Efficient centralized track initiation method for multistatic ... - IEEE Xplore
Efficient centralized track initiation method for multistatic radar Shiyou Xu, Chaojing Tang, Peiliang Jing, Zengping Chen Science and Technology on Automatic Target Recognition Laboratory, National University of Defense Technology, Changsha 410073, China. Email: [email protected] Abstract—An efficient centralized track initiation method for multistatic radar system is proposed in this paper. The method mainly consists of a same source association technique and a measurement fusion approach. The main advantage of the new method is its good performance of dealing with large measuring deviation and high false alarm rate environment, when compared with other existing track initiation methods. And this is verified by the Monte Carlo simulations. Keywords—multistatic radar; track initiation; false alarm; measuring deviation; same source association
I.
INTRODUCTION
Track initiation is the procedure by which a new target entering radar coverage is acquired by the tracking system [12]. This means there generally will be a far range between target and radar. Moreover, to detect modern military aircrafts which probably have low radar cross section (RCS), radar system need to use a low detection threshold. Thus, for track initiation procedure in such situations, target Cartesian localization will have a large measuring deviation and the false alarm rate may always be high. During the past several decades, a great deal of research has been devoted to track initiation using monostatic radar, and two main series, sequential methods and batch methods, have been proposed. But unfortunately these methods, such as heuristic rule method, logic-based method [3], Hough Transform method [4-6] and modified Hough Transform method [7], have a rather low correct track probability when dealing with targets corrupted by false alarms and/or large measuring deviation [8]. Consequently, the track maintenance methods (such as probability data association (PDA) [9] and joint probability data association (JPDA) [10]) that could perfectly treat with false alarms, is greatly restricted. This is because the exertion of the performance of track maintenance methods has to depend on the cooperating with a track initiation method with high correct track initiation probability. Then, the scheme using track association [11-13] and fusion [14-16] in a data processing center will make no sense any more in the above situations. In short, the poor performance of the existing track initiation methods is just the bottleneck of targets tracking issue in low signal-to-noise ratio and/or large measuring deviation environment.
Distinctly unlike traditional track initiation methods which use only one monostatic radar, method in this paper tries to use a T/R-R2 multistatic radar system and a centralized data processing scheme. By the authors’ knowledge, no track initiation method using multistatic radar system or centralized data processing scheme has been studied. During the devised method in this paper, at every scan, measurements from all the receivers of multistatic radar are processed by four steps. Firstly, a same source association approach, which could distinguish real targets measurements from false alarms just by the statistical character differences between them, is adopted. The measurements being recognized by same source association approach as from one same real target will make up of one association group, and the ones being recognized as from false alarms will be abandoned. Secondly, the position element of measurements from every association group will be processed by a fusion technique, the aim of which is to gain high precision target position measurement. Thirdly, a target velocity measurement extracting technique is implemented on Doppler frequency shift element of measurements from every association group. Finally, the position & velocity measurements are used to initiate tracks by unscented Kalman filter [17] and nearest neighbor (NN) logic [18]. At the end of this paper, a contrast of the performance between the new method and logic-based track initiation method (logic-based method is one excellent among the existing methods [8]) is applied. The rest of the paper is organized as follows. In Section 2, the basic working principle of multistatic radar is simply reviewed. Section 3 is devoted to the same source association technique, measurement fusion approach and velocity measurement extraction method. In Section 4, the logic of data association and track management is detailed. Simulation results are shown in Section 5. Section 6 gives conclusions of this work. II.
BASIC WORKING PRINCIPLE OF MULTISTATIC RADAR
For brevity, and without loss of generality, a 2-dimension T/R-R2 multistatic radar system is applied in this paper. The system configuration is shown in Fig. 1. The basic working principle of T/R-R2 multistatic radar at the kth scan is as follows: first T/R site transmits a signal and records the time ts k , then T/R, receiver 1 and 2 extract measurements
each scan is assumed to have Poisson distribution; and the position of false alarms is assumed to uniformly distribute in the interesting area.
Y/km
receiver 2
( x2 , y2 )
a
2k
Let ri0k ( i = 1," I k ) , r j1k ( j = 1," J k ) and rn2 k ( n = 1," N k ) represents the No. i , j and n estimated target position for T/R site, receiver 1 and 2 respectively. Because of' the nonlinear relationship between measurement ⎡⎣tr mk , α mk ⎤⎦ and target position r , unscented transform (UT) [19] is applied here to estimate the means and covariance matrixes of the distances among ri 0k , r j1k , and rn2k according to (1) and (2).
target
a
1k
receiver 1
a
( x1 , y1 )
0k
Let X/km
T/R site (0, 0)
Fig. 1. Sketch map of working principle of multistatic radar system '
⎡⎣tr 0 k , α 0 k , f d0 k ⎤⎦ (signal arrival time, local azimuth, and ' ⎡⎣tr 1k , α 1k , f d1k ⎤⎦ Doppler frequency shift), and 2k 2k 2k ' ⎡⎣tr , α , f d ⎤⎦ from the three reflecting or scattering signals respectively. '
Let r k = ⎡⎣ x k，y k ⎤⎦ represents the position vector of target ' at scan k , Rm = [ xm，ym ] ( m = 0,1, 2 ) represents the location vector of receiver m ( m = 0 represents T/R site), and c denotes the speed of light. Then for T/R site and each receiver, the local target position can be computed by:
r
r
mk
(
= ( tr
III.
