Tracking and Localizing Moving Targets in the Presence ... - IEEE Xplore
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Tracking and Localizing Moving Targets in the Presence of Phase Measurement Ambiguities Yongqiang Cheng, Xuezhi Wang, Terry Caelli, Fellow, IEEE, and Bill Moran, Member, IEEE
Abstract—When tracking a target using phase-only signal returns, range ambiguities due to the modulo 2 in measured signal phases are a major issue. Standard approaches to such problems would typically involve the use of Diophantine equations. In this paper, a simpler and robust solution is examined which uses look-up tables defined between the phase measurement and target location spaces to determine the phase measurement mapping. We show, first, that when the target motion is significant between data sampling intervals the location ambiguity can be resolved over time via known target-in-cluster tracking techniques. This method determines the optimal allocation of location with respect to phase measurements relative to the quantization of look up table values. Second, when the target is undergoing micromotions (jitter) which results in the same collection of candidate locations from phase measurements over time, the location ambiguity can be resolved using a novel phase distribution discrimination method. In this method a probability density function of the ambiguous phase-only measurement is derived that takes both sensor noise and target motion distributions into account based on directional statistics. Optimal locations are inferred from such distributions. Examples are given to demonstrate the effectiveness of these proposed methods. Index Terms—Data association-based filtering, distributed sensor localization, phase-only measurement, range ambiguity, target tracking.
I. INTRODUCTION
ANY measurement methods in physics and engineering result in a set of ambiguous phase signals from which the unknown quantity (for example, a distance or angle) has to be calculated by a combination of phase measurements [1]. While phase measurements provide higher accuracy than time
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Manuscript received November 17, 2010; revised March 10, 2011; accepted April 26, 2011. Date of publication May 10, 2011; date of current version July 13, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ta-Hsin Li. This work was supported in part by the Melbourne Systems Laboratory, Department of Electrical and Electronic Engineering, the University of Melbourne and by the Queensland Research Laboratory, National ICT Australia (NICTA). Y. Cheng is with the School of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan, China (e-mail: [email protected]). X. Wang and B. Moran are with the Department of Electrical and Electronic Engineering, University of Melbourne, Australia (e-mail: [email protected]; [email protected]). T. Caelli is with the Victoria Research Laboratory, National ICT Australia. He is also with the Department of Computer Science and Software Engineering, University of Melbourne, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2011.2152399
delay measurements, the problem of integer ambiguity of the unknown quantity is inevitably introduced [2]. Specially, in the ranging/positioning applications of sensor networks, the ranging techniques comprised of two or more frequencies known as dual-frequency or multiple-frequency continuous waves (MFCW) [3], [4] are often employed to increase the maximum unambiguous detection range. An example of such applications is the Radio Interferometric Positioning System (RIPS) [5], [6], which utilizes frequency interference by measuring the relative phase offset of the interference signal in multiple frequencies to obtain a combination of distance differences among four motes. Similar approaches have been applied to the Global Positioning System (GPS) for high resolution positioning in differential GPS [7]. However, it is impossible to extend the maximum unambiguous detection range while, at the same time, improving ranging accuracy [8], [9]. More complex techniques such as stagger frequency differences may be adopted to deal with such limitations but these approaches present enormous practical challenges in terms of computational and constructional cost, sensing ability, and accuracy when applied to the resource-limited wireless sensors [10]. Single frequency continuous waves greatly facilitate the hardware configuration of wireless sensors where high frequency continuous wave returns from a target are often available at different locations. When they are used to measure the distances between the sensors and target, multiple values can be identified as a result of the received signal phase shifts modulo . The measurement model involving integer ambiguities of target locations can be ascribed to a class of Diophantine Equations [11]. A Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems usually have fewer equations than unknown variables and involve finding the integers that satisfy all equations [12], [13]. The problem is also known as the ambiguity due to phase wrapping in the signal processing community [14]. There are several approaches to resolve the Diophantine Equations caused by the underlying measurement model of sensor networks. For example, Wenzler [1] proposed a nonlinear processing approach based on the nonius principle in order to achieve an optimized determination of the unknown quantity from ambiguous measurements. Using this method all possible solutions to the Diophantine equations can be found. However, in the context of tracking or localizing a target, solutions to the Diophantine equations cannot always remove the ambiguity of target locations. Other approaches to solving ambiguous phase measurement is phase unwrapping such as [14] where the frequency and phase of unknown signal are found by applying lattice theory and the
