PASSIVE RADAR TARGET TRACKING USING CHIRPLET TRANSFORM

Farzad FarhadZadeh , Hamidreza Amindavar

Amirkabir University of Technology, Department of Electrical Engineering, Tehran, Iranfarhadzadeh@aut.ac.ir, harnidarni@aut.ac.ir

ABSTRACT

In this paper, we utilize chirplet transformation to estimate the differ­ential delays-Dopplers in an array of sensors. After chirplet model­ing ofthe received signals from each sensor we use extended Kalmanfiltering (EKF) for tracking the targets by estimating the differentialdelays and differential Dopplers. This new approach is particularlyuseful in passive radar and sonar for target tracking. Chirplet model­ing is crucial since the received signals are non-stationary in nature.

1. INTRODUCTION

An important issue in RADAR and SONAR is target tracking. Inthe active scenario we extract the appropriate information by com­paring the radiated and return signals. On the other hand, in passiveRADAR and SONAR a radiating source can be tracked by observingits signal at three or more spatially separated receivers. Most of thesystems which have been analyzed depend on the differential timedelay of the signal wavefront between pairs of sensors determinebearing and range, [4, 3, I, 2]. When the terget is moving relative tothe receiving array, the various signal components are not only time­delayed but also Doppler time-compressed relative to each other.Measurement of these differential Dopplers provides important ad­ditional information about source speed and heading. So, we needboth differential delay and Dopplers to track targets. Our primaryinterest here is simply in tracking, and we restrict ourselves solely tothat objective. Historically, several approaches have been used forsignal tracking, either by working directly with the raw signals or byusing some transformation of it. Many of the various methods forthe tracking of signals make use of the Fourier transform. Althoughthis transform is extremely useful and well established, it does havedrawbacks-principal difficulties in analyzing short-term transient. Inaddition, alternatives to the short time Fourier transform with bettertime-frequency localization have been suggested; for example, theWigner distribution and its variants. Our justification for wishingto use chirplet transformation is simply that many signals presentin the space radiated by broadcasting system have linear frequencymodulation(LFM) and the same property as chirp signals. There­for, we use chirplet transform as an adaptive transform to extractappropriate information from signals scattered by targets. The res­olution of the chirplet transformation has local adaptivity, and thispotentially enables us to zoom in on irregularities and characterizethenl more specifically than is possible with Fourier transformations.In addition, extracting the differential delay-Doppler is not a sim­ple problem and some researchers have proposed some algorithmswhich are complicated , these parameters can be estimated by firstorder or higher order ambiguity function [I, 10] in Maximum Likeli­hood Estimation(MLE) sense. But in this manner, when the numberof targets are two or more, finding the maximum of first or higherorder ambiguity function is not feasible. Since ambiguity function

is a quadratic function, cross terms are appeared and noise and inter­ference becalue amplified. So, It is not possible to estimate differ­ential delay and Dopplers. The purpose of this paper is to providea new passive approach to target tracking by estimating the differ­ential delays and Dopplers. We also describe a tracking scheme forlocating the targets. We use chirplet transformation to generate im­ages of differential delay-Doppler radar returns from targets. Weintroduce a simple new method to extract proper information fromthe received signals. In this method, we use received signals fromtargets for tracking by having some preconception about scatteredsignals. Since the broadcasting signal from a TV/RADIO station,with linear frequency modulation (LFM) has a non-stationary char­acteristics, hence, it can be estimated by sum of some chirplets, it iswell understood that chirp is one of the most important functions inFM and TV signals. We approximate a received signal by a weightedsum of chirplets [7] parameterized by the location in time, locationin frequency, chirp rate and the chirp duration. Such four-parameterchirplet approximation offers more efficient parametric representa­tions of many signals of interest than the representations obtainedusing short time Fourier transform, Gabor transform, wavelet trans­form, and wavelet packets, the literature on this subject is vast, butnaming a few [7, 8, 9]. Wavefronts at each sensor of the array areapproximated by parameterized chirplet transform, and then differ­ential delays and Dopplers are estimated by subtracting the locationof frequency and time in the same chirplets, finally extracted infor­mation is used to track targets by EKF. This paper is organized asfollows, in section 2, a discussion about estimation of differentialDoppler and delay is presented, in section 3, we discuss tracking byEKF, and in section 4, we provide the simulations and results, andsome concluding remarks at the end.

