Multitarget tracking in cluttered environment for a multistatic passive radar system under the DAB/DVB network
 Yi Fang Shi^{1},
 Seung Hyo Park^{1} and
 Taek Lyul Song^{1}Email author
https://doi.org/10.1186/s1363401704454
© The Author(s) 2017
Received: 25 September 2016
Accepted: 13 January 2017
Published: 23 January 2017
Abstract
The target tracking using multistatic passive radar in a digital audio/video broadcast (DAB/DVB) network with illuminators of opportunity faces two main challenges: the first challenge is that one has to solve the measurementtoilluminator association ambiguity in addition to the conventional association ambiguity between the measurements and targets, which introduces a significantly complex threedimensional (3D) data association problem among the targetmeasurement illuminator, this is because all the illuminators transmit the same carrier frequency signals and signals transmitted by different illuminators but reflected via the same target become indistinguishable; the other challenge is that only the bistatic range and rangerate measurements are available while the angle information is unavailable or of very poor quality.
In this paper, the authors propose a new target tracking algorithm directly in threedimensional (3D) Cartesian coordinates with the capability of track management using the probability of target existence as a track quality measure. The proposed algorithm is termed sequential processingjoint integrated probabilistic data association (SPJIPDA), which applies the modified sequential processing technique to resolve the additional association ambiguity between measurements and illuminators. The SPJIPDA algorithm sequentially operates the JIPDA tracker to update each track for each illuminator with all the measurements in the common measurement set at each time. For reasons of fair comparison, the existing modified joint probabilistic data association (MJPDA) algorithm that addresses the 3D data association problem via “supertargets” using gate grouping and provides tracks directly in 3D Cartesian coordinates, is enhanced by incorporating the probability of target existence as an effective track quality measure for track management. Both algorithms deal with nonlinear observations using the extended Kalman filtering. A simulation study is performed to verify the superiority of the proposed SPJIPDA algorithm over the MJIPDA in this multistatic passive radar system.
Keywords
1 Introduction
In a multistatic passive radar system, illuminators of opportunity such as radio or television transmitters can be used. The transmitted signals are not under the control of the receiver; thus, the receiver can remain hidden. One can measure the time difference of arrival (TDOA) and Doppler shift between signal received directly from the illuminator and delayed copies from potential targets. As the receiver and transmitter in the multistatic passive radar system are completely separated, the receiver therein just needs to process the received signals while it has to service the transmitter in a feedback manner and consumes much more power in some other radar systems; therefore, the multistatic passive radar system is more economic. Due to the noncooperative illuminator transmits low RF signals, the chance to detect stealth and low altitude targets increases [1]; however, this causes problem in that measurement of the azimuth angle is often of very poor quality or not even available at some extreme situations. Therefore, in this multistatic passive radar system, the range and range rate are measured while the angle information is assumed to be unavailable, due to the low RF frequencies of the illuminating signals.
In this paper, we focus on target tracking from preprocessed detections originating from the television or radio broadcasting signals that are modulated according to the digital audio broadcasting (DAB) or digital video broadcasting (DVB) standards [2, 3]. The using of the DAB/DVB signals delivers numbers of advantages compared to that of analog signals: such as, the improved detection performance [4], the more effective signal processing process [5], and the more easily estimated multipath. As a result, a couple of widely spaced transmitters that broadcast the DAB/DVB signals on the same carrier frequency with each responsible for a small and overlapping subscriber footprint, are used to cover a large surveillance space.
However, there are two main challenges in this multistatic passive radar system with a DAB/DVB network. One is that there is a new association ambiguity between the measurements and the illuminators on top of the conventional ambiguity between the measurements and targets, which results in threedimensional (3D) data association and adds significant data association complexity. This is because all illuminators in this system transmit the same frequency broadcasting signals and it is not available for the receiver to differentiate the received signals from the different illuminators’ signals. The other challenge is that more than one illuminator is needed to locate a single target due to the absence of angle information, which inevitably generates a lot of ghosts.
There has been numbers of research focused on the problems in this multistatic passive radar system with DAB/DVB networks. The trackbeforedetect (TBD) algorithms [6, 7] that estimate the target positions directly from the unprocessed DAB/DVB signals can be used to solve the association measurements since the target is observed over several consecutive scans and hence reducing the number of possible associations. The TBD algorithms ensure better detection and estimation performance than conventional algorithms at the price of an increased computational load, while the empirical techniques can be adopted to reduce the complexity of the TBD algorithms, one can refer to [8–10], therein, the proposed algorithms have a complexity linear in the number of integrated scans and in the time on target. As opposed to from the unprocessed signals, most other researchers focus on the target position estimation from the preprocessed detections (measurements), which require less computation source over the TBD algorithms but have to address the data association problem. The authors in [11] propose a multidimensional assignment approach to solve the transmitter ambiguity for bistatic range, range rate, and precise azimuth measurements. In [12], the authors propose a multihypothesis tracking (MHT)based threestage approach that includes primary tracking directly on the measurements and a twodimensional (2D) estimate to address the association problem that is later resolved into 3D. Another algorithm employing the likelihood ratio test to remove “ghost” tracks is investigated in [13]. Recently, Choi et al. [14, 15] propose two groups of algorithms for multitarget tracking directly in the Cartesian coordinates. One group consists of the extended Kalman filter (EKF) and unscented Kalman fitler (UKF) [16] based modified joint probabilistic data association (MJPDA) to resolve the additional ambiguity between measurements and illuminators, while the other group consists of the bootstrap particle filter (BPF) and auxiliary particle filter (APF) [17] based data association under the probabilistic multihypothesis tracker (PMHT) measurement model. However, these preprocesseddetectionbased tracking algorithms neglect providing an effective track quality measure for track management, which motivates the authors to consider the probability of target existence as a track quality measure for multitarget tracking from preprocessed detections.
