PASSIVE RADAR TARGET TRACKING USING CHIRPLET TRANSFORM Farzad FarhadZadeh , Hamidreza Amindavar Amirkabir University of Technology, Department of Electrical Engineering, Tehran, Iran
[email protected],
[email protected] ABSTRACT In this paper, we utilize chirplet transformation to estimate the differential delays-Dopplers in an array of sensors. After chirplet modeling of the received signals from each sensor we use extended Kalman filtering (EKF) for tracking the targets by estimating the differential delays and differential Dopplers. This new approach is particularly useful 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. In the active scenario we extract the appropriate information by comparing the radiated and return signals. On the other hand, in passive RADAR and SONAR a radiating source can be tracked by observing its signal at three or more spatially separated receivers. Most of the systems which have been analyzed depend on the differential time delay of the signal wavefront between pairs of sensors determine bearing and range, [4, 3, 1, 2]. When the terget is moving relative to the 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 need both differential delay and Dopplers to track targets. Our primary interest here is simply in tracking, and we restrict ourselves solely to that objective. Historically, several approaches have been used for signal tracking, either by working directly with the raw signals or by using some transformation of it. Many of the various methods for the tracking of signals make use of the Fourier transform. Although this transform is extremely useful and well established, it does have drawbacks–principal difficulties in analyzing short-term transient. In addition, alternatives to the short time Fourier transform with better time-frequency localization have been suggested; for example, the Wigner distribution and its variants. Our justification for wishing to use chirplet transformation is simply that many signals present in the space radiated by broadcasting system have linear frequency modulation(LFM) and the same property as chirp signals. Therefor, we use chirplet transform as an adaptive transform to extract appropriate information from signals scattered by targets. The resolution of the chirplet transformation has local adaptivity, and this potentially enables us to zoom in on irregularities and characterize them 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 algorithms which are complicated , these parameters can be estimated by first order or higher order ambiguity function [1, 10] in Maximum Likelihood Estimation(MLE) sense. But in this manner , when the number of targets are two or more, finding the maximum of first or higher order ambiguity function is not feasible. Since ambiguity function
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is a quadratic function, cross terms are appeared and noise and interference became amplified. So, It is not possible to estimate differential delay and Dopplers. The purpose of this paper is to provide a new passive approach to target tracking by estimating the differential delays and Dopplers. We also describe a tracking scheme for locating the targets. We use chirplet transformation to generate images of differential delay-Doppler radar returns from targets. We introduce a simple new method to extract proper information from the received signals. In this method, we use received signals from targets for tracking by having some preconception about scattered signals. 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 is well understood that chirp is one of the most important functions in FM and TV signals. We approximate a received signal by a weighted sum of chirplets [7] parameterized by the location in time, location in frequency, chirp rate and the chirp duration. Such four-parameter chirplet approximation offers more efficient parametric representations of many signals of interest than the representations obtained using short time Fourier transform, Gabor transform, wavelet transform, and wavelet packets, the literature on this subject is vast, but naming a few [7, 8, 9]. Wavefronts at each sensor of the array are approximated by parameterized chirplet transform, and then differential delays and Dopplers are estimated by subtracting the location of frequency and time in the same chirplets, finally extracted information is used to track targets by EKF. This paper is organized as follows, in section 2, a discussion about estimation of differential Doppler and delay is presented, in section 3, we discuss tracking by EKF, and in section 4, we provide the simulations and results, and some concluding remarks at the end. 2. ESTIMATING DIFFERENTIAL-DELAY AND DIFFERENTIAL-DOPPLER In this section we estimate differential delay-Doppler that exist between signals received from sensors of an array by the means of chirplet 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. 1, then the signal received at the ith sensor is presented as [10] Si (k) = w(k − τi )ejdi + νi (k)
(1)
