Q,k-1}^{T}$$$$\begin{array}{c}\hfill {\mathcal{X}}_{k|k-1}^{*}=\frac{1}{\sqrt{m}}[{\mathbf{X}}_{1,k|k-1}^{*}-{\hat{\mathbf{x}}}_{k|k-1}\phantom{\rule{1.em}{0ex}}{\mathbf{X}}_{2,k|k-1}^{*}-{\hat{\mathbf{x}}}_{k|k-1}\dots \\ \hfill {\mathbf{X}}_{m,k|k-1}^{*}-{\hat{\mathbf{x}}}_{k|k-1}]\end{array}$$
**Tria**represents triangularization algorithm for matrix decomposition. Here, QR decomposition algorithm has been used. - Measurement update: Calculate the propagated cubature points and update the state and square-root error covariance estimates.$${\mathbf{X}}_{i,k|k-1}={\mathbf{S}}_{k|k-1}{\xi}_{i}+{\hat{\mathbf{x}}}_{k|k-1}$$$${\mathbf{Z}}_{i,k|k-1}^{*}=\mathbf{h}\left({\mathbf{X}}_{i,k|k-1}\right)$$$${\hat{\mathbf{z}}}_{k|k-1}=\frac{1}{m}\sum _{m=1}^{M}{\mathbf{Z}}_{i,k|k-1}^{*}$$$${\hat{\mathbf{x}}}_{k|k}={\hat{\mathbf{x}}}_{k|k-1}+{\mathbf{W}}_{k}({\mathbf{z}}_{k}-{\hat{\mathbf{z}}}_{k|k-1})$$$${\mathbf{S}}_{k|k}=\mathbf{Tria}([{\mathcal{X}}_{k|k-1}-{\mathbf{W}}_{k}{\mathfrak{J}}_{k|k-1}\phantom{\rule{1.em}{0ex}}{\mathbf{W}}_{k}{\mathbf{S}}_{R,k}])$$$${\mathbf{S}}_{zz,k|k-1}=\mathbf{Tria}([{\mathfrak{J}}_{k|k-1}\phantom{\rule{1.em}{0ex}}{\mathbf{S}}_{R,k}])$$$${\mathbf{R}}_{k}={\mathbf{S}}_{R,k}{\mathbf{S}}_{R,k}^{T}$$$${\mathbf{P}}_{xz,k|k-1}={\mathcal{X}}_{k|k-1}{\mathfrak{J}}_{k|k-1}^{T}$$$$\begin{array}{c}\hfill {\mathcal{X}}_{k|k-1}=\frac{1}{\sqrt{m}}[{\mathbf{X}}_{1,k|k-1}-{\hat{\mathbf{x}}}_{k|k-1}\phantom{\rule{1.em}{0ex}}{\mathbf{X}}_{2,k|k-1}-{\hat{\mathbf{x}}}_{k|k-1}\dots \\ \hfill {\mathbf{X}}_{m,k|k-1}-{\hat{\mathbf{x}}}_{k|k-1}]\end{array}$$$$\begin{array}{c}\hfill {\mathfrak{J}}_{k|k-1}=\frac{1}{\sqrt{m}}[{\mathbf{Z}}_{1,k|k-1}-{\hat{\mathbf{z}}}_{k|k-1}\phantom{\rule{1.em}{0ex}}{\mathbf{Z}}_{2,k|k-1}-{\hat{\mathbf{x}}}_{k|k-1}\dots \\ \hfill {\mathbf{Z}}_{m,k|k-1}-{\hat{\mathbf{z}}}_{k|k-1}]\end{array}$$$${\mathbf{W}}_{k}=({\mathbf{P}}_{xz,k|k-1}/{\mathbf{S}}_{zz,k|k-1}^{T})/{\mathbf{S}}_{zz,k|k-1}$$