0k
mk
( tr =
0k
0k − ts k ) c ⎡ cos (α ) ⎤ ⎥ ⋅⎢ 2 ⎢sin (α 0 k ) ⎥ ⎣ ⎦
− ts ) ⋅ c − r k
0k
)
⎡cos (α mk ) ⎤ ⎥ + Rm ⋅⎢ ⎢sin (α mk ) ⎥ ⎣ ⎦
(1)
⎧uij01k = E ⎡ ri0 k − r1j k ⎤ ⎣ ⎦ ⎪ ⎨ 01k 0k 1k 01k 0k 1k 01k ⎪Qij = E ⎡⎣ ri − rj ⎤⎦ − uij ⋅ ⎡⎣ ri − r j ⎤⎦ − uij ⎩
)}
(3)
⎧ uin02 k = E ⎡ ri 0 k − rn2 k ⎤ ⎣ ⎦ ⎪ ⎨ 02 k 0k 2k 02 k 0k 2k 02 k ⎪Qin = E ⎡⎣ ri − rn ⎤⎦ − uin ⋅ ⎡⎣ ri − rn ⎤⎦ − uin ⎩
)}
(4)
⎧unj21k = E ⎡ rn2 k − rj1k ⎤ ⎣ ⎦ ⎪ ⎨ 21k 2k 1k 21k 2k 1k 21k ⎪Qnj = E ⎡⎣ rn − rj ⎤⎦ − unj ⋅ ⎡⎣ rn − r j ⎤⎦ − unj ⎩
)}
(5)
{(
)(
{(
)(
{(
)(
(2)
(6)
din02 k ( uin02 k ) ⋅ ( Qin02 k ) ⋅ uin02 k
(7)
d nj21k ( unj21k ) ⋅ ( Qnj21k ) ⋅ unj21k
(8)
'
−1
−1
−1
'
At scan k, assume T/R site, receiver 1 and 2 extracts Ik, Jk and Nk measurements respectively. In fact, some of the measurements originate from real targets, and others originate from false alarms which seriously affect the performance of track initiation. It is intuitive that measurements originating from the same real target are just corrupted by measuring errors. Thus the mahalanobis distance among the local positions derived from these measurements could be used as a statistic to test whether they come from one same real target. Assume the estimation deviation of the reflecting or scattering signal arrival time and azimuth at every site are given the same as Gaussian distribution N ( 0, δ tr2 ) (s) and N ( 0, δα2 ) (degree) for brevity. Both the number and position of false alarms are assumed to be random and statistically independent among different receivers from scan to scan. In particular, the number of false alarms from every receiver in
'
dij01k ( uij01k ) ⋅ ( Qij01k ) ⋅ uij01k '
SAME SOURCE ASSOCIATION AND MEASUREMENT FUSION TECHNIQUE
A. Same source association
'
Then, the square mahalanobis distance matrixes D 01k = ⎡⎣ dij01k ⎤⎦ , D 02 k = ⎡⎣ din02 k ⎤⎦ and D 21k = ⎡⎣ d nj21k ⎤⎦ Ik × J k I k × Nk Nk × J k could be given by '
m = 1, 2
'
'
'
If ⎡⎣tri 0 k , α i0 k , f di0 k ⎤⎦ , ⎡⎣trj1k , α 1j k , f dj1k ⎤⎦ and ⎡⎣trn2 k , α n2 k , f dn2 k ⎤⎦ originate from one same real target, according to a predetermined probability pa and the chi squared distribution, a threshold ε could be used to make sure ⎧ p {dij01k < ε } = pa ⎪ ⎪ 02 k ⎨ p {din < ε } = pa ⎪ 21k ⎪⎩ p {d nj < ε } = pa
(9)
The scheme of same source association is as follows. z
Step 1: Find the association groups among which all the three square mahalanobis distances are smaller than ε .
The detail of one round selection is shown below. First initialize i = j = n = 0 and d min = 3ε . And for every combination of i , j and n , if (10) is satisfied, then update i , j , n and d min as (11)
⎧⎪ dij01k < ε ; din02 k < ε ; d nj21k < ε ; ⎨ 01k 02 k 21k ⎪⎩( dij + din + d nj ) < d min
(10)
⎧⎪ i = i; j = j; n = n; ⎨ 01k 02 k 21k ⎪⎩ d min = dij + din + d nj
(11)
After the round of updating, the final d min is compared with 3ε . And if d min is smaller than 3ε , an association group consisting of ⎡⎣tri0 k , α i0 k , f di0 k ⎤⎦ , ⎡⎣trj1k , α 1j k , f di1k ⎤⎦ , and ⎡⎣trn2 k , α n2 k , f di2 k ⎤⎦ is derived. Then clear the effect of this association group ( just let d ij01k = dij01k = d in02 k = din02 k = d nj21k = d nj21k = 2ε ( i = 1, 2,..., I k , j = 1, 2,..., J k , n = 1, 2,..., N k )), and repeat the above operations in step 1 to find the next association group until after one round of updates, the value of d min is not smaller than 3ε . z
Step 2: Find the association groups among which two of the distances are smaller than ε .
The detail is similar to step 1, but the updating of d min is processed as (12) d min = dij01k + din02 k + d nj21k − max ⎡⎣ dij01k , din02 k , d nj21k ⎤⎦
(12)
(
B. Position measurement fusion For every association group, the target Cartesian position r k could be derived with high precision by the weighted sum of the Cartesian position of the three measurements. The Cartesian position of the measurement could be computed by UT method according to (1) and (2). And the weights are proportional to the fisher information matrix of the measurements. Without loss of generality, x1 and y2 are chosen to be 100km, x2 and y1 are chosen to be 0, and the square area in the first quadrant is considered for an example. The average errors for 500 times Monte Carlo simulations of the target position derived from two methods (method 1: the way that only one receiver T/R site is used; method 2: position fusion method) are shown in Fig. 3. Fig. 3 shows that fusion position measurement has a rather higher precision than position measurement extracted by only T/R site. C. Velocity measurement extraction
And the condition of updating is ⎧ min ⎡ d 01k , d 02 k , d 21k ⎤ < ε in nj ⎦ ⎣ ij ⎪ ⎪ 01k 02 k 21k ⎨ m id ⎣⎡ dij , din , d nj ⎦⎤ < ε ⎪ 01k 02 k 21k 01k 02 k 21k ⎪⎩ d min > dij + din + d nj − max ⎡⎣ dij , din , d nj ⎤⎦
Take δ tr = 1× 10−7 and δ α = 0.5 for an example, Fig. 2 gives the average ghost elimination effect as a function of average number of false alarms λ over 500 times tests.