where τi , and di denote the delay and Doppler shift and νi (k) represents zero-mean sensor noise. Our objective is to estimate the differential delay-Doppler parameters at a pair of sensors, i.e. ∆τij = τi − τj and ∆dij = di − dj . Also for a received signal by an ith sensor Si (k), we have the following representation [7]. Si (k) =
q X
p=1
ai ejφp s(k; tp , ωp , cp , dp ) + υk (k)
(2)
where tp ,ωp , and cp are real numbers and dp is a positive real number. The parameters tp , ωp , cp and dp represent, respectively, location in time, location in frequency, chirp rate, and the duration of the chirped signal, respectively, and υi (k) is the receiver noise and/or the statistical modeling mismatch. s(·) is defined as √ 1 s(k; tp , ωp , cp , dp ) = ( 2πd)− 2 k − tp 2 c exp[−( ) + j (k − tp )2 + jω(k − tp )]. (3) 2d 2 Therefor, we have received signals Si (k) {i = 1, . . . N }, at each sensor of N -element array. Then, we approximate the received signals by weighted sum of chirplets. In our experiments, q, the number of chirplets is equal the number of targets and do not vary with i, an easily justifiable assumption if the observation time of the observation space are both constant. We have N group that each group has q-chirplets described by {ap , φp , tp , ωp , cp and dp } that are illustrated in Fig. 1. In the next step, we extract the most similar chirplets for the pairs of sensors that each sensor has a group of q-chirplets. For instance, the ith and j th sensors are selected as a pair of sensor. The important issue in the most similar chirplets is that two most important properties of chirplets, chirp rate and duration, are fixed. In the forgoing instance, mth and nth chirplets are selected from ij th pair of sensors respectively. Therefor, we can extract similar chirplets with these properties, and finally differential-delay and Doppler can be estimated by subtracting the location in time, (i.e. ∆τij = tmi − tnj ,tmi is selected from ith group and tnj from j th group of chirplets), and the location in frequency; i.e., ∆dij = ωmi −ωnj ,ωni is selected from ith group and ωnj from j th group of ,q chirplets, of the selected chirplets. Where {ti,p , ωi,p , ci,p , di,p }N i=1,p=1 are the estimated chirplet parameters. Our algorithm consists of the following steps. 1. The received signals should be approximated by the means of q-chirplets (q is the number of targets). 2. After selection of pairs of sensors that belong to the array, the best similar chirplets are extracted from pairs of sensors. 3. Differential-delay and differential-Doppler are estimated by subtracting time-location and frequency-location of selected chirplets.
Next, we discuss the tracking of targets using the estimated differential delay-Doppler in the estimation phase. 3. EXTENDED KALMAN FILTER-BASED TRACKING By using the Kalman prediction and update equations [12], [11], we demonstrate the system and measurement model for tracking based on differential delay-Doppler. We begin by assuming a linear dynamic system model, then, the standard discrete-time Kalman state equation takes the form xk+1 = Fk xk + uk ;
(4)
where, xk is a vector of kinematic components, uk is the process noise, and Fk is our (possibly) time-varying state matrix. Since our application will not involve a target which is maneuvering, we adopt a constant-acceleration motion model. Thus, in three dimensions, xk will contain 6 components xk = [px,k vx,k ax,k py,k vy,k ay,k pz,k vz,k az,k ]
(5)
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Fig. 1. A set-up for signal measurement.
where px,k , vx,k and ax,k denote position,velocity, and acceleration respectively, along the x-axis at time k. The components for the y and z axes are defined in a similar manner. The Newtonian system matrix then assumes a block diagonal form F˜ 0 0 (6) F = 0 F˜ 0 0 0 F˜ where motion along each coordinate axis evolves independently according to 1 ∆t ∆t2 /2 F˜ = 0 1 ∆t (7) 0 0 1
In this case, as long as the intersample interval T is a constant, F and F˜ do not vary with k. For this constant-acceleration dynamic system, the process noise model uk changes in the state due to the changes in the underlying acceleration increment sequence. We assume that {uk } is a zero-mean white Gaussian sequence with known covariance, E[ui u′j ] = Qδ(i − j), where δ is the Kronecker delta function and Q is a block diagonal ˜ 0 0 Q ˜ 0 Q=0 Q (8) ˜ 0 0 Q Since this is a formulation in discrete time, each random acceleration ˜ can be increment acts upon the state for the sample period T . Q evaluated as, 5 ∆t /20 ∆t4 /8 ∆t3 /6 ˜ = ∆t4 /8 ∆t3 /3 ∆t2 /2 σu2 Q (9) ∆t3 /6 ∆t2 /2 ∆t where σu2 is the variance of the noise sequence modeling the acceleration increment process. In the measurement model, we have
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Zi j, k as the set of measurements provided by the ij th pair of sensors at time stance k. Then, the multisensor measurement vector at time k is defined as the concatenation of all the current scans, or Zk = {Z1,k , · · · , ZM,k }, where M is the number of differential delay-Doppler involved to track the target. The measurement model for the standard Kalman filter can then be expressed as Zk = Hk xk + wk
(10)