#### 5.2. Interactive Multiple Model (IMM) Estimator

- Mixing probabilities calculation:$${\overline{c}}^{j}=\sum _{i=1}^{r}{p}_{ij}{\mu}_{k-1|k-1}^{i}$$$${\mu}_{k-1|k-1}^{(i,j)}=P\{{M}_{k-1}^{i}|{M}_{k}^{j},{\mathbf{Z}}_{1}^{k-1}\}=\frac{1}{{\overline{c}}^{j}}{p}_{ij}{\mu}_{k|k-1}^{i}$$
- Interaction of state mean and covariance:$${\hat{\mathbf{x}}}_{k-1|k-1}^{0j}=\sum _{i=1}^{r}{\hat{\mathbf{x}}}_{k-1|k-1}^{i}{\mu}_{k-1|k-1}^{(i,j)}$$$$\begin{array}{c}\hfill {\mathbf{P}}_{k-1|k-1}^{0j}=\sum _{i=1}^{r}{\mu}_{k-1|k-1}^{(i,j)}\{{\mathbf{S}}_{k-1|k-1}^{i}{({\mathbf{S}}_{k-1|k-1}^{i})}^{T}+\\ \hfill [{\hat{\mathbf{x}}}_{k-1|k-1}^{i}-{\hat{\mathbf{x}}}_{k-1|k-1}^{0j}]{[{\hat{\mathbf{x}}}_{k-1|k-1}^{i}-{\hat{\mathbf{x}}}_{k-1|k-1}^{0j}]}^{T}\}\end{array}$$
- State estimate update: The initial condition state estimate ${\hat{\mathbf{x}}}_{k-1|k-1}^{0j}$ and covariance matrix ${\mathbf{P}}_{k-1|k-1}^{0j}$ are fed to the SCKF algorithm described in previous section to compute the updated estimates ${\hat{\mathbf{x}}}_{k|k}^{j}$ and ${\mathbf{S}}_{k|k}^{j}$ for each filter model.
- Computing model likelihood function:$${\tilde{\mathbf{v}}}_{k}^{j}={\mathbf{z}}_{k}-\mathbf{h}({\hat{\mathbf{x}}}_{k|k-1}^{j})$$$${\mathbf{S}}_{Z,k}^{j}={\mathbf{A}}_{k}^{j}{({\mathbf{A}}_{k}^{j})}^{T}$$$${\wedge}_{k}^{j}=\frac{1}{\sqrt{|2\pi {\mathbf{S}}_{Z,k}^{j}|}}exp\{-0.5{[{\tilde{\mathbf{v}}}_{k}^{j}]}^{T}{[{\mathbf{S}}_{Z,k}^{j}]}^{-1}[{\tilde{\mathbf{v}}}_{k}^{j}]\}$$
- Updating the model probability:$$c=\sum _{j=1}^{r}{\wedge}_{k}^{j}{\overline{c}}^{j}$$$${\mu}_{k|k}^{j}=P\{{M}_{k}^{j}|{\mathbf{Z}}_{1}^{k}\}=\frac{1}{c}{\wedge}_{k}^{j}{\overline{c}}^{j}$$
- Combining state mean and covariance estimates for output:$${\hat{\mathbf{x}}}_{k|k}=\sum _{j=1}^{r}{\hat{\mathbf{x}}}_{k|k}^{j}{\mu}_{k|k}^{j}$$$${\mathbf{P}}_{k|k}=\sum _{j=1}^{r}{\mu}_{k|k}^{j}\{{\mathbf{S}}_{k|k}^{j}{({\mathbf{S}}_{k|k}^{j})}^{T}+[{\hat{\mathbf{x}}}_{k|k}-{\hat{\mathbf{x}}}_{k|k}^{j}]{[{\hat{\mathbf{x}}}_{k|k}-{\hat{\mathbf{x}}}_{k|k}^{j}]}^{T}\}$$

## 6. Initial State Estimation

## 7. Track Management Using Rule-Based M/N Logic

- State estimate ${\hat{\mathbf{x}}}_{k|k}$
- Square-root of error covariance estimate ${\mathbf{S}}_{k|k}$
- Residual covariance ${\mathbf{S}}_{Z,k}$
- Hit counter H
- Miss counter M

## 8. Performance Evaluation Simulations

#### 8.1. Performance Evaluation Metrics

- Root mean square error (RMSE) in position: Assume $[{x}_{k},{y}_{k}]$ and $[{\hat{x}}_{k},{\hat{y}}_{k}]$ represent the true and estimated positions, respectively, of a target at time instant k in $xy$-plane. The RMSE in position in terms of Cartesian coordinates at time instant k can be written as$${\mathrm{RMSE}}_{k}^{XY}=\sqrt{{({\hat{x}}_{k}-{x}_{k})}^{2}+{({\hat{y}}_{k}-{y}_{k})}^{2}}$$
- Root mean square error (RMSE) in velocity: Similarly, if $[{v}_{{x}_{k}},{v}_{{y}_{k}}]$ and $[{\hat{v}}_{{x}_{k}},{\hat{v}}_{{y}_{k}}]$ represent the true and estimated velocities, respectively, of a target at time instant k, then the RMSE in velocity at time instant k can be written as$${\mathrm{RMSE}}_{k}^{{v}_{x}{v}_{y}}=\sqrt{{({\hat{v}}_{{x}_{k}}-{v}_{{x}_{k}})}^{2}+{({\hat{v}}_{{y}_{k}}-{v}_{{y}_{k}})}^{2}}$$
- Posterior Cramer–Rao lower bound (PCRLB): PCRLB states that if both the state and measurement are random, then the state covariance matrix for an unbiased estimator is bounded as;$${\mathbf{P}}_{k|k}\ge {\mathbf{J}}_{k}^{-1}$$$$\mathrm{PCRLB}\{[{\hat{{\mathbf{x}}_{k}}]}_{j}\}={[{\mathbf{J}}_{k}^{-1}]}_{jj}$$
- Mean execution time for one data scan (${T}_{E}$): Mean execution time of algorithm for one cycle is computed by laptop computer. The specifications of computer are $1.9$ GHz processor, 4 GB RAM and Windows10 for Matlab2018. The total execution time is composed of time for data association (${T}_{DA}$), target’s state estimation by filtering (${T}_{F}$) and track management (${T}_{TM}$) steps.
- IMM mean model probabilities: IMM mean model probabilities for maneuvering targets reflect how efficiently IMM algorithm can recognize different dynamic motion models of the targets and switch accordingly.