(13)
)
The association group can be confirmed if the final d min is smaller than 2ε . The selection of association group should be continued until after one round of updates, the value of d min is still 2ε . After step 1 and 2 have been done, the measurements that have not formed any association group will be thought as false alarms. 15 False measurement detected by T/R
After forming the fusion position measurement r k of every association group, the three Doppler frequency shifts measured by T/R site, receiver 1 and receiver 2 can be used to ' estimate the 2-dimension velocity r k = ⎡⎣ x k，y k ⎤⎦ of target by ' ⎧ ⎡ r mk ⎤ ⎪ f mk = 2 f c ⋅ ⎣ ⎦ ⋅ r k ⎪ d c r mk ⎪ ⎨ ' mk 0k ' ⎪ mk f c ⎛ ⎡⎣ r ⎤⎦ ⎡⎣ r − R m ⎤⎦ ⎜ + mk ⎪ fd = ⋅ c ⎜ r 0k r − Rm ⎪ ⎝ ⎩
m=0 ⎞ ⎟ ⋅ r k ⎟ ⎠
(14) m = 1, 2
Where f c is the radar signal carrier frequency. After target velocity being derived, then a' complete target Cartesian state vector Z ( k ) = ⎡⎣ x k , y k , x k , y k ⎤⎦ can be formed, and can be used to perform the subsequent target tracking process.
True measurement detected by T/R False association group
10
1500 Average error (m)
Number
True association group
5
0
0
2
4
6
8
10
12
λ Fig. 2. The false alarms elimination effect of same source association.
1000 500 0 100
100
50 Y(km)
50 0
0
X(km)
(a) The way that only T/R site is used.
initiated. For all candidate tracks, once the track score equals 0, the track will be degraded as a dead track, which means the track is terminated. Only the track with length no less than L could be considered originating from an existing target.
40
V.
20
100
50 Y(km)
50 0
0
X(km)
(b) Position fusion method. Fig. 3. Contrast of average errors of measurement between T/R and fusion method.
IV.
DATA ASSOCIATION AND TRACK MANAGEMENT MECHANISM
The data association and track management mechanism using the new track initiation method are detailed as below. First, tracks are classified into 3 types: candidate tracks, dead tracks, and active tracks. And every track has two properties: track score (denotes the quality of a track, and is the actor to decide whether a track should be terminated or not) and track length (denotes how many times of a track being updated by the measurements, and is the actor to decide whether a track should be initiated or not).
pc =
z
z
Step 2: If one candidate track is updated successfully by measurements, the track score and length should plus 1; if not, then only the track score should minus 1. Step 3: Every Z ( k ) that updates none tracks should initialize a new candidate track as that at scan 1. Step 4: For candidate tracks, once the track length equals to a predetermined value L, the track should be set as an active track, which means the track is
(18)
Take the predetermined track length value L=4 and
δ tr = 1× 10−7 . Then performance (pc) of the two methods:
method developed in this paper and logic based method (applied on measurements from T/R), which is as a function of the average number of false alarms λ and standard azimuth deviation δ a of angle measurement, is shown as Fig. 4. 0.4 0.2 0 0
λ 5 10 0.2
0.6 0.4 δ (degree)
0.8
1
a
(a) Typical logic based track initiation method.
1
Pc
2) Scan k(k>1) z Step 1: Use Z ( k ) and R p ( k ) that has not updated any active tracks to update the existing candidate tracks states by NN logic (data association) and Kalman filter (state estimation). Then use the transition model to get state prediction and prediction covariance matrix.
nc M
Here M is the number of real targets, and nc is the number of real target tracks being initiated correctly.
1) Scan 1 ' Every Z ( k ) = ⎡⎣ x k , y k , x k , y k ⎤⎦ and its covariance matrix R p ( k ) extracted from association group are used to initialize a candidate track, whose score is a predetermined integer S and length is 1. Then Kalman filter is initialized by the state of the candidate track, and used to get the state prediction and prediction covariance matrix at the next scan.
z
SIMULATIONS AND RESULTS
To evaluate the performance of the new track initiation methods, correct track initiation probability pc is defined as follows:
0 100
Pc
Average error (m)
60
0.5 0 0 5
λ 10
0.2
0.4
0.6
0.8
δa (degree)
(b) The new track initiation method using multistatic radar. Fig. 4. Performance contrast between two methods.
The comparison between Fig. 4(a) and Fig. 4(b) intuitively shows that the new track initiation method significantly
outperforms typical logic based method. The correct track initiation probability of logic based method is always rather low, even when the parameters are perfect. While the new method exhibits rather good performance. Assume δ tr = 1× 10 , δ a = 0.6 and λ = 12 , then results of one Monte Carlo run of the two methods about a simple scenario consisting of three targets are shown in Fig. 5 to Fig. 8.
4
10
x 10
9.5 y(m)
−7
9
8.5
Fig. 5(a) shows the three real target trajectories. While Fig. 5(a)-(c) shows the measurements with false alarms detected by T/R site, receiver 1 and receiver2, during one Monte Carlo run.
8
8
8.5
4
10
y(m)
9.5
10 4
x 10
x 10
(d) Measurements with false alarms detected by receiver 2. Fig. 5. Target real positions and corrupted positions detected by 3 receivers over 20 radar scans.
9.5
9
8.5
8
9 x(m)
8
8.5
9 x(m)
9.5
10
Fig. 6 shows that logic-based track initiation method using just one receiver not only can hardly initiate all true targets correctly, but also initiates lots of false tracks. It is easy to understand that false measurements cause the false tracks. While the disability of initiating tracks of true targets are due to both the large measuring deviation and the existence of false measurements.
4
4
x 10
10
(a) Real positions of targets without error.
x 10
4
x 10
9.5 y(m)
10
y(m)
9.5
9
8.5
8.5
8
9
8
8
8.5
9 x(m)
9.5
8
8.5
9 x(m)
9.5
10 4
x 10
10
(a) T/R.
