Algorithm for multiple targets localization and dataassociation in distributed radar networks

A. SamokhinDepartment of Applied Mathematics

Moscow Institute of Radio Engineering,Electronics, and Automatics (MIREA), Russia

absamokhin@yandex.ru

I. Ivashko, A. YarovoyMicrowave Sensing, Systems and Signals (MS3)Delft University of Technology, The Netherlands

I.Ivashko@tudelft.nl, A.Yarovoy@tudelft.nl

AbstractThis paper presents the algorithm of multiple targetlocalization from the time delay in the network with omnidi-rectional radars. All possible combinations from the set of themeasured time delay values in each of the radar are processedwith this algorithm. The combination that does not correspondto the real target position is considered as a ghost target. Theproposed localization algorithm consists of the three stages, wherethe filtering of the incorrect associations, deghosting, is made.

Index Termsradar networks, localization, ghost targets, dataassociation.

I. INTRODUCTIONRadar networks are considered as a promising alternative for

the number of civil and military applications. Such radars givepossibility to cover wide areas and to estimate 3D position ofthe target with exploitation of the non-directional antennas.

Data association is an inherent part of the target localizationin the radar network []. The localization of the multiple targetshas to be performed from the data, measured locally in each ofthe radar. The incorrect data association of these measurementsleads to the false targets - ghosts. Geometrical interpretationof this type of the target is shown in the Fig. 1. The presenceof the ghost targets increase the probability of false alarm,respectively. Therefore, to ensure the efficient performanceof the multisite radar system, the correct data association,deghosting, has to be provided. Deghosting is also considered

R1

R2

R3

Ghost target T1

T2

Fig. 1: Ghost target phenomena

as an essential part of the multiple targets tracking, where the

prior information of the target state vector from the previousmoment of time is used [1]. This reduces potential numberof ghosts substantially, compared to the deghosting in thesingle moment of time. A number of works are dedicatedto the deghosting algorithms for target localization, basedon the bearing only information both for tracking and forlocalization only purposes [2], [3], [4], [5]. Compared to therange-based localization, in the bearing-only localization theobservation area of the single radar is limited with the certainangle, that corresponds to the antenna beamwidth. This allowsto decrease the number of ghost targets substantially. Paper[6] presents deghosting algorithm for automotive applications,where targets positions are estimated from the time delayinformation. In this scenario, the observation sector is limitedto the 180o with the cars bumper.

Some of the works address the problem of accurate targetlocalization, based on the range information, in the network ofmultiple sensors. There are two main issues discussed: how touse effectively the measurements from all the sensors, whenthe number of unknowns is much less than the number offunctions, and when the measurements are corrupted by noise.The first issue is usually solved with the least square (LSE)methods [7]. And the second issue is often addressed with thealgorithms, based on the projection onto convex sets (POCS)[8]. Both problems are solved, based on the assumption thatdata association has been made correctly in the previous stage.

This paper focuses on the data association problem in thesingle moment of time, e.g. without a priori knowledge of thetargets state vector. The proposed algorithm deals with thedeghosting problem in the monostatic radar network, wherethe target position is evaluated from the set of time delayvalues, measured locally in each of the omnidirectional radar.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The network of N monostatic onmidirectional radars withcoordinates: l1 = [x1, y1, z1], l2 = [x2, y2, z2], l3 =[x3, y3, z3], ln = [xN , yN , zN ] is considered. The measuredtime delays from each of the radar are forwarded to the centralprocessing unit (CPU), where the target position is evaluated.It is assumed, that all of the targets are detected (Pd = 1).Therefore, the number of targets in the scene m corresponds

to the number of detections m in each of the radar. Targetsare considered as point scatterers.

The measurements from each individual radar forms thevector of size m 1:

R(1)1

R(1)2.

.

.

R(1)m

R(2)1

R(2)2.

.

.

R(2)m

R(3)1

R(3)2.

.

.

R(3)m

R(N)1

R(N)2.

.

.

R(N)m

(1)

where R(n)m is the measured distance from the nth radar to thetarget m; n = 1, N .