2. ESTIMATING DIFFERENTIAL-DELAY ANDDIFFERENTIAL-DOPPLER

In this section we estimate differential delay-Doppler that exist be­tween signals received from sensors of an array by the means ofchirplet transform. If w(k) is a narrowband signal scattered by a tar­get at a range Ri (k) and moving with a constant velocity, see Fig. I,then the signal received at the i th sensor is presented as [10]

( I)

where Ii, and di denote the delay and Doppler shift and Vi (k) rep­resents zero-mean sensor noise. Our objective is to estimate the dif­ferential delay-Doppler parameters at a pair of sensors, i.e. a Tij =Ti - Tj and adij = di - dj. Also for a received signal by an i th

sensor Si(k), we have the following representation [7].

q

Si(k) == L aiej¢P s(k; tp , wp , cp , dp ) + vk(k) (2)p=l

978-1-4244-2241-8/08/$25.00 ©2008 IEEE 478

Fig. 1. A set-up for signal measurement.

Source moving with velocity v

where motion along each coordinate axis evolves independently ac­cording to

(7)

(6)F=[~~~]o 0 F

(}2 (}3

sensor1 it sensor2 l2 sensor3

81,1 ][ 82,1 ] [ 83,1 ]S1,2 S2,2 S3,2

S1,q S2,q S3,q

group#1 group#2 group#3

_ [1 /).t /).t2 /2]

F = 0 1 /).t

001

In this case, as long as the intersample interval T is a constant, Fand F do not vary with k. For this constant-acceleration dynamicsystem, the process noise model Uk changes in the state due to thechanges in the underlying acceleration increment sequence. We as­sume that {Uk} is a zero-mean white Gaussian sequence with knowncovariance, E[Uiuj] = Qb(i - j), where b is the Kronecker deltafunction and Q is a block diagonal

where px,k ,Vx,k and ax,k denote position,velocity, and accelerationrespectively, along the x-axis at time k. The components for the yand z axes are defined in a similar manner. The Newtonian systemmatrix then assumes a block diagonal form

where tp,wp, and Cp are real numbers and dp is a positive real num­ber. The parameters t p, Wp, Cp and dp represent, respectively, loca­tion in time, location in frequency, chirp rate, and the duration of thechirped signal, respectively, and Vi (k) is the receiver noise and/orthe statistical modeling mismatch. s (.) is defined as

s(k;tp,wp,cp,dp) = (~d)-!

exp[-( k ;dtp

)2 + j ~ (k - t p)2 + jw(k - t p)]. (3)

3. EXTENDED KALMAN FILTER-BASED TRACKING

Next, we discuss the tracking of targets using the estimated differen­tial delay-Doppler in the estimation phase.

Therefor, we have received signals Si (k) {i = 1, ... N}, at eachsensor of N -element array. Then, we approximate the received sig­nals by weighted sum ofchirplets. In our experiments, q, the numberof chirplets is equal the number of targets and do not vary with i, aneasily justifiable assumption if the observation time of the observa­tion space are both constant. We have N group that each group hasq-chirplets described by {ap, ¢p, tp,Wp, Cp and dp} that are illus­trated in Fig. 1. In the next step, we extract the most similar chirpletsfor the pairs of sensors that each sensor has a group of q-chirplets.For instance, the i th and jth sensors are selected as a pair of sen­sor. The important issue in the most similar chirplets is that twomost important properties of chirplets, chirp rate and duration, arefixed. In the forgoing instance, m th and nth chirplets are selectedfrom ijth pair of sensors respectively. Therefor, we can extract sim­ilar chirplets with these properties, and finally differential-delay andDoppler can be estimated by subtracting the location in time, (i.e./).Tij = t mi - t nj ,tmi is selected from ith group and t nj from

jth group of chirplets), and the location in frequency; i.e., /).dij =Wmi -wnj ,wni is selected from i th group and wnj fromjth group of

chirplets, ofthe selected chirplets. Where {ti,p, Wi,p, Ci,p, di,p}~i,~=1are the estimated chirplet parameters. Our algorithm consists of thefollowing steps.

1. The received signals should be approximated by the means ofq-chirplets (q is the number of targets).

2. After selection of pairs of sensors that belong to the array, thebest similar chirplets are extracted from pairs of sensors.