4
x 10
4
(b) Measurements with false alarms detected by T/R.
10
x 10
4
10
x 10
9.5 y(m)
y(m)
9.5
9
9
8.5 8.5
8 8
8
8.5
9 x(m)
9.5
10
8
8.5
9 x(m)
4
x 10
(c) Measurements with false alarms detected by receiver 1.
(b) Receiver 1.
9.5
10 4
x 10
4
10
x 10
y(m)
9.5
9
8.5
8
8
8.5
9 x(m)
9.5
10 4
x 10
(c) Receiver 2.
Fig. 6. Result of logic based track initiation applied on measurements from 3 receivers respectively. 4
x 10
10
y(m)
9.5
9
8.5
8
8
8.5
9 x(m)
9.5
Fig. 7 gives the position measurements after the same source association and position fusion procedure. It is clear that many false alarms have been eliminated, and the precision of target position has a rather great promotion when compared with single receiver. Fig. 8 reveals that the new method forms all true targets correctly, and no false tracks is produced at the same time. Of course, the good performance of the proposed track initiation method relies greatly on the same source association technique, measurement fusion and velocity measurement extraction procedure. Furthermore, it should be noted that the velocity measurement extraction could make the track state initiation just by one measurement, and so the two points differential state initiation procedure is avoided. The two points differential state initiation procedure is a necessary part for typical track initiation methods, and during which very probably lots of false tracks will be produced because of the large uncertain gate. The comparison between Fig.6 and Fig.8 distinctly shows the validity of the new centralized track initiation method for multistatic radar system. VI. CONCLUSIONS Aiming at increasing correct track initiation probability and promoting target tracking performance in large measuring deviation and high false alarm rate environment, an efficient centralized track initiation method for multistatic radar system is proposed in this paper. The devised method mainly consists of a same source association technique, a position fusion process, a velocity measurement extraction procedure and a track initiation mechanism. The simulations show that the new method has rather good performance when compared with the existing methods.
References
10 4
x 10
[1]
Fig. 7. Position fusion measurements after same source association. [2]
4
10
x 10
[3]
9.5 y(m)
[4]
9 [5]
8.5
8
8
8.5
9 x(m)
9.5
10
[6]
4
x 10
[7]
[8] Fig. 8. Result of multistatic radar track initiation method applied on fused measurements.
Matteo Sedehi, Pierfrancesco Lombardo, Alfonso Farina, “A modified M/N logic for track initiation of low observable targets using amplitude information,” IEEE Radar symposium, 2006,pp. 1-4. ZHANG Yan-hang, SU Xiao-hong and MA Pei-jun, “A Hough transform track initiation algorithm based on subtractive clustering,” JOURNAL OF HARBIN INSTITUTE OF TECHNOLOGY, 2010, 42(2),pp. 264-267. Y. Bar-Shalom, T. E. Fortmann, “Tracking and Data Association,” New York: Academic, 1988. D. J. Hunt, L. W. Nolte, and A. R. Reibman, “Hough transform and signal detection theory performance for images with additive noise,” Computer Vision, Graphics Image Processing, 1990, 52(3), pp. 386– 401. B. D. Carlson, E. D. Evans, and S. L. Wilson, “Search radar detection and track with the Hough transform, part I: System concept,” IEEE Transactions on Aerospace and Electronic System, 1994, 30(1), pp. 102– 108. Carlson B D, Evans E D, Wilson S L, “Search radar detection and track with the Hough Transform. III. Detection performance with binary integration,” IEEE Transactions on Aerospace and Electronic Systems, 1994, 30(1), pp. 116-125. Ji Chen, Leung H, Lo T, et al, “A modified probabilistic data association filter in a real clutter environment,” IEEE Transactions on Aerospace and Electronic Systems, 1996, 32(1), pp. 300-313. Zhijian Hu, Henry Leung, “Statistical Performance Analysis of Track Initiation Techniques,” IEEE Transactions on Signal Processing, 1997, 45(2), pp. 445-456.
[9] [10]
[11]
[12]
[13]
[14]
Bar-Shalom Y and Tse E, “Tracking in a cluttered environment with probabilistic data association,” Automatica, 1975, 11(9), pp. 451-460. Ahmeda. S. S, Keche. M, Harrison. I, Woolfson. M. S, “Adaptive joint probabilistic data association algorithm for tracking multiple targets in cluttered environment,” IEE Proceedings of Radar, Sonar and Navigation, 1997, 144(6), pp. 309-314. A. k. Roy, S. K. Rao, “Multi-track association and fusion,” 2011 International Conference on Communications and Signal Processing, 2011, pp. 131-135. HE You, SONG Qiang, XIONG Wei, “Track Alignment-Correlation Technique Based on Phase Correlation,” ACTA ELECTRONICA SINICA, 2010, 38(12), pp. 2718-2723. GAO Xin-bo, LI Jie, “A Feature-Weighted Fuzzy Clustering Algorithm for Track Association in Sensor Network,” ACTA ELECTRONICA SINICA, 2007, 35(12A), pp. 148-151. Zhou Yan, Li Jian-Xun, “Distributed robust fusion of quantized track for target tracking in wireless sensor networks,” Systems Engineering and Electronics, 2011, 33(3), pp. 643-649.
[15] HUIMIN CHEN, THIAGALINGAM KIRUBARAJAN, YAAKOV BAR-SHALOM, “Heterogeneous track-to-track fusion,” IEEE Transactions on Aerospace and Electronic Systems, 2003, 39(2), pp. 386-400. [16] Ting Yuan, Bar-Shalom Y., Xin Tian, “Heterogeneous track-to-track fusion,” 2011 Proceedings of the 14th International Conference on Information Fusion, 2011, pp. 1-8. [17] S. J. Julier and J. K. Uhlmann, “A New Extension of the Kalman Filter to Nonlinear Systems,” The 11th International Symposium on Aerospace/Defense Sensing, Simulation and Controls, 1997. [18] M. kosaka, S. Miyamoto and H. Ihara, “A Track Correlation Algorithm for Multisensor Integration,” The IEEE/AIAA 5th Digital Avionics Systems Conference, 1983, pp. 1-8. [19] S. J. Julier, J. K. Uhlmann and H. F. Durrant, “A New Approach for filtering nonlinear systems,” Proceedings of American Control Conference, 1995, 3, pp. 1628-1632.