In the noise free measurement model with no signal attenua-tion, the accuracy of estimation target range is limited with theradar range resolution R [9]. Consequently the real targetrange R(n)m lies in the interval:

R(n)m R

2 R(n)m R

(n)m +

R

2. (2)

When there is more than one target in the scene (m > 1),the correct association of the set of the measurements (1) hasto be made in order to find positions of the real targets. In thenoiseless measurement model the target position correspondsto the point of spheres intersection with centers l1, l2, l3,ln. When the measurements are corrupted with the noise, theposition of the real target will be in the volume V (R) ofspheres intersection with the boundaries:

(R(1)m R2)2 (R

(1)m )2 (R

(1)m +R2)2;

(R(2)m R2)2 (R

(2)m )2 (R

(2)m +R2)2;

(R(3)m R2)2 (R

(3)m )2 (R

(3)m +R2)2;

(R(4)m R2)2 (R

(4)m )2 (R

(4)m +R2)2;

(3)

The ghost target is defined by the combination of ranges thatcorresponds to the intersection of spheres from different targets(Fig.1).

In the following sections we present the three-stage deghost-ing algorithm. The first stage comprises target localization viatrilateration. The second stage is based on the geometricalanalysis of the estimated targets positions. In the third stagethe analytical analysis of the localized targets is made.

The proposed deghosting procedure is based on the filteringout the incorrect associations. In this paper we define deghost-ing as filtering procedure.

III. STAGE 1 - TARGET LOCALIZATIONIn the radar network with four radars and noiseless mea-

surement model with R = 0 the system of inequalities (3)becomes the system of equations:

(x x1)2 + (y y1)2 + (z z1)2 = (R(1))2;

(x x2)2 + (y y2)2 + (z z2)2 = (R(2))2;

(x x3)2 + (y y3)2 + (z z3)2 = (R(3))2;

(x x4)2 + (y y4)2 + (z z4)2 = (R(4))2;

(4)

where xn, yn, zn are coordinates of the nth radar; s = [x, y, z]are target coordinates.

Subtracting the second equation in from the first, the thirdequation from the first and leaving the third equation as it is,well get system of three equations with three unknowns x, yand z:

(2x x1 x2)(x2 x1) + (2y y1 y2)(y2 y1)+

+ (2z z1 z2)(z2 z1) = (R(1))2 (R(2))2;

(2x x1 x3)(x3 x1) + (2y y1 y3)(y3 y1)+

+ (2z z1 z3)(z3 z1) = (R(1))2 (R(3))2;

(x x3)2 + (y y3)

2 + (z z3)2 = (R(3))2;

(5)One on the equations in (5) is reduced to the quadratic form:

x = a1y + a2z + b1;

y = a3z + b2;

Az2 +Bz + C = 0;

(6)

with the following coefficients:

A = (a1a3 + a2)2 + a23 + 1;

B = 2(a1a3 + a2)(a1b2 + b1) 2x3(a1a3 + a2)+

+ a3b2 2y3a3 2z3;

C = (a1b2 + b1)2 x3(2a1b2 + 2b1 x3)+

+ b22 + y3(y3 2b2) + z23 (R

(3))2;

where

a1 = y21x21

; a2 = z21x21

; a3 = x31a2 +z31x31a1 +y31

;

b1 =y21y212x21

+z21z212x21

+(R(1))2 (R(2))2

2x21+x21

2;

b3 = 1

2(x31a1 +y31)

[2x31b1 x31x31

y31y31 z31z31 (R(1))2 + (R(3))2

];

where xnm = xn xm; xnm = xn + xm.The quadratic equation from (6) is solved via discriminant.

Afterwards, distances from the estimated target position(s) s1(s2) to the N 3 radars are calculated. These distances haveto match with the measured distances, e.g. R(4) . . . R(N), withthe certain accuracy R (algorithm 1). In case of noiselessmeasurements, the threshold value is R = R. In thepresence of noise the threshold has to be less tight in orderto minimize the number of real targets, that can be mitigated.So it is defined as: R = R + n, where n is the noisestandard deviation. If there is more than one solution, the twomore stages of the deghosting algorithm are applied for theiranalysis. These stages are described in the following sections.