3. Differential-delay and differential-Doppler are estimated bysubtracting time-location and frequency-location of selectedchirplets.

Since this is a formulation in discrete time, each random accelerationincrement acts upon the state for the sample period T. Qcan beevaluated as,

By using the Kalman prediction and update equations [12], [11], wedemonstrate the system and measurement model for tracking basedon differential delay-Doppler. We begin by assuming a linear dy­namic system model, then, the standard discrete-time Kalman stateequation takes the form

(4)

[Q 00]

Q= 0 Q ~o 0 Q

(8)

where, Xk is a vector of kinematic components, Uk is the processnoise, and Fk is our (possibly) time-varying state matrix. Since ourapplication will not involve a target which is maneuvering, we adopta constant-acceleration motion model. Thus, in three dimensions, Xk

will contain 6 components

(9)

where a~ is the variance of the noise sequence modeling the ac­celeration increment process. In the measurement model, we have

479

where H k is allowed to vary with time and W k is the measurementnoise sequence. The observation vector Zk for the Kalman filter willconsist of pairs of differential delay and Doppler measurements. Sowe have

Zij, k as the set of measurements provided by the ijth pair of sen­sors at time stance k. Then, the multisensor measurement vector attime k is defined as the concatenation of all the current scans, orZk == {Zl,k,··· ,ZM,k}, where M is the number of differentialdelay-Doppler involved to track the target. The measurement modelfor the standard Kalman filter can then be expressed as

(10)

8t!..dij,k = ~((Prj - Pk)(Prj - Pk)' 1 I

8p A IIPrj - Pk 113 IIPrj - Pk II

(Pr, - Pk)(Pr, ~ Pk)' + 1 1) (19)1IPri - Pk II IIPri - Pk II

We note that in (17)-(19), the position and velocity vectors wouldbe more accurately written as PkIk-1 and vkIk~ 1 since they are ob­tained from the components of the one-step prediction Xk Ik -1. This

completes our specification of ifk. For the description of the mea­surement noise sequence {wk }, we assume it is an independent zero­mean Gaussian process with known covariance E [Wiwj] == R6 (i ­j), where R is a block diagonal

where Pi and Pj are the location of the ijth pair of sensors, Ais thewavelength and II . II is the Euclidean norm. In (13), ni(Pk) is anormal vector that can be evaluated as

where (~Tij,k, ~dij,k) is the differential delay-Doppler measure­ment from the ijth pair of sensors at time k. Ifwe define Pk and Vkas the position and velocity vectors, respectively, of the target at timek, then the components of Zk are related to these state componentsby

and the measurement covariance corresponding to the ijth scan isdefined as

(20)

(21)

o

R==

R- .. - [ai Tij 0 ] 21,) - 0 2 au

a6dij

In our experiments, we assume that aiT" and aid" do not vary1-J 1-J

with i, j. With these definitions for our system and measurementmodels, we can now express the prediction and update equations forour extended Kalman filter[ 12].

(13)

(12)t!..Tij,k = Ilpk - Pri II Ilpk - Prj IIc c

~d.. - 2vk.(fh(Pk) - nj(Pk))1,),k - A

4. SIMULATION AND RESULTS

where ~x is the gradient operator expressed as a column vector. Us­ing the definitions of differential delay and Doppler from (12)-(13),we see that ifk will have the following form

where the two rows shown correspond to the differential delay-Dopplerpair from the ijth pair of sensors, and all partial derivatives are eval­uated for xklk-1 as indicated in (15). After some algebra, the fol­lowing equations for the partial derivatives are obtained