I.
INTRODUCTION
Track initiation is the procedure by which a new target entering radar coverage is acquired by the tracking system [12]. This means there generally will be a far range between target and radar. Moreover, to detect modern military aircrafts which probably have low radar cross section (RCS), radar system need to use a low detection threshold. Thus, for track initiation procedure in such situations, target Cartesian localization will have a large measuring deviation and the false alarm rate may always be high. During the past several decades, a great deal of research has been devoted to track initiation using monostatic radar, and two main series, sequential methods and batch methods, have been proposed. But unfortunately these methods, such as heuristic rule method, logic-based method [3], Hough Transform method [4-6] and modified Hough Transform method [7], have a rather low correct track probability when dealing with targets corrupted by false alarms and/or large measuring deviation [8]. Consequently, the track maintenance methods (such as probability data association (PDA) [9] and joint probability data association (JPDA) [10]) that could perfectly treat with false alarms, is greatly restricted. This is because the exertion of the performance of track maintenance methods has to depend on the cooperating with a track initiation method with high correct track initiation probability. Then, the scheme using track association [11-13] and fusion [14-16] in a data processing center will make no sense any more in the above situations. In short, the poor performance of the existing track initiation methods is just the bottleneck of targets tracking issue in low signal-to-noise ratio and/or large measuring deviation environment.
Distinctly unlike traditional track initiation methods which use only one monostatic radar, method in this paper tries to use a T/R-R2 multistatic radar system and a centralized data processing scheme. By the authors’ knowledge, no track initiation method using multistatic radar system or centralized data processing scheme has been studied. During the devised method in this paper, at every scan, measurements from all the receivers of multistatic radar are processed by four steps. Firstly, a same source association approach, which could distinguish real targets measurements from false alarms just by the statistical character differences between them, is adopted. The measurements being recognized by same source association approach as from one same real target will make up of one association group, and the ones being recognized as from false alarms will be abandoned. Secondly, the position element of measurements from every association group will be processed by a fusion technique, the aim of which is to gain high precision target position measurement. Thirdly, a target velocity measurement extracting technique is implemented on Doppler frequency shift element of measurements from every association group. Finally, the position & velocity measurements are used to initiate tracks by unscented Kalman filter [17] and nearest neighbor (NN) logic [18]. At the end of this paper, a contrast of the performance between the new method and logic-based track initiation method (logic-based method is one excellent among the existing methods [8]) is applied. The rest of the paper is organized as follows. In Section 2, the basic working principle of multistatic radar is simply reviewed. Section 3 is devoted to the same source association technique, measurement fusion approach and velocity measurement extraction method. In Section 4, the logic of data association and track management is detailed. Simulation results are shown in Section 5. Section 6 gives conclusions of this work. II.
BASIC WORKING PRINCIPLE OF MULTISTATIC RADAR
For brevity, and without loss of generality, a 2-dimension T/R-R2 multistatic radar system is applied in this paper. The system configuration is shown in Fig. 1. The basic working principle of T/R-R2 multistatic radar at the kth scan is as follows: first T/R site transmits a signal and records the time ts k , then T/R, receiver 1 and 2 extract measurements
each scan is assumed to have Poisson distribution; and the position of false alarms is assumed to uniformly distribute in the interesting area.
Y/km
receiver 2
( x2 , y2 )
a
2k
Let ri0k ( i = 1," I k ) , r j1k ( j = 1," J k ) and rn2 k ( n = 1," N k ) represents the No. i , j and n estimated target position for T/R site, receiver 1 and 2 respectively. Because of' the nonlinear relationship between measurement ⎡⎣tr mk , α mk ⎤⎦ and target position r , unscented transform (UT) [19] is applied here to estimate the means and covariance matrixes of the distances among ri 0k , r j1k , and rn2k according to (1) and (2).
target
a
1k
receiver 1
a
( x1 , y1 )
0k
Let X/km
T/R site (0, 0)
Fig. 1. Sketch map of working principle of multistatic radar system '
⎡⎣tr 0 k , α 0 k , f d0 k ⎤⎦ (signal arrival time, local azimuth, and ' ⎡⎣tr 1k , α 1k , f d1k ⎤⎦ Doppler frequency shift), and 2k 2k 2k ' ⎡⎣tr , α , f d ⎤⎦ from the three reflecting or scattering signals respectively. '
Let r k = ⎡⎣ x k，y k ⎤⎦ represents the position vector of target ' at scan k , Rm = [ xm，ym ] ( m = 0,1, 2 ) represents the location vector of receiver m ( m = 0 represents T/R site), and c denotes the speed of light. Then for T/R site and each receiver, the local target position can be computed by:
r
r
mk
(
= ( tr
III.
