A New Approach for Doppler-only Target Tracking

Transcription

1 A New Approach for Doppler-only Target Tracking G. Battistelli, L. Chisci, C. Fantacci DINFO, Università di Firenze, Florence, Italy {giorgio.battistelli, luigi.chisci, A. Farina, A. Graziano Selex ES, Rome, Italy {alfonso.farina, Abstract The paper addresses the problem of estimating position and velocity of a moving target given Doppler-shift measurements from multiple sensors located at different places. First, observability of a constant velocity motion from a single Doppler sensor is studied and an observability decomposition is introduced. Then, based on such a decomposition, a novel dualstage filter for centralized multisensor Doppler-only tracking is proposed. In the first stage, a bank of local filters, operating in the observability decomposition coordinates, estimate, for each Doppler sensor, the relative observable state. Next, in the second stage all estimates from the local filters are fused in order to estimate the overall target kinematic state. In order to overcome the well known difficulties of Doppler-only tracking in the case of imprecise prior knowledge on the initial target position, a suitable multihypothesis initialization procedure is adopted. Performance of the proposed dual-stage filter is evaluated via simulation experiments. Index Terms Target tracking; Doppler measurements; nonlinear filtering; observability analysis. I. INTRODUCTION The importance of the Doppler effect began to be appreciated shortly after World War II and became an increasingly important factor in many radar applications. Dopplershift measurements are used in aviation, sounding satellites, meteorology, radiology, and bistatic radar. Another significant application is the separation of moving targets from stationary clutter. The estimation of position and velocity of a moving object using Doppler-shift measurements from different sensors has been studied over the last few decades [] [4]. Recently, passive surveillance [5] [7] has attracted renewed interests on this problem leading to promising new results [8] []. The well known poor observability of the target position from Doppler-only measurements, however, makes target state initialization a very critical issue as witnessed by the fact that successful Doppler-only target tracking has been so far obtained either by exploiting particle filters [8], [9] or by assuming accurate prior knowledge of the initial target position []. The present paper proposes a novel computationally efficient approach to Doppler-only target tracking based on a suitable nonlinear observability decomposition and a dualstage filtering approach. More specifically, observability of a constant-velocity plane motion from a single Doppler sensor is analysed and it is shown how to derive a nonlinear coordinate transformation that allows to decompose the transformed 4- dimensional state vector into a -dimensional observable substate and an unobservable component. Precisely, the observable components are the target-sensor distance ρ, the target speed ν and the angle α between the target velocity vector and the target-sensor direction, while the target azimuth θ is clearly unobservable. Exploiting this decomposition, Doppler-only target tracking is performed by means of a dual-stage filtering approach. In the first stage, a bank of local Unscented Kalman Filters UKFs, operating in the observability decomposition coordinates, estimate, for each Doppler sensor, the relative observable state. Then, in the second stage, another UKF, operating in the Cartesian coordinates, estimates the target kinematic state exploiting the estimates from the local filters of the former stage as input measurements. The rest of the paper is organized as follows. Section II formulates the Doppler-only tracking problem of interest. Section III, concerned with observability analysis via Doppler measurements, introduces a nonlinear observability decompostion to be used for the subsequent developments. Next, section IV presents a novel dual-stage Doppler-only tracking filter based on the previously introduced observability decomposition. Then, the performance of the proposed filter is evaluated in section V by means of simulation experiments. Finally, section VI ends the paper with