Moving scanning emitter tracking by a single observer using time of interception:Observability analysis and algorithm

Yifei ZHANG,Min ZHANG,Fucheng GUO

aInformation Engineering Department,Rocket Force University of Engineering,Xi’an 710025,China

bCollege of Electronic Science and Engineering,National University of Defense Technology,Changsha 410073,China

Moving scanning emitter tracking by a single observer using time of interception:Observability analysis and algorithm

Yifei ZHANGa,Min ZHANGb,*,Fucheng GUOb

aInformation Engineering Department,Rocket Force University of Engineering,Xi’an 710025,China

bCollege of Electronic Science and Engineering,National University of Defense Technology,Changsha 410073,China

Available online 21 April 2017

*Corresponding author.

E-mail address:zhangmin1984@126.com(M.ZHANG).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2017.03.006

1000-9361©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

The target motion analysis(TMA)for a moving scanning emitter with known fixed scan rate by a single observer using the time of interception(TOI)measurements only is investigated in this paper.By transforming the TOI of multiple scan cycles into the direction difference of arrival(DDOA)model,the observability analysis for the TMA problem is performed.Some necessary conditions for uniquely identifying the scanning emitter trajectory are obtained.This paper also proposes a weighted instrumental variable(WIV)estimator for the scanning emitter TMA,which does not require any initial solution guess and is closed-form and computationally attractive.More importantly,simulations show that the proposed algorithm can provide estimation mean square error close to the Cramer-Rao lower bound(CRLB)at moderate noise levels with significantly lower estimation bias than the conventional pseudo-linear least square(PLS)estimator.

©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Cramer-Rao lower bound;

Least squares;

Observability;

Scanning emitter;

Target motion analysis;

Time of interception

1.Introduction

Determining the position and velocity of a moving source by a single or multiple observer(s),which is also referred to as target motion analysis(TMA),is essential for many applications such as radar,sonar,reconnaissance,and wireless networks.1–4For a non-cooperative source,the TMA can be based on the direction of arrival(DOA),frequency of arrival(FOA),time of arrival(TOA),time difference of arrival(TDOA)or their combinations.

The DOA-based TMA has been a classical estimation problem.4–8However,estimating DOA requires multiple receiving channels,which increases the system complexity and leads to high cost.In addition,the direction finding accuracy is vulnerable to the amplitude/phase unbalance between receiving channels.On the other hand,the FOA-based TMA which uses the Doppler effect9,10needs only a single receiver channel.This technique requires relatively high frequency estimation precision and therefore it is more suitable for tracking emitters with simple modulation types and with known carrier frequency,which cannot always be satisfied for non-cooperative source.The TOA-based TMA requires only a single receiver channel as well.However,it is mainly used for tracking emitters with known pulse repeat interval(PRI)pattern.11,12The TDOA-based TMA needs at least two observers and requires accurate time synchronization among obsevers.13,14For the scanning emitter,such as the mechanically scanning radar,which covers the surveillance area in a periodic manner using a narrow beam antenna,15the TMA problem can be resorted to the time of interception(TOI),which is the time when the peak location of the main lobe of the scanning emitter reaches the observer.Unfortunately,the TMA methods mentioned above cannot be directly used for the scanning emitter tracking using TOI measurements.

The origin of the scanning emitter localization technique can be traced back to locating mobile robots from landmark bearings,16–18where the difference of directions to the landmarks are explored to determinate the robot position.By transforming the TOI measurements into the direction difference of arrival(DDOA),the same idea can be used in the considered problem of the scanning emitter passive TMA.To obtain the target position and velocity from the DDOA measurements,the existing estimation methods,15,19–23nevertheless,all assume multiple observers and the static scanning emitter.In this work,the TOI-based tracking of a moving scanning emitter using a single observer is investigated.

By a single observer,the TOI measurement can be obtained at most once in one scan cycle.As a result,measurements should be accumulated over multiple scans for parameter or state estimation.Fewer literatures have been devoted to this topic.The static emitter localization method using a single observer has been studied,24and the performance of the geometric solution15with 3 measurements is compared at different noise levels.24However,this solution cannot be directly used in the TMA for a moving scanning emitter.Therefore,new solutions to such a TMA problem are needed.They would be nontrivial solutions because of the strictly high nonlinearity between TOI and the position and velocity of the emitter.In addition,insights into the least number of measurements required for unique identification of the scanning emitter trajectory are highly desirable,as they are indeed the observability conditions for the TMA problem in consideration.