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wrapping number of the phase is estimated by a least square estimator, and measurement density modeling methods for identifying source trajectories as discussed in [15]. This latter technique has similarities to the proposed approach for moving targets. Nevertheless, the way that we considered to solve the phase ambiguity problem is quite different from the cited works. In particular, for jittering target we consider it as a discrimination problem to identify the true target location. The main focus of this paper is to deal with the ambiguities caused by phase measurements and to elucidate how to identify and remove these ambiguities in tracking and localization context. Specifically, we combine the above approaches in terms of creating mappings between target location and phase measurement spaces so that the nonlinear and indeterminate Diophantine problem reduces to the acquisition of a finite set of possible target locations over a region of interest. We then present two tracking/localization methods which are able to resolve the ambiguity of target location in the context of: 1) tracking a target with significant motion; 2) localizing a target of random vibration or micromotion (jitter) which is described by a probability distribution. In the former case we show that the target location ambiguities can be resolved via the techniques of multitarget tracking in clutter. In the latter case a probability density discrimination method is proposed to resolve the ambiguities in target location space. The analytical phase measurement distribution for a fluctuating target is derived and an inference method for prediction target locations is developed. Following the Introduction, the range ambiguity problem caused by measuring signal phases in the context of target tracking is described in Section II. The look-up table which maps phase measurements into target location space is introduced in Section III. In Section IV, we demonstrate how the location ambiguities due to phase-only measurements can be resolved for moving target tracking. The approach for localizing a target of jitter described by a distribution is given in Section V, where the derivation of phase measurement distribution, the localization procedure and technical justification of the proposed method are presented. Finally, conclusions are given in Section VI. II. PROBLEM Consider the problem of tracking a target using phase-only measurements as shown in Fig. 1. A continuous wave of single frequency , which can be modeled as (1) is transmitted by each of the three sensors. The received return signal at the sensor from the target with range is represented by (2) where the signal phase is measured by the sensor, and is the speed of propagation. Denoted by , the target state at time will typically consist of position and velocity components, although only target location is considered in this work.
Fig. 1. Tracking a target (x; y ) with three distributed continuous wave phaseonly sensors located at ( ; ), i = 1, 2, 3. The distances between two adjacent circles correspond to a single wavelength.
The target dynamics are assumed to follow a Markov process subject to a random fluctuation , i.e. (3) repwhere is the system transition (dynamical) model and resents process noise, which can often be usefully approximated . by a Gaussian distribution with zero mean and covariance The measurement of the system at time is modeled as the noisy phase vector of return signals transmitted by the three active sensors, i.e. (4) where
,
,
1, 2, 3, and
(5) describes the relationship between the measurewhere and the target state with the th component repment resents the phase of the return signal transmitted from and reis the ceived at the th sensor, and measurement noise, which is a zero mean wrapped Gaussian . , , 2, 3 denotes distribution with covariance the location of three sensors and mod is the modulo operation. Without loss of generality, the frequencies of the three sensors in this paper are assumed to be the same. One of the most critical problems in tracking and localization applications in such a measurement system is the ambiguity of the measurement due to the modulo of , which results in a collection of possible target locations. III. LOOK-UP TABLE Equation (5) indicates that for a given target state, a unique vector valued phase measurement can be found while a single phase measurement may correspond to a finite set of target locations in the region of interest. In this paper, a mapping between and phase measurement space target location space
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Fig. 3. Target location ambiguities. A phase measurement 8 generated by a target at location A corresponds to a set of possible target locations.