0k
mk
( tr =
0k
0k − ts k ) c ⎡ cos (α ) ⎤ ⎥ ⋅⎢ 2 ⎢sin (α 0 k ) ⎥ ⎣ ⎦
− ts ) ⋅ c − r k
0k
)
⎡cos (α mk ) ⎤ ⎥ + Rm ⋅⎢ ⎢sin (α mk ) ⎥ ⎣ ⎦
(1)
⎧uij01k = E ⎡ ri0 k − r1j k ⎤ ⎣ ⎦ ⎪ ⎨ 01k 0k 1k 01k 0k 1k 01k ⎪Qij = E ⎡⎣ ri − rj ⎤⎦ − uij ⋅ ⎡⎣ ri − r j ⎤⎦ − uij ⎩
)}
(3)
⎧ uin02 k = E ⎡ ri 0 k − rn2 k ⎤ ⎣ ⎦ ⎪ ⎨ 02 k 0k 2k 02 k 0k 2k 02 k ⎪Qin = E ⎡⎣ ri − rn ⎤⎦ − uin ⋅ ⎡⎣ ri − rn ⎤⎦ − uin ⎩
)}
(4)
⎧unj21k = E ⎡ rn2 k − rj1k ⎤ ⎣ ⎦ ⎪ ⎨ 21k 2k 1k 21k 2k 1k 21k ⎪Qnj = E ⎡⎣ rn − rj ⎤⎦ − unj ⋅ ⎡⎣ rn − r j ⎤⎦ − unj ⎩
)}
(5)
{(
)(
{(
)(
{(
)(
(2)
(6)
din02 k ( uin02 k ) ⋅ ( Qin02 k ) ⋅ uin02 k
(7)
d nj21k ( unj21k ) ⋅ ( Qnj21k ) ⋅ unj21k
(8)
'
−1
−1
−1
'
At scan k, assume T/R site, receiver 1 and 2 extracts Ik, Jk and Nk measurements respectively. In fact, some of the measurements originate from real targets, and others originate from false alarms which seriously affect the performance of track initiation. It is intuitive that measurements originating from the same real target are just corrupted by measuring errors. Thus the mahalanobis distance among the local positions derived from these measurements could be used as a statistic to test whether they come from one same real target. Assume the estimation deviation of the reflecting or scattering signal arrival time and azimuth at every site are given the same as Gaussian distribution N ( 0, δ tr2 ) (s) and N ( 0, δα2 ) (degree) for brevity. Both the number and position of false alarms are assumed to be random and statistically independent among different receivers from scan to scan. In particular, the number of false alarms from every receiver in
'
dij01k ( uij01k ) ⋅ ( Qij01k ) ⋅ uij01k '
SAME SOURCE ASSOCIATION AND MEASUREMENT FUSION TECHNIQUE
A. Same source association
'
Then, the square mahalanobis distance matrixes D 01k = ⎡⎣ dij01k ⎤⎦ , D 02 k = ⎡⎣ din02 k ⎤⎦ and D 21k = ⎡⎣ d nj21k ⎤⎦ Ik × J k I k × Nk Nk × J k could be given by '
m = 1, 2
'
'
'
If ⎡⎣tri 0 k , α i0 k , f di0 k ⎤⎦ , ⎡⎣trj1k , α 1j k , f dj1k ⎤⎦ and ⎡⎣trn2 k , α n2 k , f dn2 k ⎤⎦ originate from one same real target, according to a predetermined probability pa and the chi squared distribution, a threshold ε could be used to make sure ⎧ p {dij01k < ε } = pa ⎪ ⎪ 02 k ⎨ p {din < ε } = pa ⎪ 21k ⎪⎩ p {d nj < ε } = pa
(9)
The scheme of same source association is as follows. z
Step 1: Find the association groups among which all the three square mahalanobis distances are smaller than ε .
The detail of one round selection is shown below. First initialize i = j = n = 0 and d min = 3ε . And for every combination of i , j and n , if (10) is satisfied, then update i , j , n and d min as (11)
⎧⎪ dij01k < ε ; din02 k < ε ; d nj21k < ε ; ⎨ 01k 02 k 21k ⎪⎩( dij + din + d nj ) < d min
(10)
⎧⎪ i = i; j = j; n = n; ⎨ 01k 02 k 21k ⎪⎩ d min = dij + din + d nj
(11)
After the round of updating, the final d min is compared with 3ε . And if d min is smaller than 3ε , an association group consisting of ⎡⎣tri0 k , α i0 k , f di0 k ⎤⎦ , ⎡⎣trj1k , α 1j k , f di1k ⎤⎦ , and ⎡⎣trn2 k , α n2 k , f di2 k ⎤⎦ is derived. Then clear the effect of this association group ( just let d ij01k = dij01k = d in02 k = din02 k = d nj21k = d nj21k = 2ε ( i = 1, 2,..., I k , j = 1, 2,..., J k , n = 1, 2,..., N k )), and repeat the above operations in step 1 to find the next association group until after one round of updates, the value of d min is not smaller than 3ε . z
Step 2: Find the association groups among which two of the distances are smaller than ε .
The detail is similar to step 1, but the updating of d min is processed as (12) d min = dij01k + din02 k + d nj21k − max ⎡⎣ dij01k , din02 k , d nj21k ⎤⎦
(12)
(
B. Position measurement fusion For every association group, the target Cartesian position r k could be derived with high precision by the weighted sum of the Cartesian position of the three measurements. The Cartesian position of the measurement could be computed by UT method according to (1) and (2). And the weights are proportional to the fisher information matrix of the measurements. Without loss of generality, x1 and y2 are chosen to be 100km, x2 and y1 are chosen to be 0, and the square area in the first quadrant is considered for an example. The average errors for 500 times Monte Carlo simulations of the target position derived from two methods (method 1: the way that only one receiver T/R site is used; method 2: position fusion method) are shown in Fig. 3. Fig. 3 shows that fusion position measurement has a rather higher precision than position measurement extracted by only T/R site. C. Velocity measurement extraction
And the condition of updating is ⎧ min ⎡ d 01k , d 02 k , d 21k ⎤ < ε in nj ⎦ ⎣ ij ⎪ ⎪ 01k 02 k 21k ⎨ m id ⎣⎡ dij , din , d nj ⎦⎤ < ε ⎪ 01k 02 k 21k 01k 02 k 21k ⎪⎩ d min > dij + din + d nj − max ⎡⎣ dij , din , d nj ⎤⎦
Take δ tr = 1× 10−7 and δ α = 0.5 for an example, Fig. 2 gives the average ghost elimination effect as a function of average number of false alarms λ over 500 times tests.