concluding remarks and possible future research directions. II. PROBLEM FORMULATION A Doppler sensor measures the frequency shift f d of a received target echo. By the Doppler effect, such a shift is proportional to the radial speed ρ see fig. according to the relationship f d = ft f R = λ ρ = λ ν cos α where: f T and f R denote the transmitted and, respectively, received signal frequencies; λ = c/f T is the transmitted signal wavelength, c being the light speed; ρ = ν cos α, ν being the target speed and α the angle between the target direction and the sensor-target line, referred to in the sequel as the relative heading. Let us consider a target, moving in the p x p y plane, with kinematic state x = [p x, ṗ x, p y, ṗ y ] evolving in time according to the discrete-time motion model x t+ = f x t + w t The Doppler-only tracking problem of interest in this paper consists of estimating, at each time t, the kinematic state vector

2 p y Doppler sensor Target p x Fig. : Geometric view of Doppler tracking in given Doppler measurements yt i = h i x t + vt, i i D h i x = px p i x ṗx + p y p i y ṗy 4 λ p x p i x + p y p i y where: D is the set of Doppler sensors; w t and vt i are mutually uncorrelated white Gaussian noises with zero mean and covariances Q > and σdoppler i >, i D, respectively; p i = [p i x, p i y] is the known position vector of sensor i D. Notice from that a Doppler measurement is a nonlinear function of both target position and velocity. Hence, even in the absence of noise, there actually exist many 4-dimensional kinematic states x = [p x, v x, p y, v y ] compatible with a single Doppler measurement y = h x where h is given by. To give an insight view on the high sensitivity of the measurement function with respect to the target position, fig. plots the contour lines of the function h i p x, v x, p y, v y in the p x, p y plane, for fixed constant velocity ṗ x = ṗ y = of the target and sensor position x i = y i =. Because of the aforementioned problem, as [Hz] Fig. : Contour plot of the measurement function having constant velocity ṗ x = ṗ y = of the target and the sensor located at position x i = y i =. also stated in [8], [9], traditional unimodal filters, e.g. Kalman Filter KF, Extended KF EKF or Unscented KF UKF, are inappropriate for state estimation of a moving object with Doppler-only measurements and hence multimodal filters such as, for instance, particle filters are by far preferable. Particle filters, however, have high computational payload and, hence, are not eligible for low cost, low power Doppler sensors. In order to significantly reduce the computational burden, the authors propose a novel approach based on a dual-stage filter. Such an approach relies on: the observability analysis of a constant velocity motion model from a single Doppler sensor; a smart target initialization technique; a dual stage filter for target tracking. The observability analysis will show that from measurements of a single Doppler sensor it is possible to observe the range ρ, the target speed ν and the previously defined relative heading angle α. Once good estimates of these local coordinates have been obtained for each Doppler sensor by means of a first filtering stage, they can be fused in a second filtering stage so as to estimate the desired Cartesian coordinates of target s position and velocity. III. OBSERVABILITY ANALYSIS Recall from system theory that a dynamical system is observable whenever, in the absence of noise, it is possible to uniquely determine the initial state given present and future observations. The focus of this section is on observability analysis of the discrete-time system { xt+ = f x t 5 y t = h x t for the specific linear state transition function T s f x = Ax = T s x, T s > 6 and nonlinear observation function h x = x x + x x 4 x + x Notice that this amounts to studying observability of a target, moving in a plane at constant velocity, from a single Doppler sensor located at the origin. Notice that x = p x, x = p y and x = ṗ x, x 4 = ṗ y represent the target Cartesian position and, respectively, velocity components while T s denotes the sampling interval, i.e. the interval between subsequent Doppler measurements. Clearly, full observability cannot be guaranteed with a single fixed Doppler sensor. Hereinafter, it will be shown what actually can be observed in such a situation. A. Observability decomposition For the subsequent developments, it is convenient to represent target position and velocity by the scalar complex numbers { p = p x + ı p y = ρe ı θ 8 7 ṗ = ṗ x + ı ṗ y = νe ı θ+α 9