This paper focuses on the multitarget tracking algorithms directly in threedimensional (3D) Cartesian coordinates using the preprocessed detections. Motivated by the techniques presented in [18] and [19, 20]. The authors propose a new algorithm entitled sequential processingjoint integrated probabilistic data association (SPJIPDA) which provides tracks directly in 3D Cartesian coordinates, and enable the track management using the probability of target existence as a track quality measure. To the best knowledge of the authors, the only existing multitarget tracking algorithms directly in 3D Cartesian coordinates under the multistatic DAB/DVB passive radar system are proposed in [14, 15], they are the modified joint probabilistic data association (MJPDA) and the particle filtering under the probabilistic multiple hypothesis (PMHT) model, with the MJPDA delivers more robust tracking performance and requires less computational resources. However, both of the exiting algorithms neglect providing an efficient track quality measure for track management. Therefore, in order to compare the proposed algorithm (SPJIPDA) to the exiting algorithms, the MJPDA algorithm is enhanced by incorporating the probability of target existence as a track quality measure for track management, and termed modified joint integrated probabilistic data association (MJIPDA). The proposed SPJIPDA algorithm avoids the extra data association ambiguity between the measurements and illuminators, and moreover, the measurement information from various illuminators is utilized in a more effective way and delivers much better tracking performance compared with the MJIPDA algorithm.
The paper is organized as follows. Section 2 describes the problem statement, and the modified JIPDA algorithm is presented in Section 3. Section 4 investigates the sequential processing JIPDA algorithm and the simulation is implemented in Section 5, with the conclusions in Section 6.
2 Problem statement
The tracking algorithms presented in this paper are based on the infinite sensor resolution and point target assumptions. Denote a track or a potential target followed by a certain track by superscript τ, of which the interpretation is clear from the context.
2.1 Target
where T is the sampling interval between two consecutive scans, ⊗ denotes the Kronecker product and I _{3} is the identity matrix of size 3. For reasons of simplicity, as well as better concentrating on the essence of problems caused by multitarget tracking in clutter for a multistatic passive radar system under the DAB/DVB network, we assume a constant velocity target trajectory model. There have two feasible methods to face with the possible target trajectory changes: one is to increase the value of target trajectory plant noise covariance matrix (Q _{ k }) when the target trajectory changes slightly, therein, the increased plant noise covariance is able to account for certain amount of mismatch between the assumed trajectory model and the actual one. However, if the target trajectory changes dramatically or even maneuvers among different motion models, one can resort to the interactive multiple model (IMM) algorithm [24], which performs well on systems characterized by multiple models of target behavior.
where \({\mathbf {z}_{k}}\,=\,{[{\gamma _{k}}\;\;{\dot \!\! \gamma _{k}}]^{T}}, h_{k}^{s}({\mathbf {x}_{k}^{\tau }},\mathbf {x}_{s})={[{\gamma _{k}}({\mathbf {x}_{k}^{\tau }},{\mathbf {x}_{s}})\;\;{\dot \gamma _{k}}({\mathbf {x}_{k}^{\tau }},{\mathbf {x}_{s}})]^{T}}\), and w _{ k } is the white Gaussian noise sequence with zero mean and covariance R _{ k }, uncorrelated with the plant noise sequence \(\mathbf {v}_{k}^{\tau }\). At each time k, the receiver receives a random set of measurements Z _{ k } without prior information on the origin of each measurement. Each measurement has only one resource, either a target or clutter, but it can be originated from any illuminator. Denote the set of measurements up to and including time k by Z ^{ k }={Z ^{ k−1}, Z _{ k }}, and let Z _{ k,i } denote the ith measurement of Z _{ k }.
2.2 Clutter
where the clutter measurement density \(\rho _{k,i}=\frac {P_{fa}}{V_{rc}}\) [24, 25], with P _{ fa } and V _{ rc } denote the probability of false alarm and the sensor resolution cell volume. As can be easily seen, the probability of false alarm impacts on the number of clutter measurements significantly, that is, if the probability of false alarm P _{ fa } increases, the clutter measurement density ρ _{ k,i } also increases, and the mean number of clutter measurements in the surveillance space V _{ k } increases, which results in a increased number of clutter measurements at each time k. In target tracking, the clutter measurement density is either a priori known or estimated adaptively based on the current measurements [26].
2.3 Multistatic passive radar in a DAB/DVB network
As the focus of this paper is on target tracking from detections of the preprocessed signals in the multistatic passive radar systems of a DAB/DVB network, there mainly exist two challenges. One is that there is a new association ambiguity between measurements and illuminators besides the conventional one between measurement and targets, which results in 3D data association among targetmeasurement illuminators. This is because each of the multiple illuminators transmits the same frequency digital signal, albeit with different delays due to the illuminatortargetreceiver geometry, the fact that received signals are composed of multiple unlabeled delays per target, and that there is no useful information to discriminate the origin of the measurement. As the number of 3D association events increases dramatically [15] even given small number of measurements, targets and illuminators, it is computationally unrealistic to directly implement conventional 2D data association approaches to resolve the 3D data association problem in this passive radar system.
The other challenge is due to the broadcasting signal frequencies for passive radars and the type of receivers, the angle information is of realistically poor quality and target tracking using only the range and range rate without angle information inevitably generates ghost tracks. A bistatic range measurement can locate a target at an ellipsoid in 3D Cartesian coordinates, but for the intersection of two ellipsoids, which is an ellipse, a third bistatic range measurement is necessary to possibly locate the target, resulting in the generation of multiple ghosts. Furthermore, in the presence of measurement noise and the clutter measurement, the situation is even more severe.
In the following two sections, two track maintenance algorithms directly in 3D Cartesian coordinates are introduced. The first presented in Section 3 is the modified JIPDA algorithm, which is enhanced by incorporating the probability of target existence as a track quality measure. The second algorithm, proposed in Section 4 is the sequential processing JIPDA algorithm, which is our main contribution of this paper.
3 Modified JIPDA (MJIPDA)
3.1 Supertarget and gate grouping
The JPDA algorithm [24] enumerates and probabilistically evaluates every validated measurement to the target association event, and the target states are estimated by using the marginal association probability. In this multistatic passive radar system, as discussed in Section 2.3, the computational cost of evaluating all possible 3D association events is much higher than the cost for 2D association events. The authors in [15] proposed a suboptimal idea of supertarget \(\tilde \tau = \{ \tau,s\}\), which is a hypothetical target consisting of a pair of target τ and illuminator s, and succeeded to recast the 3D association among measurements, targets and illuminators to a 2D listmatching problem between measurements and supertargets.
with the predicted measurement \(h_{k}^{s}(\mathbf {x}_{kk  1}^{\tau },\mathbf {x}_{s})\) and its associated covariance \(\mathbf {S}_{k}^{\tau,s}\) with respect to supertarget \(\tilde \tau = \{ \tau,s\}\), where κ is the gating threshold.
The supertargets sharing at least one measurement in their corresponding validation gates are classified as one group. Thus, all of the 3D association events can be separated as groups of 2D association events, which decreases the number of association events on a large scale. Note that the concept of a group here is slightly different from the cluster, in the sense that the supertargets in different groups are possibly from the same target since one target can create N _{ s } supertargets via combining with N _{ s } illuminators.
Example of supertargets
\(\tilde \tau _{1}={\left (\tau _{1}, s_{1}\right)}\)  \(\tilde \tau _{2} = \left ({ \tau _{1},s_{2}}\right)\)  \(\tilde \tau _{3} = \left ({ \tau _{2}, s_{1}}\right)\)  \(\tilde \tau _{4} = \left ({ \tau _{2},s_{2}}\right)\)  