(13)
)
The association group can be confirmed if the final d min is smaller than 2ε . The selection of association group should be continued until after one round of updates, the value of d min is still 2ε . After step 1 and 2 have been done, the measurements that have not formed any association group will be thought as false alarms. 15 False measurement detected by T/R
After forming the fusion position measurement r k of every association group, the three Doppler frequency shifts measured by T/R site, receiver 1 and receiver 2 can be used to ' estimate the 2-dimension velocity r k = ⎡⎣ x k，y k ⎤⎦ of target by ' ⎧ ⎡ r mk ⎤ ⎪ f mk = 2 f c ⋅ ⎣ ⎦ ⋅ r k ⎪ d c r mk ⎪ ⎨ ' mk 0k ' ⎪ mk f c ⎛ ⎡⎣ r ⎤⎦ ⎡⎣ r − R m ⎤⎦ ⎜ + mk ⎪ fd = ⋅ c ⎜ r 0k r − Rm ⎪ ⎝ ⎩
m=0 ⎞ ⎟ ⋅ r k ⎟ ⎠
(14) m = 1, 2
Where f c is the radar signal carrier frequency. After target velocity being derived, then a' complete target Cartesian state vector Z ( k ) = ⎡⎣ x k , y k , x k , y k ⎤⎦ can be formed, and can be used to perform the subsequent target tracking process.
True measurement detected by T/R False association group
10
1500 Average error (m)
Number
True association group
5
0
0
2
4
6
8
10
12
λ Fig. 2. The false alarms elimination effect of same source association.
1000 500 0 100
100
50 Y(km)
50 0
0
X(km)
(a) The way that only T/R site is used.
initiated. For all candidate tracks, once the track score equals 0, the track will be degraded as a dead track, which means the track is terminated. Only the track with length no less than L could be considered originating from an existing target.
40
V.
20
100
50 Y(km)
50 0
0
X(km)
(b) Position fusion method. Fig. 3. Contrast of average errors of measurement between T/R and fusion method.
IV.
DATA ASSOCIATION AND TRACK MANAGEMENT MECHANISM
The data association and track management mechanism using the new track initiation method are detailed as below. First, tracks are classified into 3 types: candidate tracks, dead tracks, and active tracks. And every track has two properties: track score (denotes the quality of a track, and is the actor to decide whether a track should be terminated or not) and track length (denotes how many times of a track being updated by the measurements, and is the actor to decide whether a track should be initiated or not).
pc =
z
z
Step 2: If one candidate track is updated successfully by measurements, the track score and length should plus 1; if not, then only the track score should minus 1. Step 3: Every Z ( k ) that updates none tracks should initialize a new candidate track as that at scan 1. Step 4: For candidate tracks, once the track length equals to a predetermined value L, the track should be set as an active track, which means the track is
(18)
Take the predetermined track length value L=4 and
δ tr = 1× 10−7 . Then performance (pc) of the two methods:
method developed in this paper and logic based method (applied on measurements from T/R), which is as a function of the average number of false alarms λ and standard azimuth deviation δ a of angle measurement, is shown as Fig. 4. 0.4 0.2 0 0
λ 5 10 0.2
0.6 0.4 δ (degree)
0.8
1
a
(a) Typical logic based track initiation method.
1
Pc
2) Scan k(k>1) z Step 1: Use Z ( k ) and R p ( k ) that has not updated any active tracks to update the existing candidate tracks states by NN logic (data association) and Kalman filter (state estimation). Then use the transition model to get state prediction and prediction covariance matrix.
nc M
Here M is the number of real targets, and nc is the number of real target tracks being initiated correctly.
1) Scan 1 ' Every Z ( k ) = ⎡⎣ x k , y k , x k , y k ⎤⎦ and its covariance matrix R p ( k ) extracted from association group are used to initialize a candidate track, whose score is a predetermined integer S and length is 1. Then Kalman filter is initialized by the state of the candidate track, and used to get the state prediction and prediction covariance matrix at the next scan.
z
SIMULATIONS AND RESULTS
To evaluate the performance of the new track initiation methods, correct track initiation probability pc is defined as follows:
0 100
Pc
Average error (m)
60
0.5 0 0 5
λ 10
0.2
0.4
0.6
0.8
δa (degree)
(b) The new track initiation method using multistatic radar. Fig. 4. Performance contrast between two methods.
The comparison between Fig. 4(a) and Fig. 4(b) intuitively shows that the new track initiation method significantly
outperforms typical logic based method. The correct track initiation probability of logic based method is always rather low, even when the parameters are perfect. While the new method exhibits rather good performance. Assume δ tr = 1× 10 , δ a = 0.6 and λ = 12 , then results of one Monte Carlo run of the two methods about a simple scenario consisting of three targets are shown in Fig. 5 to Fig. 8.
4
10
x 10
9.5 y(m)
−7
9
8.5
Fig. 5(a) shows the three real target trajectories. While Fig. 5(a)-(c) shows the measurements with false alarms detected by T/R site, receiver 1 and receiver2, during one Monte Carlo run.
8
8
8.5
4
10
y(m)
9.5
10 4
x 10
x 10
(d) Measurements with false alarms detected by receiver 2. Fig. 5. Target real positions and corrupted positions detected by 3 receivers over 20 radar scans.
9.5
9
8.5
8
9 x(m)
8
8.5
9 x(m)
9.5
10
Fig. 6 shows that logic-based track initiation method using just one receiver not only can hardly initiate all true targets correctly, but also initiates lots of false tracks. It is easy to understand that false measurements cause the false tracks. While the disability of initiating tracks of true targets are due to both the large measuring deviation and the existence of false measurements.
4
4
x 10
10
(a) Real positions of targets without error.
x 10
4
x 10
9.5 y(m)
10
y(m)
9.5
9
8.5
8.5
8
9
8
8
8.5
9 x(m)
9.5
8
8.5
9 x(m)
9.5
10 4
x 10
10
(a) T/R.
4
x 10
4
(b) Measurements with false alarms detected by T/R.
10
x 10
4
10
x 10
9.5 y(m)
y(m)
9.5
9
9
8.5 8.5
8 8
8
8.5
9 x(m)
9.5
10
8
8.5
9 x(m)
4
x 10
(c) Measurements with false alarms detected by receiver 1.