3 where: ı is the imaginary unit such that ı = ; ρ = p = p x + p y is the range; ν = v = ṗ x + ṗ y is the speed; θ = p = atan p y, p x is the azimuth; α = ṗ p = atan ṗ y, ṗ x atan p y, p x is the difference of the heading angle minus the azimuth, i.e. relative heading. Then, the constant velocity motion model 5-6 over IR 4 is more compactly rewritten over IC as { pt+ = p t + T s ṗ t ṗ t+ = ṗ t Next, the aim is to re-express the constant velocity motion model in the transformed state vector z = [ρ, ν, α, θ] instead of the original state vector x. The geometrical meaning of such transformed coordinates is clearly depicted in fig.. From - and the definitions of the transformed coordinates, the following transformed state equations are obtained: ρ t+ = p t+ = p t + T s ṗ t = ρ t e ı θt + T s ν t e ı θt+αt = ρ t + T s ν t e ı αt e ı θt = ρ t + T s v t cos α t + ı sin α t [ = ρ t + T s ν t cos α t + T s ν t sin α t ] = [ ρ t + T s ν t + T s ρ t ν t cos α t ] ν t+ = ṗ t+ = ṗ t = ν t α t+ = ṗ t+ p t+ = ṗ t p t + T s ṗ t = θ t + α t ρ t + T s ν t e ı αt e ı θt = θ t + α t θ t ρ t + T s ν t e ı αt = α t atan T s ν t sin α t, ρ t + T s ν t cos α t θ t+ = p t+ = p t + T s ṗ t = ρ t + T s ν t e ı αt e ı θt = θ t + ρ t + T s ν t e ı αt = θ t + atan T s ν t sin α t, ρ t + T s ν t cos α t 4 5 Further, in the transformed state, the Doppler observation equation takes the form: y t = λ ν t cos α t 6 Looking at the equations -6 it is immediate to see that the Doppler observation y is not affected at all by the azimuth angle θ, i.e. θ is unobservable from y. As far as the other transformed coordinates ρ, ν, α are concerned, their observability from Doppler observations of a single sensor should be proved either by performing an observability check on the nonlinear state-output map -6, e.g. via differential geometric methods [], or by finding an observation procedure to find ρ, µ, α given y, y,.... Due to lack of space, the observability check will be omitted while an observation procedure [] will be reviewed in the next subsection. Such an observation procedure not only proves the observability of ρ, ν, α from a single Doppler sensor but will also be exploited in the multisensor Doppler-only tracking algorithm proposed in section IV. Following the above derivations and the existence of an observation procedure for ρ, ν, α, it can be stated that the vector z t = ρ t ν t α t θ t [ ] = zo,t, z o,t = zō,t ρ t ν t α t, zō,t = θ t 7 provides a canonical observability decomposition wherein z o,t and zō,t are, respectively, the observable and unobservable states. B. Observation procedure In [] the author proposes a procedure to solve - 6 for {ρ k, ν k, α k } k given y, y, y. As reported in table I, such a procedure denoted by ρ, ν, α = OBSERVATIONy, y, y, T operates as follows: given three Doppler measurements y, y, y from the same sensor and equally spaced in time by T, it returns z o, = [ρ, ν, α ], i.e. the observable state at the intermediate time instant. It is worth pointing out that the relative heading angle α can actually be determined only up to the sign. In fact, it is easy to see from -6 that changing the sign of the initial condition θ while leaving unchanged the other initial conditions, yields the same sequence of measurements. Hence, a single Doppler sensor, besides being unable to observe the azimuth angle θ, cannot distinguish the sign of the relative heading angle α though it can observe its absolute value α. IV. DUAL STAGE FILTERING Let us now address the problem of interest in this paper, i.e. tracking of a single target via multiple Doppler sensors deployed over the surveillance area by means of a centralized approach wherein all Doppler measurements are gathered and processed by a fusion center. It is common experience that an EKF or UKF directly working on the Doppler measurements, fails to provide acceptable tracking performance whenever there is no or imprecise prior knowledge on the target s initial position. On the other hand, particle filters [8], [9] can yield satisfactory performance at the price, however, of a higher computational burden. In order to avoid excessive computational cost, the idea suggested from the previous analysis is to adopt a dual stage filter DSF: in the first stage a Local Filter LF estimates, for each Doppler sensor i D, the observable state z i o,t while in the second stage a Cartesian Tracking Filter CTF fuses all estimates ẑ i o,t, i D, in order to estimate the desired target state x t. The overall dual-stage