z _{1}  ×  ✓  ×  ✓ 
z _{2}  ✓  ×  ✓  × 
z _{3}  ×  ✓  ×  ✓ 
z _{4}  ✓  ×  ✓  × 
3.2 Modified JIPDA using supertargets
The trajectory state pdf \(p(\mathbf {x}_{k}^{\tau } \chi _{k}^{\tau },{\mathbf {Z}^{k  1}})\) is only conditioned on the target existence \(\chi _{k}^{\tau }\), and for the rest of this paper, this conditioning is only implicit.
3.2.1 Data association
where P _{ G } is the gating probability, c _{ k } is the normalizing constant. \(P(\chi _{k}^{\tilde \tau }{\mathbf {Z}^{k  1}}) = P(\chi _{k}^{\tau } {\mathbf {Z}^{k  1}})\) due to \(\tilde \tau = \{ \tau,s\}\).
3.2.2 Track state update
where E K F _{U} is the extended Kalman filter update procedure, and the predicted measurement function is \(h_{k}^{s} = h_{k}^{s}\left (\hat {\mathbf {x}}_{kk  1}^{\tau },{\mathbf {x}_{s}}\right)\).
4 Sequential processing JIPDA (SPJIPDA)
The authors propose another methodology named sequential processing JIPDA (SPJIPDA) for target tracking in clutter for this multistatic passive radar system, which avoids the association between measurements and illuminators by sequentially processing the tracks for each illuminator with all the measurements in the common measurement set, and then recursively calculates the probability of target existence as a track quality measure for track management. Note that the concept of the supertarget is not needed in this approach.
4.1 Sequential processing framework
The sequential implementation for multisensor multitarget tracking in clutter is introduced in [19, 20], which processes the measurements from one sensor at a time, and the measurements of the next sensor are then used to further improve the intermediate state estimation. However, the measurements of different illuminators in the multistatic passive radar system in a single frequency (DAB/DVB) network are indistinguishable, the typical sequential approach needs to be modified slightly. The sequential processing method proposed in this section is that the hybrid state of each track is updated sequentially for each illuminator with all the measurements in the common measurement set.
4.2 Sequential processing JIPDA implementation