Algorithm 1: Find coordinates of the targetsinput : Combinations of the measured range values from

three radars (1)output: Estimated set of the targets coordinatesfor i 2 to C - number of range combinations do

if D 0 thens1 l4 = R

(N3)s1 ;

s2 l4 = R(N3)s2 ;

ifR(N3)s1 R(N3)m R orR(N3)s2 R(N3)m R thens1 or s2 can be a real target and has to befurther analyzed;

elses1 or s2 is eliminated as a ghost target;

endelse

the ranges R(1), R(2), R(3) are measured to thedifferent targets;

endend

IV. STAGE 2 - DEGHOSTING, BASED ON THE GEOMETRYANALYSIS

This stage of deghosting is based on the analysis of eachcombination of ranges R(1), R(2), R(2), . . . R(N), that hasbeen used for target localization in the previous stage. Thebasic idea of this procedure is to find whenever the N sphereswith radiuses R(1), R(2), R(2), . . . R(N) intersects or not. Ifthey intersects, then intersection point corresponds to the realtarget position.

Not all of the ranges from the given combinationR(1), R(2), R(2), . . . R(N) are analyzed simultaneously. Thisset is subdivided into the combinations of three radars andeach of them is analyzed separately.

Without loss of generality, let us to explain this algorithmwithin the 2D spatial model and three radars in the network.In the noiseless measurement model there will be one pointO of intersection of there circles that correspond to the targetposition; in 3D model - two points. If to account the radarrange resolution, there will be an area of intersection of thespheres. Within the following algorithm we check wheneverthese spheres intersect or not. The model is shown on theFigure 2, where A, C, F are positions of the radars; B andD are points of intersection of circles from radars A and C;O is the point of intersection of circle from radar F with theline BD. If these three circles intersects, then O has to lie onthe line BD.

The task is to find 1 = FD and 2 = FB - distancesfrom radar F to the points of intersection of two spheres fromradars A and C.

Solution:BO = OD; AB = R(1); BC = R(2); R(1) and R(2)are distances from the target to the radars A(1) and C(2)

A (1) C (2)

F (3)

B

HO

D

G

1R2R

12

Fig. 2: Schematic illustration of the target localization

respectively;AC = d12; AF = d13; FC = d23 are distances between theradars;12BO d12 = Sd12,R(1),R(2) ,where Sd12,R(1),R(2) is the area of the triangle ABC, calcu-lated using Herons formula:

Sd12,R(1),R(2) =1

4

4(R(1))2(R(2))2 ((R(1))2 + (R(2))2 d212)

2.

BO =2S

d12,R(1),R(2)

d12. Similarly: FH = GO = 2Sd12,d13,d23d12 ,

where Sd12,d13,d23 is the area of triangle ACF .CH2 = FC2 FH2 = d223 FH

2;OC2 = BC2 BO2 = (R(2))2 BO2;OH = OC CH (CH - from triangle FCH , where FCis known, FH has been found);GF = OH;GD = GO OD = FH BO; FD2 = GF 2 +GD2;FB2 = GF 2 + (FH +BO)2.

1 and 2 are functions of the R(1) and R(2): 1 =f(R(1), R(2), d12, d13, d23); 2 = f(R(1), R(2), d12, d13, d23).Taking into account the radar range resolution, the parameters1 and 2 can take the following values:

1 =

11121314

=

f(R1 +

R2 , R2 +

R2 )

f(R1 R2 , R2

R2 )

f(R1 +R2 , R2

R2 )

f(R1 R2 , R2 +

R2 )

2 =

21222324

=

f(R1 +

R2 , R2 +

R2 )

f(R1 R2 , R2

R2 )

f(R1 +R2 , R2

R2 )

f(R1 R2 , R2 +

R2 )

(7)

Then we find the maximum and the minimum values fromthe vectors 1 and 2, that corresponds to the possiblemaximum and the minimum distances from the third radarF to the points B and D, one of which corresponds to the

real target position:

1max = max(1);2max = max(2);1min = min(1);2min = min(2).

(8)

where 1min (1max); 2min (2max) are the minimum (max-imum) distances from the radar F to the points D and Brespectively.

If the real target is located in the close proximity to thepoints D or B and R(3) is the measured target range from thethird radar F to the target, then at least one of the followinginequalities has to be satisfied:

1min < R(3) + R2 < 1max

1min < R(3) R2 < 1max

2min < R(3) + R2 < 2max

2min < R(3) R2 < 2max

(9)

Graphical interpretation of this condition is shown in the Fig.3, where the real target is located in the close proximity to thepoint D.