4 TABLE I: Observation procedure proposed in [] exploiting Doppler measurements procedure OBSERVATIONy, y, y, T k = y, k = y y y ψ = k k + k k k k γ = acos ψ if y < then if k then π u = cos γ else if < k then u = cos γ else if k < then [ u = ψ [ ψ ] + ψ + ] ψ else if y > then if k then π u = cos γ else if < k then π u = cos + γ else if k < then u = [ ψ ψ ] d k = k +u ν = yλ + d y λ α = acos ν ρ = ν T d + u return: z i o, = [ρ ν α] end procedure [ ψ + ] ψ filter proposed for Doppler-only tracking is schematized in the block-diagram of fig.. The two stages of the DSF, i.e. LF y t Doppler sensor Local filter ẑ o,t, P o,t... Cartesian tracking filter ˆx t, P t... y D t Doppler sensor D Local filter ẑ D o,t, P D o,t Fig. : Block diagram of the dual-stage filter. and CTF, will be described separately in subsections IV-A and IV-B. A. Local filter The task of this filter is to estimate, for each Doppler sensor i D, the observable state z i o,t exploiting Doppler measurements yk i for k =,,..., t. To this end, an UKF [4] could be used, the main problem however being how to properly initialize such a filter in case of no prior knowledge on the initial target position. A possible idea is to initialize ẑ i o,t at time t = given the first three measurements y, i y, i y i exploiting the observation procedure of table I. Unfortunately, however, the observation procedure is highly sensitive to measurement noise in that small errors on the Doppler measurements imply large errors on the observable state components ρ, ν, α, especially on ρ. To overcome this practical difficulty, a multi-hypothesis approach can be undertaken, i.e., by generating in some way several alternative hypotheses for the initial conditions; then, for each hypothesis, the associated estimate, covariance and probability can be propagated by separate filters and the unlikely hypotheses, i.e. the ones with small probabilities, can be pruned. The LF algorithm relative to a generic Doppler sensor i D is reported in Table II. First notice that the LF algorithm starts to run at a suitably chosen time t 4 so that a sufficient number of Doppler measurements is available for initialization of the observable coordinates. Hereinafter, the various steps of the LF will be briefly described.. Initialization - At time t 4, given the available measurements { y, i y, i..., yt } i, the i-th local filter generates multiple hypotheses for the initial condition z i o,t. Each hypothesis, indexed by j H where H is the hypothesis set, is characterized by the estimate z i,j o,t, its covariance Pi,j o,t and a probability µ i,j t. The specific procedure among the many available options adopted in this paper to generate the hypotheses operates through the following steps.. Generation of triplets - All possible triplets of equally and maximally spaced in time Doppler measurements are generated. For instance, if t = 7, the resulting triplets are: y, i y, i y5, i y, i y4, i y6. i. Generation of pseudo-measurements - For each triplet y IR, a set of pseudo-measurements y IR ±mσ y e n is generated, where: m {,,, }, n {,, }, σ y is the standard deviation of Doppler measurements, e n denotes the n-th vector having in position n as unique non-zero element. Notice that in this way, 9 different pseudo-measurements are obtained for a single triplet. A pictorial view of the pseudomeasurement generation is given in fig. 4.. Generation of initial conditions - Each pseudomeasurement yτ k i, yi τ k, yi τ generates an hypothesis for the initial condition z i o,t exploiting the observation procedure and a prediction step from discrete time τ k up to time t..4 Hypothesis probability initialization - For each hypothesis, the probability is initially set equal to the same value, i.e. µ i,j t = / H where H is the cardinality of the hypothesis set.