Track state propagation

Measurement selection and likelihood evaluation

Multitarget data association

Track state update
4.2.1 Track state propagation
Here p _{1,1} is the probability of target existence at time k given that it exists at time k−1 [21], and \(P(\chi _{k}^{\tau } (s  1){\mathbf {Z}^{k}})\) is the posterior probability of target existence with respect to illuminator s−1.
where E K F _{ P } denotes the extended Kalman filter prediction procedure, and \(\hat {\mathbf {x}}_{kk}^{\tau } (s  1)\) and \(\mathbf {P}_{kk}^{\tau } (s  1)\) are the posterior mean and covariance of the track trajectory state with respect to the illuminator s−1 at time k.
4.2.2 Measurement selection and likelihood evaluation
4.2.3 Multitarget data association
4.2.4 Track update
This recursive procedure with respect to each track τ at time k operates for each illuminator s=1,…,N _{ s }.
4.2.5 Track output
5 Simulation validation
In this section, the numerical experiments for multitarget tracking in a multistatic passive radar system under the DAB/DVB network are discussed, in which the superiority of the proposed algorithm (SPJIPDA) over the algorithm (MJIPDA) is validated and the track management in both algorithms using the probability of target existence as a track quality measure turns out to be efficient.
5.1 Scenario description in the DAB/DVB network
5.2 Performance measure criterion
Due to the association ambiguity between measurements and illuminators as well as the unavailability of angle information, the track initiation using measurements in the multistatic passive radar with the DAB/DVB network is significantly computationally expensive and usually fails to give satisfactory performance, thus, is outside the scope of this paper; preliminary work can be found in [29], wherein the track initiation performance is quite sensitive to the clutter and target detection probability, and works only with very small number of clutter measurements and high target detection probability.
In this simulation, the target ground truth information is used to initiate a track [15]. The initiated track position and velocity are generated from the target truth position and velocity with a certain disturbance (assumed to be a Gaussian distribution), i.e., the initiation position of target τ is \(\hat {\mathbf {x}}_{0}^{\tau,p} = \mathbf {x}_{0}^{\tau,p} + N\left (p_{0};0,\sigma _{I}^{2}\right)\), and the initiation velocity is \(\hat {\mathbf {x}}_{0}^{\tau,v} = \mathbf {x}_{0}^{\tau,v} + N\left (v_{0};0, {10^{4}}\sigma _{I}^{2}\right)\), where \(\mathbf {x}_{0}^{\tau,p}\) and \(\mathbf {x}_{0}^{\tau,v}\) are the true initial target position and velocity of target τ, respectively.
5.2.1 Track management
An important performance measure for target tracking in clutter is the track management. In this simulation, both the MJIPDA and the SPJIPDA algorithms calculate the probability of target existence as the track quality measure for track management. Each initiated track is given an initial value as the probability of target existence, which is recursively updated by the measurements in the subsequent scans. The track management procedure confirms a track if its updated probability of target existence is greater than a predefined confirmation threshold, and it maintains the confirmed status until terminated when the updated probability of target existence falls below a predefined termination threshold.
where \({N_{c}^{i}}\) is the number of confirmed tracks at ith Monte Carlo run. For fair performance comparison of track maintenance, both SPJIPDA and MJIPDA algorithms are given the same track management parameters (the initial probability of target existence, the track confirmation threshold and termination threshold).
5.2.2 Trajectory average estimation error
where \(\mathbf {x}_{k}^{\tau }\) is the trajectory estimation of track τ at time k and x is the true target state.
5.3 Numerical results