We are now ready to show the tracking performance of this sys­tem through Monte Carlo simulations.Our intended application isa passive radar system for tracking.To uniquely locate a target in3-dimensional space, differentia-delay measurements at least fromthree different transmitters are needed. The same applies to theunique determination ofa targets 3-dimensional velocity vector frommeasurements of Doppler shift.Of course, in the presence of noise,we could use more than three transmitters to improve tracking per­formance.As such, we will use the 6 number of transmitters in oursimulations.To facilitate analysis tracking results, we will considerthe simplest target scenario:one aircraft.The specific scenario wewill consider is shown in Fig.2 .The six receivers are identified inthe plot. The true trajectory of the target is drawn.In every step the real received signal is approximated by 1 chirplets, thus we have 6 unique groups of chirplets.We consider the first re­ceiver as a reference ofthe other receivers (i.e, i == 1, j == {2, ... ,6}),we have 15 combinations to select the pairs of sensors, and twobest chirplts are selected from every pair of groups and differen­tial delay-Doppler, arij, adij are estimated by subtracting theirtime and frequency 10cations.In Fig.3 results of estimating differen­tial delay-Doppler between 1st and 3rd sensor versus real differen­tial delay-Doppler is illustrated. Finally the estimated differentialdelay-Doppler is given to the EKF and the target is tracked. In Fig.4the simulated tracked motion of the target is illustrated versus thereal motion.

(16)

(15)

(14)

From (12)-(13), we see that the relationship between Xk and Zk forour application is nonlinear(i.e., Zk == hk(Xk)). Thus, we haveto use an extended Kalman filter, where the nonlinear measurementfunction hk is linearized about the one-step state prediction Xk Ik -1.

The resulting Jacobian matrix can be written as

8~Tij,k nj(Pk) - ni(Pk)8p c

8~dij,k == 2ni(Pk) - nj(Pk)8v A

(17)

(18)

5. CONCLUSION

In this paper, we introduce a new method to track targets using dif­ferential delay-Doppler by· a multiple sensor system. Our tracking

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6. REFERENCES

Fig. 3. The differential-delay and differential Doppler between 1st

and 3rd is estimated in SNR= 5dB.

method is based on EKF by means of chirplet transform. The suc­cess of the approach is stemmed in the parametric modeling of tar­gets using chirplet transformation that make viable the delays andDopplers.

[7] J. C. O'Neill, P. Flandrin, " Chirp hunting," in Proc. IEEEInt. symp. Time-Frequency Time-Scale Anal., pp. 425428, Oct.1998.

[8] A. Bultan, "em A four-parameter atomic decomposition ofchirplets," IEEE Trans. Signal Process., vol. 47, no. 3, pp.731745, Mar. 1999.

[9] R. Gribonval, "Fast ridge pursuit with multiscale dictionary ofgaussian chirps," IEEE Trans. Signal Process., vol. 49, no. 5,pp. 9941001, May 2001.

[10] A. V. Dandawate, G. B. Giannakis "Differential delay-Dopplerestimation using second and higher-order ambiguity func­tions," lEE PROCEEDINGS-F, vol. 140, no. 6, Dec. 1993.

[11] S. Herman, P. Moulin, "A particle filtering approach to FM­band passive radar tracking and automatic target recognition,"IEEE Aerospace Conf Proceed., 2002.

[12] Y. Bar-Shalom, X. R. Li, "Multitarget-multisensor tracking:principles and techniques." Storrs, CT: YBS Publishing, 1995.

[13] http://tfd.sourceforge.net/.

Fig. 4. Plots of various extended Kalman filter estimates for thetarget versus its true values (except for the position error plot wherethe "true" values would be zero). The legend in the lower right paneapplies to all four plots(SNR= 5dB).

50 60 70time(sec)

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[1] C. H. Knapp and G. C. Carter, "Estimation of time delay in thepresence of source or receiver motion," 1. Acoust. Soc. Amer.,vol. 61, June 1977.

[2] E. Weinstein, " Optimal source localization and tracking frompassive array measurment ", IEEE Trans. on acoustic, speech,and signal procesing , vol. ASSP-30, no. 1, febraury 1982.

[3] W. R. Hahn and S. A. Tretter,"Optimum processing for delayvector estimation in passive signal arrays," IEEE Trans. In­form. Theory, vol. IT-19, no. 5, pp. 608-614,1973.

[4] W. R.Hahn, "Optimum signal processing for passive sonarrange and bearing estimation," J, Acoust. Soc. Amer., vol. 58,no. 1, 1975.

[5] T. Ura, R. Bahl, M. Sakata, J. Kojima, T. Fukuchi., et 'ai," Acoustic tracking of sperm whales using two sets of hy­drophone array," Int'l symp. on underwater technology, 2004.

[6] 1. Cui, W. Wong, "The adaptive chirplet transform and visualevoked potentials, IEEE Trans. on Biomedical Engineering,vol. 53, no. 7, Page(s):1378 - 1384, July 2006.

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