Derivation of the observability conditions of the TMA problem is intractable due to the nonlinear characteristic of the problem.Two methods,the nonlinear differential equation method25and the linearization-based method,26have been applied to the observability analysis for the nonlinear system.These methods are mathematically complicated and do not provide useful insights as well.When the emitter is fixed,the observability condition can be derived easily using simple geometrical method since the observer and the emitter are located on a positioning circle defined by the TOI measurement.15However,it becomes more complex for the moving emitter because it is hard to determine the positioning circle nonuniquely,due to the emitter motion.To analyze the observability in presence of the high nonlinearity between the measurements and the target state,a re-parameterization method is introduced in this work in order to obtain an equivalent analytic model.Some necessary observability criteria are established on the basis of this model and physical interpretations for the obtained conditions are given.

A possible approach for the scanning emitter TMA is to recast it into a nonlinear least square(NLS)problem by using the maximum likelihood(ML)solution.21–23However,NLS estimation requires a proper initial guess close to the true solution,which may not be easy to find in practice.To make use of the geometrical characteristic intrinsically in the measurements,we transform the TOI into DDOA measurements.19A pseudo-linear least square(PLS)estimator is then proposed by considering the first DOA as a nuisance parameter to be identified jointly with the emitter motion trajectory.However,it is known that the PLS estimator suffers from the presence of significant bias.5,8,19To reduce the estimation bias caused by the correlations between the regressor and regressand,an instrumental variable(IV)method is proposed.A weighted IV(WIV)estimator is also developed to reduce the estimation variance.The proposed method attains performance very close to the Cramer-Rao lower bound(CRLB)at moderate noise levels.

The rest of this paper is organized as follows.Section 2 establishes the DDOA model for the scanning emitter TMA using TOI only with multiple scans.Section 3 conducts the observability analysis and gives the physical interpretations.Section 4 presents the closed-form PLS estimator as well as the WIV estimator.Section 5 presents the simulation results to verify the proposed estimators.Finally,conclusions are drawn in Section 6.

2.Problem formulation

Without the loss of generality,we consider the TMA problem in a 2D plane.As shown in Fig.1,an emitter is moving at a constant velocity˙x=［vTx，vTy］T.There is a mechanically scanning antenna with a known constant scan rate ω at the emitter.Its main beam periodically sweeps across a single maneuvering observer,which intercepts the signals and records the interception time of the beam peaks.15It is assumed that at timetk,the observer,located atsk= ［xok，yok］T,intercepts the main beam signal,which was transmitted by the emitter atxk= ［xTk，yTk］T.

According to Fig.1,the TOI measurement of the scanning signal is

wherek=1，2，...，N.Nis the total number of the interceptions.cis the speed of light.Tn=2π/ω is the scanning period,which is known because ω is known.t0and β0are the initialsignal transmission time and azimuth of the rotating antenna.denotes the signal propagation range between the observer and the emitter,whererk=sk-xkrepresents the vector from the emitter to the observer.βk=arctan（yk/xk）is the angle between vectorrkand thex-axis anti-clockwise,wherexk=xok-xTkandyk=yok-yTkare the relative coordinates of each axis.ηkis a zero mean Gaussian white noise with the covariance σ2.Parameterst0and β0are unknowns,which make emitter localization difficult by usingdirectly.To eliminate the influence of these unknowns,a straightforward way is to evaluate the time difference of interception(TDOI)between two TOI measurements.13The obtained TDOI is equal to

wherei≥j,i，j=1，2，...，N;βij= βi- βjis the difference of the radiating angle between two intercepted signals,equaling to the DDOA;ηij= ηi- ηjis the measurement error.

For a distant emitter,it is reasonable to apply the approximation15thatri≈rj,which eliminates the influence of the term （ri-rj）/c.The TDOI is then given by

After multiplying ω on both sides of Eq.(3),the measured DDOA becomes

wheretijis the true value ofand δij= ω（ηi- ηj）the corresponding measurement error.In this way,measured TOIs are converted to approximate DDOA measurements.

The DDOA between thekth interception and the first one with respect to the initial location and velocity of the scanning emitteru1=

wherek=2，3，...，N,x1is the emitter position where itsfirst intercepted signal is radiated,and

To facilitate the following theoretical and algorithm development,6–8we assume that the moving process of the scanning emitter is noiseless,and makeu1as the unknown to be estimated in this TMA problem.For simplicity,the indexkis presumed to run from 2 toNin the sequel unless otherwise specif i ed.It is important to notice thattk1in Eq.(6)is not available.An alternative way is to useinstead.This makes the scanning emitter TMA problem more intractable than the conventions.5–8However,should be treated as a known constant rather than a random variable in Eq.(6).The assumption is reasonable under small measurement noise condition.