Case 1: detection probability P _{ D }=0.9, measurement noise deviation σ _{ r }=10 m and \(\sigma _{\dot r}=0.1\;m/s\), clutter measurement density ρ=0.001 (n u m b e r/m ^{2}.s ^{−1}), Gaussian disturbance deviation σ _{ I }=10 m.

Case 2: detection probability P _{ D }=0.9, measurement noise deviation σ _{ r }=30 m and \(\sigma _{\dot r}=0.3\;m/s\), clutter measurement density ρ=0.001 (n u m b e r/m ^{2}.s ^{−1}), Gaussian disturbance deviation σ _{ I }=40 m.
As a consequence, the SPJIPDA algorithm delivers much better tracking performance in terms of both track management and trajectory estimation. This can be explained by that the track state at each scan in SPJIPDA algorithm is improved multiple times via sequential updating with respect to various illuminators, wherein the target measurements detected by different pair of illuminatorreceivers are updated by incrementally accurate predicted track states at single scan; therefore, the likelihood of target measurements obtained by the SPJIPDA algorithm is higher than that by the MJIPDA algorithm, which gives a faster increasing rate of the probability of target existence.
Execution time [sec.]
Case 1  Case 2  

MJIPDA  750  832 
SPJIPDA  620  698 
Entire simulation time  12,000  12,000 
6 Conclusions
This paper investigates two solutions for multitarget tracking in clutter directly in 3D Cartesian coordinates using multistatic passive radar with a DAB/DVB network, where both algorithms are capable of track management using the probability of target existence as the track quality measure.
The MJIPDA algorithm is developed by incorporating the probability of target existence into the MJPDA algorithm as a track quality measure for track management, and the MJPDA algorithm addresses the targetmeasurement illuminator association ambiguity via “supertargets” using gate grouping. The SPJIPDA algorithm sequentially operates the JIPDA tracker to update each track for each illuminator with the common measurement set at each scan. Compared with the MJIPDA algorithm, the SPJIPDA enhances the target’s track multiple times only at single scan by sequentially processing with respect to various illuminators; therefore, the target measurements can be utilized in a more efficient way to update the target’s track state. The simulation validates the efficiency of the proposed algorithm and also shows the superiority performance of SPJIPDA over the MJIPDA algorithm.
There have several aspects worthy of further work: the availability and robustness of the proposed algorithms are worthwhile to be validated based on the real data obtained from a realistic setup of multistatic passive radar system under the DAB/DVB network; the more computationally efficient versions of the proposed multitarget tracking algorithm are attractive to be developed since the proposed multitarget tracking algorithms employ the optimal Bayesian multitarget joint data association approach which may suffer from the numerical explosion when numbers of close targets presented in the surveillance space; the extension of the proposed algorithms to multiple maneuver targets tracking in cluttered environment is also valuable.
7 Nomenclature
7.1 A. List of acronyms
 DAB/DVB
Digital audio/video broadcast
 JPDA
Joint probabilistic data association
 JIPDA
Joint integrated probabilistic data association
 MJIPDA
Modified joint integrated probabilistic data association
 SPJIPDA
Sequential processingjoint integrated probabilistic data association
 TDOA
Time difference of arrival
 EKF
Extended Kalman filter
 UKF
Unscented Kalman filter
 BPF
Bootstrap particle filter
 APF
Auxiliary particle filter
 TBD
Trackbeforedetect
 PMHT
Probabilistic multihypothesis tracker
 FJE
Feasible joint event
 JMTDA
Joint multitarget data association
 ANCT
Average number of confirmed tracks
 RMSE
Root mean square error
7.2 B. List of symbols