#### 8.2. Parameter Selection and Simulated Data Generation

#### 8.3. Scenario 1: Three Non-Maneuvering Targets Moving Tangential to the Radar Broadside

#### 8.4. Scenario 2: Two Non-Maneuvering Targets with Circular Motion

#### 8.5. Scenario 3: Two Maneuvering Targets with Crossing Track Patterns

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**The block diagram of multi-target tracking algorithm based on 2-D velocity measurement function.

**Figure 4.**Track management block of multi-target tracker algorithm based on 2-D velocity measurement function.

Targets | Initial States of Targets | ||||
---|---|---|---|---|---|

x (m) | ${\mathit{v}}_{\mathit{x}}$ (m s${}^{-1}$) | y (m) | ${\mathit{v}}_{\mathit{y}}$$(\mathbf{m}\mathbf{s}{}^{\mathbf{-}\mathbf{1}})$ | $\mathbf{\Omega}$ (rad s${}^{-1}$) | |

1 | −4 | 7.9 | 6.6 | 4.8 | −1.2 |

2 | 4 | −6.6 | 5.5 | 4.8 | 1.2 |

3 | 3 | −9.5 | 4.4 | 6.4 | 2.1 |

Targets | Initial States of Targets | ||||
---|---|---|---|---|---|

x (m) | ${\mathit{v}}_{\mathit{x}}$ (m s${}^{-1}$) | y (m) | ${\mathit{v}}_{\mathit{y}}$$(\mathbf{m}\mathbf{s}{}^{-1})$ | $\mathbf{\Omega}$ (rad s${}^{-1}$) | |

1 | −1 | 3 | 3.8 | 0 | 4.8 |

2 | 1.5 | −2.2 | 3 | 0 | −4.8 |

Targets | Initial States of Targets | ||||
---|---|---|---|---|---|

x (m) | ${\mathit{v}}_{\mathit{x}}$ (m s${}^{-1}$) | y (m) | ${\mathit{v}}_{\mathit{y}}$$(\mathbf{m}\mathbf{s}{}^{-1})$ | $\mathbf{\Omega}$ (rad s${}^{-1}$) | |

1 | −2.25 | 3 | 3.5 | 0 | 4.8 |

2 | 1.5 | −2 | 3.5 | 0 | −4.8 |

Scenario | Execution Time | |||
---|---|---|---|---|

${\mathit{T}}_{\mathit{E}}$$(\mathbf{ms})$ | ${\mathit{T}}_{\mathbf{DA}}$ (ms) | ${\mathit{T}}_{\mathit{F}}$ (ms) | ${\mathit{T}}_{\mathbf{TM}}$ (ms) | |

1 (IMM-SCKF) | 18.8 | 9.37 (50%) | 6.02 (32%) | 3.41 (18%) |

2 (IMM-SCKF) | 16.3 | 8.61 (53%) | 4.56 (28%) | 3.13 (19%) |

3 (NCV-SCKF) | 14.1 | 7.61 (54%) | 3.52 (25%) | 2.96 (21%) |

4 (IMM-SCKF) | 19.4 | 9.90 (51%) | 6.50 (33%) | 2.98 (16%) |

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**MDPI and ACS Style**

Ishtiaq, S.; Wang, X.; Hassan, S.
Multi-Target Tracking Algorithm Based on 2-D Velocity Measurements Using Dual-Frequency Interferometric Radar. *Electronics* **2021**, *10*, 1969.
https://doi.org/10.3390/electronics10161969

**AMA Style**

Ishtiaq S, Wang X, Hassan S.
Multi-Target Tracking Algorithm Based on 2-D Velocity Measurements Using Dual-Frequency Interferometric Radar. *Electronics*. 2021; 10(16):1969.
https://doi.org/10.3390/electronics10161969

**Chicago/Turabian Style**

Ishtiaq, Saima, Xiangrong Wang, and Shahid Hassan.
2021. "Multi-Target Tracking Algorithm Based on 2-D Velocity Measurements Using Dual-Frequency Interferometric Radar" *Electronics* 10, no. 16: 1969.
https://doi.org/10.3390/electronics10161969