The measurement of DDOA corrupted by the noise is written in terms ofu1as

According to the distribution of ηk,it can be deduced that δk1is zero mean and the covariance is

The corresponding measurement errors are Gaussians and they follow N（0，ω2σ2Σ）,where N（μ，Ω）denotes the Gaussian distribution with mean μ and variance Ω.

Accordingly,the scanning emitter TMA problem can be transformed to an optimization problem,which is to estimate the initial stateu1from a batch of DDOA measurements given by Eq.(4)with Eqs.(6)and(7)being the dynamic and observation models.

3.Observability analysis

In principle,the observability of a system is irrelevant to the measurement noise,and the observability conditions are usually derived under the noise-free model.26To make use of the geometrical characteristic intrinsically in the measurements,wefirst recast the nonlinear system into a linear form,and then establish the observability based on the obtained model.For this purpose,we write the cotangent of βk1as

It can be seen that once θkis determined,cotβ1can be deduced from （θk）（5）,and the state of the emitter can be obtained fromuk=）（1:4）,where （·）（i）denotes theith element of a vector.Furthermore,u1can be determined fromukvia Eq.(6).

Although introducing variable β1increases the unknowns to estimate the state,the intrinsic relation β1=arctan（y1/x1）counteracts this change.Therefore,the observability criteria of the scanning emitter TMA problem can be acquired by checking whether θkhas a unique solution.

From Eq.(6),it can be derived that

identity matrix with dimensioni.We further notice thatR1Φk1=Φk1R1,and by multiplyingR1on both sides of Eq.(13),the transition function of θkis given by In dynamic system theory,27a linear discrete system is locally observable,if its observability matrix ΓNhas full column rank,viz.rank（ΓN）=n,whereNis the total number of measurements andnthe number of unknown parameters to be estimated.After re-parameterization,the linear system in our problem is obtained using Eqs.(12)–(14).The sufficient and necessary criterion of the observability for the scanning emitter TMA problem in consideration is

The first lineh1is fixed and introduced here because β11=0 and Φ11=I5.It can be seen as the intrinsic constraint between θ1and β1from Eq.(10)whenk=1.It is difficult to find the geometry meanings directly from Eq.(15).Some necessary conditions are then resorted to and the physical interpretations of the observability conditions are made visible.

Necessary condition 1.The observer cannot be stationary or moving at a constant velocity.

When the observer is stationary or moving at a constant velocity,the velocity of the observer is assumed as ［vox，voy］T,wherevoxandvoyare constants.The location of the observer attk, relative tos1, can be expressed assk1= ［xk1，yk1］T= ［tk1vox，tk1voy］T,wherexk1=xok-xo1andyk1=yok-yo1.Putting it back to Eq.(16)and applying some manipulations yield

where λiis theith column of ΓN.It would lead to rank（ΓN）≤ 4.

Necessary condition 2.The DDOA measurements cannot be all the same.

When the DDOA are all the same,βij≡,whereis a constant andi≠j.Putting it back to Eq.(16)yields

which also makes rank（ΓN）≤ 4.

A typical example is that all the positions of the observer at the interception times are on the radial direction of the emitter motion,which makes all the DDOA measurements equal to zero.Another situation is depicted in Fig.2.All the vectors from the emitter to the observer at the interception times are parallel except one,such asr1.This makes all the DDOA measurements equal,and the system cannot be identified uniquely.Other situations,which make the DDOA measurements all the same,also exist and make the system unobservable.

Necessary condition 3.The vectors from the emitter to the observer at the interception times cannot be co-circular.

After some elementary transformations of ΓN,Eq.(15)is recast into

where

wherexk1=xk-x1andk=1，2，3.According to the cocircular judgment theorem given in Appendix A,whenx1are on the same circle O,the angle relations can be given as

In the same way,whenk=4，5，...，Nandare on the same circle O,Eq.(23)can also be satisfied.At this time,it makes allrk=sk-xk=-x1co-circular and results in rank（ΓN）=rank（EN）≤ 4.

The discussions above are based on the DDOA measurements relative to the first intercepted scan angle β1.As a matter of fact,any scan angle βk∉ ｛0，π｝ is suitable in place of β1,so the intrinsic constraints on cotβ1can be satisfied to obtain Eqs.(18)and(19).