5 TABLE II: Local filter algorithm for the Doppler sensor i for t = t, t +,... do. INITIALIZATION if t = t then j =. GENERATION OF TRIPLETS t k = for all τ {ϕ N < ϕ k < ϕ k < ϕ < t } do. GENERATION OF PSEUDO-MEASUREMENTS l = j + j = j + ȳ l = [ y i τ k, yi τ k, ] yi τ for m {,,,,, } do for n {,, } do j = j + ȳ j = ȳ l + mσ y e n. GENERATION OF INITIAL CONDITIONS for all triplet r = l to j do ẑ i,r o,τ k = observationȳr, T P i,r o,τ k = diagσρ, σν, σα for all initial conditions r = l to j do ẑ i,r o,t, Pi,r o,t = prediction ẑ i,r o,τ k, Pi,r o,τ k, t τ + k T s.4 PROBABILITY INITIALIZATION H = {,,..., N z} µ i,j t = N z, j H. FILTERING. PREDICTION AND CORRECTION for all j H do ẑ i,j o,t t, Pi,j o,t t ẑ i,j o,t, Pi,j o,t, li,j t = prediction = correction. PROBABILITY UPDATE for all j H do µ i,j t = µi,j t li,j t µ i,r t li,r t r H. HYPOTHESIS PRUNING if t t p then γ µ = { H H = j } µ i,j t γ µ.4 HYPOTHESIS SELECTION r = arg max j H return µ i,j t ẑ i,r o,t, Pi,r o,t ẑ i,j o,t, Pi,j o,t, Ts ẑ i,j o,t t, Pi,j o,t t, yi t, σ i Doppler. Filtering - Recursive filtering starts at time t and operates according to the following steps.. Prediction & correction - The estimate and covariance of each hypothesis are updated, i.e. one-step-ahead predicted with the model -4 and then corrected with the current Doppler measurement yt, i via an UKF. The correction step also provides the hypothesis likelihood defined as follows l i.j t = N e i,j t ;, S i,j o,t where e i,j t = yt i ˆν i,j t t cos ˆαi,j t t is the innovation provided by the filter associated to Doppler sensor i and hypothesis j while N ; m, Σ denotes the normal Gaussian PDF with mean m and covariance Σ.. Hypothesis probability update - The hypothesis probabilities are updated exploiting the previously computed likelihoods.. Hypothesis pruning - Whenever t t p, t p t being a preset time instant, all hypotheses with probabilities below the threshold γ µ = / H are discarded..4 Hypothesis selection - Whenever t t e, t e being a preset time instant at which the CTF starts to run, the most likely hypothesis defined as the one with largest probability, is selected and the associated estimate and covariance are passed to the second, CTF, filtering stage. third Doppler shift measurement second Doppler shift measurement first Doppler shift measurement Fig. 4: Pictorial view of the proposed method to create pseudo measurements. The solid black corresponds to the initial triplet of measurements. The solid red, green and blue represent, respectively, ±, ± and ± values starting from the initial triplet. B. Cartesian tracking filter Starting at the preset time instant t e, the estimates ẑ i o,t = i [ˆρ t, ˆν t, i ˆα t] i and relative covariances [ ] P i R i o,t = t, R i t R from the LFs are used as input measurements and, respectively, measurement noise covariances by the CTF algorithm of table III. Although the CTF is a standard nonlinear filter, e.g. EKF or UKF, operating in the Cartesian coordinates with a linear dynamics and nonlinear measurements, a few remarks on it are in order.