∥x∥ The euclidean norm of vector x

x ^{ T } The transpose of vector x

⊗ The Kronecker product

I _{3} The identity matrix of size 3

s A illuminator indexed by s

τ A track or a target followed by certain track by superscript τ

\(\tilde {\tau }\) A hypothetical target consisting of a pair of target τ and illuminator s

\(\chi _{k}^{\tau }\) Event that target τ exists at time k

\(\chi _{k}^{\tau }(s)\) Event that target τ exists at time k for illuminator s

\(\mathbf {x}_{k}^{\tau }\) The trajectory state in 3D Cartesian coordinates of target τ at time k

\(\mathbf {x}_{k}^{\tau,p}\) The position component of \(\mathbf {x}_{k}^{\tau }\)

\(\mathbf {x}_{k}^{\tau,v}\) The velocity component of \(\mathbf {x}_{k}^{\tau }\)

\(\mathbf {x}_{k}^{\tau }(s)\) Trajectory state in 3D Cartesian coordinates of target τ at time k for illuminator s

x _{ s } The position of illuminator s in 3D Cartesian coordinates

x _{ r } The position of the only receiver in 3D Cartesian coordinates

N _{ s } The number of entire illuminators

γ _{ k } Sensor received bistatic range measurement at time k

\({\dot \gamma _{k}}\) Sensor received bistatic rangerate measurement at time k

F Target trajectory transition matrix

Q _{ k } Target trajectory plant noise covariance matrix

R _{ k } Measurement noise covariance matrix

P _{ D } Target detection probability

P _{ G } Gating probability that the (true) measurement will fall in the gate

Z _{ k } Sensor received set of measurements at time k

Z _{ k,i } The ith measurement of Z _{ k }

Z ^{ k } Set of measurements up to and including time k

ρ _{ k,i } Clutter measurement density of Z _{ k,i }

\(\mathbf {z}_{k}^{\tilde \tau }\) Set of selected measurements at time k with respect to supertarget \(\tilde \tau \)

\(N_{k}^{\tilde \tau }\) The cardinality of \(\mathbf {z}_{k}^{\tilde \tau }\)

\(\mathbf {z}_{k,i}^{\tilde \tau }\) The ith measurement of \(\mathbf {z}_{k}^{\tilde \tau }\)

ξ _{ j } The jth feasible joint event

\(\mathbf {z}_{k}^{\tau }(s)\) Set of selected measurements at time k with respect to track τ for illuminator s

\(m_{k}^{\tau }(s)\) The cardinality of \(\mathbf {z}_{k}^{\tau }(s)\)

\(\mathbf {z}_{k,i}^{\tau }(s)\) The ith measurement of \(\mathbf {z}_{k}^{\tau }(s)\)

T _{0}(ξ _{ j }) The set of supertargets allocated no measurement in ξ _{ j }

T _{1}(ξ _{ j }) The set of supertargets allocated one measurement in ξ _{ j }

\(\Xi (\tilde \tau,i)\) The set of FJEs which allocate measurement i to supertarget \(\tilde {\tau }\)

\(\chi _{k,i}^{\tilde \tau }\) Event that the selected measurement i is the detection of supertarget \(\tilde {\tau }\) at time k

\(\beta _{i\tilde {\tau }}\) Posterior association probability measurement i is supertarget \(\tilde {\tau }\) detection

\(\beta _{k,i}^{\tau }(s)\) Posterior association probability measurement i is target τ detection for s

E(τ) The set of supertargets originated from track τ

p _{1,1} The probability of target existence at time k given that it exists at time k−1

T The sampling interval between two consecutive scans

P _{ fa } The probability of false alarm

V _{ rc } The sensor resolution cell volume
Declarations
Acknowledgements
This work was supported by the Agency for Defense Development, Republic of Korea (Grant UD160001DD).
Authors’ contributions
YS made the main contributions to conception and tracking algorithms’ design, as well as drafting the article. SP mainly designed the simulation validation and results analysis. TS offered critical suggestions on the algorithms’ design, provided significant revising for important intellectual content and gave final approval of the current version to be submitted. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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