It is clear that if β1is known,the DDOA is equivalent to the DOA measurement.It implies that the observability of the DDOA-based TMA is comparable to that of the DOA-based TMA25,26under condition 1 as well as part of condition 2.However,it is more intractable for the DDOA problem because of the unknown β1.Therefore,more criteria,condition 3,should be satisfied for the DDOA-based TMA.

4.Estimation algorithms

4.1.Pseudo-linear least square estimator

It can be seen from above that the nonlinear estimation problem can be recast into a pseudo-linear one by introducing an extra variable β1.The pseudo-linear form of the DDOA measurement is given by Eq.(12).However the true values of the DDOA inhkare not available,and only the noise corrupted measurements can be used instead.The PLS estimator5can be carried out for this problem.

Substituting the noise corrupted DDOA measurements Eq.(7)into Eq.(10)yields

Referring to the procedure from Eqs.(10)–(12),the pseudomeasurement function is

υk=rksinδk1/sinβ1.Furthermore,rkis not available,and in order to get a closed-form solution,υkcan be approximated by υk≈r0δk1as δk1→ 0 under the same assumption in Eq.(3),wherer0is an unknown constant and it will not affect the normalized computational algorithm.5–7

Substituting Eq.(14)into Eq.(26)yields

We rewrite all the measurements in a compact form as

However,due to the approximation made in υk,the weighted matrix has serious deviation,which makes improving the performance ofdifficult.This can be seen clearly in the simulations.Another way to choose the weighted matrix is by usingobtained fromto form the matrix13,nevertheless,it is also inefficient to improve the performance due to the inaccuracy of

4.2.Weighted instrumental variable estimator

Although the PLS estimator is in closed form,it contains significant estimation bias.The bias ofis given by

which is obtained according to Slutsky’s theorem.29Since the measurement noise δk1is presented both in^Γ andv,we havewhich implies that the PLS estimator is biased even forN→∞.In order to reduce the severe bias of the PLS estimator,we developed an IV estimator.30

The normal equation for the PLS estimator is

whereGis the IV matrix.is the IV estimator,which is derived by

The choice of the IV matrix is problem dependent.31–35The optimal choice ofGis,but it is not available.A suboptimal choice is taken as follows.First,the DDOA evaluated from the PLS estimator is given as

Third,replacing βk1andtk1withIV matrix as

In order to minimize the error covariance of,a weighted matrix is introduced and derived as follows.The noise term in Eq.(26)is re-approximated by υk≈rkδk1/sin β1.Then,^v= ［r2δ21，r3δ31，...，rNδN1］T/sin β1is used to approximatevin Eq.(28).It has zero mean and variance with

Parameterrkis still unavailable,and is estimated as= ‖sk-‖2,whereis derived from Eqs.(6)and(33)when=.Then,the weighted matrix is chosen as

5.Simulation results

The moving scanning emitter tracking geometry is considered in the simulation and is shown in Fig.4.An emitter located at［0，0］Tkm moves at a constant velocity ［0，120］Tm/s.The mechanically scanning radar with 10 s scan cycle is working clockwise.In order to satisfy the observability,a maneuver pattern is adopted7that the observer travels with three legs.For each leg,the observer moves 240 s at a constant speed of 300 m/s and takes a course change of 90°at the end of the leg.The observer first intercepts the scanning signal at［-100，-200］Tkm with the course 135°.The total scenario is about 720 s and produces 72 TOI measurements.The true value of TDOI of every adjacent TOI and the true value of the DDOA relative to the first interception angle β1are shown in Fig.5.

The bias and the root mean square error(RMSE)are obtained after 2000 times Monte-Carlo simulations.The bias of the estimator is defined by the distance between the mean estimation and the true value,36

Wefirst examine the performance of the PLS estimator,the IV estimator as well as their weighted version.The ML estimator,which is implemented using the Gauss-Newton algorithm and is initialized to the true value with 2 iterations,is also presented.The RMSEs of the estimators are compared with the CRLB at different noise levels.The TOI noise standard variance ranges from 0 to 6 ms with 1 ms as the interval.The RMSE and bias of the initial state are calculated at the last interception of the scenario.The derivation of the CRLB and the ML estimator are both given in Appendix B.

The simulation results in Fig.6 show that all the estimators perform the same trend in the RMSE and the bias of position and velocity estimation.The ML estimator is approximately unbiased and reaches the CRLB.The PLS estimator presents significant variance and bias,and the WPLS estimator performs exactly the same as the PLS estimator.The IV estimator reduces the errors both in RMSE and bias,but still has disparity with the ML estimator.The WIV estimator is comparative to the ML estimator and asymptotically attains the CRLB even when σ=6 ms.It alleviates the bias caused by the PLS estimator,and is also asymptotically unbiased in moderate measurement noises.