6 TABLE III: Cartesian Tracking Filter algorithm at time t + if t t e then y t = col {ˆρ i t, } ˆνi t i D { } R t = block-diag R i t i D ˆxt t, P t t = predictionˆxt, P t, T s ˆx t, P t = correction ˆx t t, P t t, y t, R t Only the range and speed estimates ˆρ i t and, respectively, ˆν i t are used in the CTF stage, while the relative heading estimates ˆα i t are ignored due to their sign ambiguity. Though the estimates of the various LFs are certainly correlated, they are assumed uncorrelated for the sake of simplicity. V. PERFORMANCE EVALUATION To assess performance of the proposed DSF algorithm described in section IV, a -dimensional planar tracking scenario is considered over a surveillance area of 5 5 [km ], wherein three Doppler sensors are deployed as shown in fig. 5. Four different target trajectories shown in fig. 6 have been considered. m m x 4 x 4 Dopplersensors Surv. Area Doppler 4 5 m Target Target Fig. 5: Network with Doppler sensors. Target 4 5 m Nodes: 4 DOP Target 4 Fig. 6: Target trajectories. The start/end point for each trajectory is denoted, respectively, by / The motion of the target is modeled according to the nearlyconstant velocity model [5] defined by and 6 with zero x 4 x 4 mean process noise w t having covariance: 4 T 4 s T s Q = σw T s Ts 4 T s 4 T s Ts T s where σ w =.[m/s ] and the sampling interval is T s = 5. Doppler sensors are characterized by carrier frequency f c =.9[GHz], i.e. wavelength λ =., and standard deviation of measurement noise equal to σ Doppler =.5[Hz]. No prior knowledge on the target initial position is assumed. Hence, the CTF is initialized with an estimate in the center of the surveillance area, i.e. ˆx = [ 5,, 5, ], and associated covariance matrix P = diag 8, 4, 8, 4. Performance is evaluated for both LFs and the CTF. As far as local filters are concerned, performance is measured for each Doppler sensor i D by the Root Mean Square Error RMSE for the three components respectively range, speed and relative heading angle of the observable state. In particular, the RMSE of the angle α t is evaluated without considering its sign because of the sign ambiguity described in section III-B. For the CTF, performance is evaluated by the Position RMSE PRMSE and Velocity RMSE VRMSE. The reported metrics have been averaged over Monte Carlo trials carried out with different, independently generated, measurement noise realizations for each considered target trajectory. The duration of each simulation trial is fixed to 5 4 samples. The parameters of the DSF have been chosen as follows: the time instant for LF initialization is t = 7, for a total of 8 initial condition hypotheses for each sensor; the starting time of pruning is t p = ; the starting time of CTF is t e = 6; the initial variances for ρ, ν, α are respectively σ ρ =, σ ν = 5[m/s], σ α = [ ]. The performance of DSF is evaluated in the case of Doppler sensors located as shown in fig. 5; notice that at least three non aligned Doppler sensors are required for target observability. Figs. 7-9 display the performance of the three LFs for each component of the observable state ẑ i o,t = [ˆρ i t, ˆν t, i ˆα t] i, i D = {,, }. Here, solid line squares represent averages, at each time instant, over the Monte Carlo trials. Fig. shows the performance of the CTF reporting both PRMSE and VRMSE averaged over all Monte Carlo trials and considered target trajectories As it can be seen from figs. 7-9, each LF is capable of estimating ẑ o,t with high accuracy. In particular the local filters start at time t = 5 t = 7 and hence the first 6 measurements are used to generate the initial conditions; after 8, the range error is in the order of hundreds of meters and constantly decreases reaching an error in the order of tens of meters; speed and relative heading errors start, respectively, with values in the order of few [m/s] and, respectively,