where Hk is allowed to vary with time and wk is the measurement noise sequence. The observation vector Zk for the Kalman filter will consist of pairs of differential delay and Doppler measurements. So we have Zk = {(∆τ1,k , ∆d1,k ), · · · , (∆τN,k , ∆dN,k )}
(11)
where (∆τij,k , ∆dij,k ) is the differential delay-Doppler measurement from the ij th pair of sensors at time k. If we define pk and vk as the position and velocity vectors, respectively, of the target at time k, then the components of Zk are related to these state components by kpk − prj k kpk − pri k − c c vk .(ˆ ni (pk ) − n ˆ j (pk )) ∆dij,k = 2 λ ∆τij,k =
(12) (13)
where pi and pj are the location of the ij th pair of sensors, λ is the wavelength and k · k is the Euclidean norm. In (13), n ˆ i (pk ) is a normal vector that can be evaluated as (14)
From (12)-(13), we see that the relationship between xk and Zk for our application is nonlinear(i.e., Zk = hk (xk )). Thus, we have to use an extended Kalman filter, where the nonlinear measurement function hk is linearized about the one-step state prediction ˆ xk|k−1 . The resulting Jacobian matrix can be written as ˜ k , [hk (x)∆′ x ]x=ˆx H k|k−1
(15)
where ∆x is the gradient operator expressed as a column vector. Using the definitions of differential delay and Doppler from (12)-(13), ˜ k will have the following form we see that H
∂∆τij ∂px ˜ Hk = ∂∆dij ∂px
0
0
∂∆dij ∂vx
0
.. . 0
∂∆τij ∂py ∂∆dij ∂∆dij ∂py ∂vy
0 0
−
(pri − pk )(pri − pk )′ 1 + I) kpri − pk k3 kpri − pk k
(19)
We note that in (17)-(19), the position and velocity vectors would be more accurately written as pk|k−1 and ˆ vk|k−1 since they are obtained from the components of the one-step prediction xk|k−1 . This ˜ k . For the description of the meacompletes our specification of H surement noise sequence {wk }, we assume it is an independent zeromean Gaussian process with known covariance E[wi w′j ] = Rδ(i − j), where R is a block diagonal ˜ R1 0 . . . 0 ˜2 0 R R= . (20) . .. .. ˜M 0 R and the measurement covariance corresponding to the ij th scan is defined as 2 ˜ ij = σ∆τij 20 σu2 (21) R 0 σ∆dij 2 2 do not vary and σ∆d In our experiments, we assume that σ∆τ ij ij with i, j. With these definitions for our system and measurement models, we can now express the prediction and update equations for our extended Kalman filter[12].
4. SIMULATION AND RESULTS
p ri − p k n ˆ i (pk ) = kpri − pk k
′ ∂∆dij,k 2 (pr − pk )(prj − pk ) 1 = ( j − I ∂p λ kprj − pk k3 kprj − pk k
∂∆τij 0 ∂pz ∂∆dij ∂∆dij ∂pz ∂vz
.. .
0 (16) 0
where the two rows shown correspond to the differential delay-Doppler pair from the ij th pair of sensors, and all partial derivatives are evaluated for ˆ xk|k−1 as indicated in (15). After some algebra, the following equations for the partial derivatives are obtained n ˆ j (pk ) − n ˆ i (pk ) ∂∆τij,k = ∂p c n ˆ i (pk ) − n ˆ j (pk ) ∂∆dij,k =2 ∂v λ
We are now ready to show the tracking performance of this system through Monte Carlo simulations.Our intended application is a passive radar system for tracking.To uniquely locate a target in 3-dimensional space, differentia-delay measurements at least from three different transmitters are needed. The same applies to the unique determination of a targets 3-dimensional velocity vector from measurements 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 our simulations.To facilitate analysis tracking results, we will consider the simplest target scenario:one aircraft.The specific scenario we will consider is shown in Fig.2 .The six receivers are identified in the 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 of the other receivers (i.e, i = 1, j = {2, . . . , 6}), we have 15 combinations to select the pairs of sensors, and two best chirplts are selected from every pair of groups and differential delay-Doppler, ∆τij , ∆dij are estimated by subtracting their time and frequency locations.In Fig.3 results of estimating differential delay-Doppler between 1st and 3rd sensor versus real differential delay-Doppler is illustrated. Finally the estimated differential delay-Doppler is given to the EKF and the target is tracked. In Fig.4 the simulated tracked motion of the target is illustrated versus the real motion.
(17)
5. CONCLUSION
(18)
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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Fig. 2. The flight path and locations of six receivers. Fig. 4. Plots of various extended Kalman filter estimates for the target versus its true values (except for the position error plot where the “true” values would be zero). The legend in the lower right pane applies to all four plots(SNR= 5dB).
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[7] J. C. O’Neill, P. Flandrin, “ Chirp hunting,” in Proc. IEEE Int. symp. Time-Frequency Time-Scale Anal., pp. 425428, Oct. 1998.
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[8] A. Bultan, “em A four-parameter atomic decomposition of chirplets,” IEEE Trans. Signal Process., vol. 47, no. 3, pp. 731745, Mar. 1999.
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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 and Dopplers.
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