A (1)

3R

C (2)

F (3)

23RR +

23RR

min1max1D

Fig. 3: Schematic illustration of condition (9)

The set of inequalities (9) is applied in the followingway. Each of the combinations is defined by the N valuesof measured distances to the target, that corresponds to thenumber of radars in the network. So the condition (9) ischecked for each of C3N =

N(N3)!3! possible combination

of three radars.In case of four radars, the estimated target position is

s = f(R(1), R(2), R(3), R(4)

). If it is a real target, then

condition (9) has to be satisfied for each combination of threefrom the measured ranges:

R(1), R(2), R(3)

R(1), R(3), R(4)

R(2), R(3), R(4)

R(1), R(2), R(4)

(10)

V. STAGE 3 - DEGHOSTING, BASED ON THE ANALYTICALANALYSIS

This stage of the deghosting algorithm is based on theanalysis of the consecutive numbers of the range values,measured in the single radar. We call measurements in eachradar as radar marks.

Consecutive numbers in the radar correspond to the num-bered measurements of the target ranges. Numbering conse-quence is not important here. The main reason of numberingis to make a distinction between successive measurementsin the radar. So each combination of ranges that is definesthe target position is characterized with these consecutivenumbers. For example, for the given combination of rangevalues R(1/3), R(2/1), R(3/5), R(4/m) it means that marks #3,#1, #5 and #m from the first, second, third and fourth radarswere used to localize the target. This consecutive numbering isused to compute qnm, that is the number of times that mark mfrom the nth radar has been used for target localization. The al-gorithm 2 is the third stage of the deghosting procedure, basedon the analytical analysis of the radar marks. Application of

Algorithm 2: Analytical deghosting algorithminput : Estimated targets with consecutive numbers of

the range values from each of the radaroutput: Filtered set of targetsfor j 1 to M - number of estimated target positionsdo

if qnm > 1 thencalculate qn+1,m ;if There are targets left, that are not classified asreal then

Analyze the marks from all of the radars inthe potential ghost targets;if Marks, that is used for localization of thepotential ghost, has not been used forlocalization of any real target then

jth target position is not resolved;else

jth measurement is ghostend

endelse

targets, for which there are no repeating numbersare real

endend

the analytical algorithm is demonstrated with the followingexamples. Table I shows scenario, when consecutive numbersof the measurements from the first radar are used two times:for localization of the second and the eighth targets. The restof the targets, for which radar marks are not repeated, arerecognized as the real ones. Then we analyze the radar marksfrom the second radar, that has been used for the localizationof the second and the eighth targets. It is observed, that mark

#4 in the eight target has been used already for localizationof the fourth target, that was previously defined as the realone. While mark #1 from the second radar has not been usedfor target localization. This means that the second target is thereal one, and the eighth target is the false one (Table II). So,the eighth target is mitigated.

Scenario with remained unresolved measurements is givenin the Table III. These measurements are analyzed additionally.Mark #4 in the first unresolved measurement has been usedfor localization of real target. So, the first unresolved targetis referred as a ghost. Marks #6 and #8 from the first radarhas not been used for localization of the real targets. So, radarmarks of these targets are further analyzed. After such analysisthe remaining measurements are still not resolved.

RadarNo.

1 2 3 4 Real/False

Rad

arm

arks

No

.

1 3 3 3 R2 2 2 2 R3 1 2 4 ?2 2 1 2 R4 4 4 1 R5 6 5 6 R6 7 7 5 R7 5 6 7 R3 4 1 5 ?

TABLE I: Analysis of the measurements from the first radar

RadarNo.

1 2 3 4 Real/False

Rad

arm

arks

No

.

1 3 3 3 R2 2 2 2 R3 1 2 4 R2 2 1 2 R4 4 4 1 R5 6 5 6 R6 7 7 5 R7 5 6 7 R3 4 1 5 F

TABLE II: Analysis of the measurements from the secondradar

RadarNo.

1 2 3 4 Real/False

Rad

arm

arks

No

.

1 1 1 1 R2 2 2 2 R3 3 3 3 R4 4 4 4 R5 5 5 5 R7 7 7 7 R4 7 8 3 ?6 6 6 6 ?6 8 8 8 ?8 6 6 6 ?8 8 8 8 ?