7 5 Range error 5 Range error a Range error. Speed error a Range error. Speed error [m/s] [m/s] b Speed error Relative heading error b Speed error Relative heading error [rad] [rad] c Relative heading error. Fig. 7: Performance of the LF in the Doppler sensor c Relative heading error. Fig. 8: Performance of the LF in the Doppler sensor. [rad] and such errors constantly decrease to lower values. Fig. shows that the CTF is capable of estimating ˆx t with high accuracy. In particular the CTF starts at time t = 8 t e = 6; PRMSE and VRMSE exhibit good performance since the first steps, reaching higher accuracy, respectively in the order of few hundreds of and few tens of [m/s] at t = 9; at steady state, i.e. since time t =, the CTF exhibits remarkable performance with PRMSE and VRMSE in the order of and, respectively,.[m/s]. VI. CONCLUSIONS The main contribution of this paper has been to analyse observability of a target, moving in a plane at constant velocity, from a single Doppler sensor. This analysis has allowed to devise a suitable nonlinear coordinate transformation such that the transformed state is decomposed into observable and unobservable parts. This observability decomposition provides a valuable tool in order to deal in an effective way with the difficult problem of Doppler-only target tracking. In fact, it has been found that a simple nonlinear filter e.g. EKF or UKF operating in such transformed coordinates and equipped with a smart, multi-hypothesis, initialization procedure is capable of accurately estimating the observable state from measurements of a single Doppler sensor. Then, estimates of such observable states obtained from multiple Doppler sensors placed at different locations can be appropriately fused so as to estimate the overall kinematic state of the target of interest. Exploiting these ideas, it has been possible to develop a novel centralized multisensor Doppler-only singletarget tracker featuring a good tradeoff between tracking performance and computational load. Extensions of this work

8 4 Range error Position error a Range error. Speed error a Position RMSE. Velocity error [m/s] [m/s] b Speed error Relative heading error b Velocity RMSE. Fig. : Performance of the CTF with Doppler sensors. [rad] c Relative heading error. Fig. 9: Performance of the LF in the Doppler sensor. are possible in many directions such as, for instance, observability decompositions for other types of target motions e.g., coordinated-turn and/or -dimensional motions, distributed and/or multitarget tracking. ACKNOWLEDGMENTS This work has been partially supported by SELEX ES under research contract COLB/CTR//7/A. REFERENCES [] S.N. Salinger and J. J. Brandstatter, Application of recursive estimation and Kalman filtering to Doppler tracking, IEEE Trans. on Aerospace and Electronic Systems, pp , 97, doi:.9/taes [] A. Farina and F. A. Studer, Radar data processing, vol. I: Introduction and tracking. John Wiley, 985. [] A. Farina and F. A. Studer, Radar data processing, vol. II: Advanced topics and applications. Researches Studies Press, England, 986. [4] Y.-T. Chan and F. L. Jardine, Target localization and tracking from Doppler-shift measurements, IEEE Journal of Oceanic Engineering, pp. 5-57, 99. [5] D. C. Torney, Localization and observability of aircraft via Doppler shifts, IEEE Trans. on Aerospace and Electronic Systems, pp. 6-68, 7. [6] I. Shames, A. N. Bishop, M. Smith and B. D. O. Anderson, Analysis of target velocity and position estimation via Doppler-shift measurements, Australian Control Conference AUCC, pp. 57-5,. [7] M. B. Guldogan, D. Lindgren, F. Gustafsson, H. Habberstad and U. Orguner, Multiple target tracking with Gaussian mixture PHD filter using passive acoustic Doppler-only measurements, 5th Int. Conf. on Information Fusion FUSION, pp. 6-67,. [8] A. Farina and B. Ristic, Recursive Bayesian state estimation from Doppler-shift measurements, Seventh Int. Conf. on Intelligent Sensors, Sensor Networks and Information Processing ISSNIP, pp ,. [9] A. Farina and B. Ristic, Joint detection and tracking using multi-static Doppler-shift measurements, IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, pp ,. [] C. Fantacci, G. Battistelli, L. Chisci, A. Farina and A. Graziano, Multiple-model algorithms for distributed tracking of a maneuvering target, 5th Int. Conf. on Information Fusion FUSION, pp. 8-5,. [] B. Ristic, B.-T Vo, B.-N. Vo and A. Farina, A tutorial on Bernoulli filters: Theory, particle implementation and applications, IEEE Trans. Signal Processing, doi:.9/tsp , to appear. [] S. Sastry, Nonlinear systems: analysis, stability and control. Springer, 999. [] R. J. Webster, An exact trajectory solution from Doppler shift measurements, IEEE Trans. on Aerospace and Electronic Systems, pp. 49-5, 98. [4] S. J. Julier and J. K. Uhlmann, Unscented filtering and nonlinear estimation, Proceedings of the IEEE, pp. 4-4, 4. [5] Y. Bar-Shalom, X. Li and T.Kirubarajan, Estimation with applications to tracking and navigation. John Wiley,.