We take the noise standard variance σ=3 ms to examine the influence of different interception numbersNfor the PLS estimator,the WIV estimator and the ML estimator.The RMSE and the bias are also derived to evaluate the estimators.The ML estimator uses the same way as mentioned above.

According to the observability conditions,the emitter is unobservable during the first leg.The simulation results shown in Fig.7 are presented from the second leg.The RMSE and bias of every estimator decrease asNincreases.The ML estimator always reaches the CRLB while the system is observable.WhenN＞35,the WIV estimator attains the CRLB.However,the PLS estimator cannot reach the CRLB even at the end of the scenario.The bias of the ML estimator is the smallest,and whenN＞32,it is steadily unbiased.The bias of the WIV estimator asymptotically reaches 0 whenN＞45.The bias of the PLS estimator does not vanish asNincreases.We can also notice that whenN＜35,the RMSEs of the WIV and the PLS are smaller than the CRLB.This is because the CRLB is only used for the unbiased estimators.36However,the estimators are biased when the number of interception is less than 35 as seen from Fig.7(b)and(d),and their RMSE may outperform the CRLB.

6.Conclusions

The moving scanning emitter tracking by a single observer has been discussed in this paper.The DDOA model is established based on the TOI measurements in multiple scans.The observability for the scanning emitter TMA is analyzed and three necessary conditions as well as the physical interpretations are given.The closed-form PLS estimator and the WIV estimator are proposed to solve the nonlinear problem.The WIV estimator reduces the bias and variance caused by the PLS estimator explicitly.Simulation results show that the WIV estimator reaches the CRLB with moderate TOI measurement errors.

Acknowledgements

The authors are grateful to associate professor Yang Le in Jiang Nan University,Wuxi,China for discussions and language review.This study was co-supported by the Shanghai Aerospace Science and Technology Innovation Fund of China(No.SAST2015028)and the Equipment Prophecy Fund of China(No.9140A21040115KG01001).

Appendix A.Co-circular judgment theorem

Co-circular judgment theorem:4 different points｛Xi｜Xi= （xi，yi）T，i=0，1，2，3，Xi∈ R2｝ are co-circular if and only if every 3 of them are not co-linear,and

Proof.Necessity:

As shown in Fig.A1,when the 4 different pointsXiare on the same circleO,it is obvious that 3 of them are not co-linear,which is necessary for a line cross a circle at 2 points at most.

Connect every pair ofXi.Drawn a line fromX1,paralleled with thex-axis,and intersect circleOat pointA.Drawn 2 lines fromX2,X3paralleled with they-axis,intersect circleOatBandC,and intersect lineX1AatDandE,respectively.Drawn another 2 lines fromX2，Bparalleled with thex-axis,and intersect lineX3CatFandG,respectively.

According to the properties of circle, α21= π-∠X2AX1=π-∠X2BX1, α31=∠X3AX1=∠X3CX1andX3F=GC.

By using the trigonometry,it can be obtained that

Substitute Eq.(A2)into the left side of Eq.(A1),and the necessary criterion is proofed.

Suff i ciency:

Assume that the angle relationships amongX1，X2，X3andX′0satisfy Eq.(A1),butis not on the circleO,as shown in Fig.A1.Referring to the necessary criteria,any pointX0on the circleOsatisfies Eq.(A1)withX1，X2，X3,so the angle relationships α31= ∠X3X′0X1= ∠X3X0X1and α21=∠X21= ∠X2X0X1can be deduced.According to the circle properties,is on the circumcircle of ΔX0X1X2and ΔX0X1X3simultaneously.It is noticed thatX0，X1，X2，X3are co-circular,sois co-circular withX1，X2，X3,and the assumption is not established.For other situations such asthe same results can be obtained directly.

End.□

Appendix B.Cramer-Rao lower bound and maximum likelihood estimator

According to the model established in Section 2,the measurements of TOI（k∈ ［1，N］）,which are corrupted by independent identical distributed Gaussian white noise, are transformed to（k∈ ［2，N］）with equivalent noises followed N（0，ω2σ2Σ）.So the CRLB obtained from the TOI is rigorously equivalent to the CRLB obtained from the DDOA with relevant noises.15

The CRLB is equal to the inverse of the Fisher matrix,36which is given by

The likelihood function for DDOA measurements is given by the joint probability density function conditioned on the initial stateu1of the scanning emitter.

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20 May 2016;revised 3 August 2016;accepted 19 December 2016