TABLE III: Additional filtration of potential ghosts

VI. CASE STUDYThis section is organized as following. Proposed three-stage

deghosting algorithm is applied to evaluate 3D position of

the multiple targets in the radar network. Target localizationis based on the time delay, measured locally in each of theradars. Four monostatic radars with omnidirectional anten-nas and the following coordinates: l1 = [1000, 1000, 5],l1 = [1000, 1000, 30], l1 = [1000, 1000, 10], l1 =[1000, 1000, 15], are used for target localization.

Figure 4 presents results on the number of ghosts vs radarrange resolution R at the different target number in the sceneassuming noiseless measurement model. R is assumed tobe the same for all radars in the network. Targets has beengenerated randomly in the surveillance volume 100010001000 [m]. The results are averaged over the 1000 MonteCarlo simulations. It is observed, that with decreasing radarrange resolution, number of ghost targets increase. Two timesworse resolution leads to the doubling of the ghost targetsnumber. Moreover, higher number of targets leads to thehigher number of ghosts, as the number of range combinationsincreases. It can be clearly seen that for the target numberm > 6, the number of ghosts increases drastically. Effects ofeach step of the deghosting procedure in the process of thelocalization of ten targets is shown in the Fig. 5a. One cansee that the application of the deghosting algorithms leads tothe subsequent decreasing of the ghosts number.

Better performance of the deghosting algorithms is demon-strated for the surveillance volume 1000 1000 100 [m],where the targets positions have been randomly generated(Fig.5b). In the previous scenario (Fig.5b), the vertical dis-crepancies of the radars and their altitudes are relativelysmall compared to the targets heights of about 1000 m.This gives less spatial diversity in resolving ghost targets.Whereas compared to the target altitudes of about 100 mvertical diversities of the radars are bigger. This allows tomitigate more false targets.

2 4 6 8 100

20

40

60

80

100

R, [m]

Num

ber o

f gho

sts

m=2m=4m=6m=8m=10

Fig. 4: Number of ghosts vs range resolution - noiselessmeasurements (100010001000 [m] - surveillance volume)

Figure 6 reports the results on the system performance ofmultiple targets localization in the presence of noise. Thenoise is modeled as normally distributed random value withzero-mean and variance 2n, N (0, 2n). The presence of thenoise might be interpreted as an effect of multipath signal

2 4 6 8 100

10

20

30

40

50

60

70

80

90

R, [m]

Num

ber o

f gho

sts

Localization onlyGeometrical filtrationGeometrical and analytical filtrations

(a) 1000 1000 1000 [m] - surveillance volume

2 4 6 8 100

10

20

30

40

50

60

R, [m]

Num

ber o

f gho

sts

Localization onlyGeometrical filtrationGeometrical and analytical filtrations

(b) 1000 1000 100 [m] - surveillance volumeFig. 5: Number of ghosts for different filtrations (m = 10 -number of targets)

propagation. Meanwhile it is assumed that this error does notaffect the target detection. The results demonstrate substantialincreasing of the number of ghost targets with increasingvalue of the noise variance both for high and low resolutionmeasurements.

2 4 6 8 100

20

40

60

80

100

120

R, [m]

Num

ber o

f gho

sts

n=0 [m]

n=3 [m]

n=10 [m]

Fig. 6: Number of ghosts at different values of the measure-ment error 2n (m = 10 - number of targets; 1000 1000 1000 [m] - volume of targets positions)

VII. CONCLUSIONSIn this paper the three-stage deghosting algorithm for mul-

tiple targets localization in distributed radar network withomnidirectional antennas has been proposed. It has beenshown, that efficiency of the deghosting procedure depends onthe number of targets, their constellation in the scene, radarrange resolution and noise variance. The more targets has tobe resolved in the scene, the higher radar range resolution isrequired to minimize number of ghosts. In particular, for 3Dlocalization of ten targets with four radars, the range resolutionR of 2m leads to the two times less number of ghosts.

Future work will be directed on the algorithms developmentfor multiple targets localization in the radar network withwide-beam directional antennas.

ACKNOWLEDGMENTThe authors would like to thank Dr. O.A.Krasnov for giving

comments on this paper.

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