(b) Receiver 1.
9.5
10 4
x 10
4
10
x 10
y(m)
9.5
9
8.5
8
8
8.5
9 x(m)
9.5
10 4
x 10
(c) Receiver 2.
Fig. 6. Result of logic based track initiation applied on measurements from 3 receivers respectively. 4
x 10
10
y(m)
9.5
9
8.5
8
8
8.5
9 x(m)
9.5
Fig. 7 gives the position measurements after the same source association and position fusion procedure. It is clear that many false alarms have been eliminated, and the precision of target position has a rather great promotion when compared with single receiver. Fig. 8 reveals that the new method forms all true targets correctly, and no false tracks is produced at the same time. Of course, the good performance of the proposed track initiation method relies greatly on the same source association technique, measurement fusion and velocity measurement extraction procedure. Furthermore, it should be noted that the velocity measurement extraction could make the track state initiation just by one measurement, and so the two points differential state initiation procedure is avoided. The two points differential state initiation procedure is a necessary part for typical track initiation methods, and during which very probably lots of false tracks will be produced because of the large uncertain gate. The comparison between Fig.6 and Fig.8 distinctly shows the validity of the new centralized track initiation method for multistatic radar system. VI. CONCLUSIONS Aiming at increasing correct track initiation probability and promoting target tracking performance in large measuring deviation and high false alarm rate environment, an efficient centralized track initiation method for multistatic radar system is proposed in this paper. The devised method mainly consists of a same source association technique, a position fusion process, a velocity measurement extraction procedure and a track initiation mechanism. The simulations show that the new method has rather good performance when compared with the existing methods.
References
10 4
x 10
[1]
Fig. 7. Position fusion measurements after same source association. [2]
4
10
x 10
[3]
9.5 y(m)
[4]
9 [5]
8.5
8
8
8.5
9 x(m)
9.5
10
[6]
4
x 10
[7]
[8] Fig. 8. Result of multistatic radar track initiation method applied on fused measurements.
Matteo Sedehi, Pierfrancesco Lombardo, Alfonso Farina, “A modified M/N logic for track initiation of low observable targets using amplitude information,” IEEE Radar symposium, 2006,pp. 1-4. ZHANG Yan-hang, SU Xiao-hong and MA Pei-jun, “A Hough transform track initiation algorithm based on subtractive clustering,” JOURNAL OF HARBIN INSTITUTE OF TECHNOLOGY, 2010, 42(2),pp. 264-267. Y. Bar-Shalom, T. E. Fortmann, “Tracking and Data Association,” New York: Academic, 1988. D. J. Hunt, L. W. Nolte, and A. R. Reibman, “Hough transform and signal detection theory performance for images with additive noise,” Computer Vision, Graphics Image Processing, 1990, 52(3), pp. 386– 401. B. D. Carlson, E. D. Evans, and S. L. Wilson, “Search radar detection and track with the Hough transform, part I: System concept,” IEEE Transactions on Aerospace and Electronic System, 1994, 30(1), pp. 102– 108. Carlson B D, Evans E D, Wilson S L, “Search radar detection and track with the Hough Transform. III. Detection performance with binary integration,” IEEE Transactions on Aerospace and Electronic Systems, 1994, 30(1), pp. 116-125. Ji Chen, Leung H, Lo T, et al, “A modified probabilistic data association filter in a real clutter environment,” IEEE Transactions on Aerospace and Electronic Systems, 1996, 32(1), pp. 300-313. Zhijian Hu, Henry Leung, “Statistical Performance Analysis of Track Initiation Techniques,” IEEE Transactions on Signal Processing, 1997, 45(2), pp. 445-456.
[9] [10]
[11]
[12]
[13]
[14]
Bar-Shalom Y and Tse E, “Tracking in a cluttered environment with probabilistic data association,” Automatica, 1975, 11(9), pp. 451-460. Ahmeda. S. S, Keche. M, Harrison. I, Woolfson. M. S, “Adaptive joint probabilistic data association algorithm for tracking multiple targets in cluttered environment,” IEE Proceedings of Radar, Sonar and Navigation, 1997, 144(6), pp. 309-314. A. k. Roy, S. K. Rao, “Multi-track association and fusion,” 2011 International Conference on Communications and Signal Processing, 2011, pp. 131-135. HE You, SONG Qiang, XIONG Wei, “Track Alignment-Correlation Technique Based on Phase Correlation,” ACTA ELECTRONICA SINICA, 2010, 38(12), pp. 2718-2723. GAO Xin-bo, LI Jie, “A Feature-Weighted Fuzzy Clustering Algorithm for Track Association in Sensor Network,” ACTA ELECTRONICA SINICA, 2007, 35(12A), pp. 148-151. Zhou Yan, Li Jian-Xun, “Distributed robust fusion of quantized track for target tracking in wireless sensor networks,” Systems Engineering and Electronics, 2011, 33(3), pp. 643-649.
[15] HUIMIN CHEN, THIAGALINGAM KIRUBARAJAN, YAAKOV BAR-SHALOM, “Heterogeneous track-to-track fusion,” IEEE Transactions on Aerospace and Electronic Systems, 2003, 39(2), pp. 386-400. [16] Ting Yuan, Bar-Shalom Y., Xin Tian, “Heterogeneous track-to-track fusion,” 2011 Proceedings of the 14th International Conference on Information Fusion, 2011, pp. 1-8. [17] S. J. Julier and J. K. Uhlmann, “A New Extension of the Kalman Filter to Nonlinear Systems,” The 11th International Symposium on Aerospace/Defense Sensing, Simulation and Controls, 1997. [18] M. kosaka, S. Miyamoto and H. Ihara, “A Track Correlation Algorithm for Multisensor Integration,” The IEEE/AIAA 5th Digital Avionics Systems Conference, 1983, pp. 1-8. [19] S. J. Julier, J. K. Uhlmann and H. F. Durrant, “A New Approach for filtering nonlinear systems,” Proceedings of American Control Conference, 1995, 3, pp. 1628-1632.