Fig. 2. (a) Look-up table, where the value of a phase measurement vector is represented by the color/gray scale of the “resolution grid.” (b) The phase measurement space that corresponds to the target location space in (a).
for a given “resolution grid” in the region of interest is proposed in look-up table form. An example of the phase measurement look-up table is given in Fig. 2(a), where at each target location “cell” the corresponding vector valued phase measurement is represented in the color/gray scale of the cell. Clearly, a phase measurement (vector) can be uniquely identified by its color/gray scale. Multiple locations may be found from the look-up table for a given phase measurement (color). The average resolution of the measurement corresponding to a look-up table is related to the size of grid in the target location space, i.e. (6) is the grid value. where Remarks: 1) The look-up table is defined by the mapping via the in. As shown in Fig. 2(a), verse function of (5), i.e., a particular vector valued phase measurement is mapped into a set of ambiguous locations of identical colors in the look-up table. Around each of these locations, the “color distribution” may not be the same as that of other locations in the set. This indicates that if the underlying target mo-
tion is significant (rather than just jittering), the resulting set of ambiguous locations will not all follow the target trajectory but “jump” to the locations of identical colors with the target true location. 2) The finer the grid of the look-up table is, the higher the target localization accuracy will be. However, this will not alter the ambiguity of target location problem. corre3) In general, a vector valued phase measurement sponds to a set of possible target locations and they can be identified from the look-up table. For example, in Fig. 3 corresponds to a phase a target located at . Multiple locations at which the target measurement yields the same phase measurement are found by using the look-up table so providing an alternative solution to the phase Diophantine approach.
IV. MOVING TARGET TRACKING When the underlying target motion (with known dynamics) is significant, large variations in the possible target location measurements occur from the observed phase sequence over time. In this case the target trajectory can be estimated via target tracking in clutter techniques such as the Probabilistic Data Association Filter [16]. The principle of measurement ambiguity removal via data association is illustrated in Fig. 4, where the of the target at time is vector valued phase measurement , mapped into a set of possible target locations via the look-up table of , while the measurement at time results in another set of possible target locations , and so on. Only the trajectory connected by the true target locations over time will be consistent with the underlying target kinematic dynamics. An example of tracking a target of constant acceleration is given below. In this case, the location of the target at is described by (7)
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Fig. 4. Illustration of target location ambiguity removal via data association.
where the initial state of the target is , the velocity and acceleration of the target are and , respectively. In this example the standard deviation of the system process noise in (7) is and the standard deviations of measurement noise in (5) rad. is A sequence of phase measurements in 50 scans is mapped into the target location space via the look-up table as illustrated in Fig. 5, where the phase measurements of the first five scans ,” respectively. are marked in order with symbols “ The underlying target trajectory can be clearly identified from the mapped location measurements over time. Fundamentally, in the target location space the distributions of target measurements and “clutter” measurements are different and can be differentiated via stochastic filtering techniques, such as particle filters [17] which is used to deal with nonlinear non-Gaussian like the underlying problems. Many other nonlinear filters are available in the literature like [18], [19]. Therefore, once phase data is mapped into target location space in terms of virtual location data, conventional multitarget tracking techniques can be used to resolve the phase measurement ambiguity and estimate the target trajectories. V. LOCALIZATION OF MICROMOTION TARGETS When the underlying target is jittering (micromotion), the ambiguous target locations mapped from a sequence of phase measurements can no longer be resolved via a target tracking technique because the yield set of possible target locations essentially have no significantly delineated change over time.
In this case phase measurement ambiguities can also be resolved using a different approach. First, we demonstrate the situation using a target of motion type described by a Gaussian distribution. Fig. 6(a) shows three possible target locations which have the same phase measurement. When the target motion at these locations follows identical Gaussian distributions, as illustrated in Fig. 6(b), the phase measurement distributions are different and distinguishable. Another example is illustrated in Fig. 7, where a target has identical microcircular motions at all ambiguous locations in the target location space and the resulting phase measurement trajectories in phase measurement space can be clearly identified. Inspired by the above observations we derive a phase measurement noise distribution by taking target “noise” into account, which serves as “ground truth” in the target localization process to compare with the one summarized from the received phase measurement sequence. The target location ambiguity problem can then be resolved using one of the density based distance measures such as the Kullback–Leibler divergence (KLD) by identifying the location of minimum density distance to its “ground truth” from the location candidate collection. A. Phase Measurement Distribution be the target micromotion distribution Let such that the underlying target state at is given by (8)
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Fig. 5. Demonstration of resolving phase measurement ambiguity via filtering technique with data association in the target location domain, where in (b) the possible target locations due to the first five phase measurements are marked with symbols “ ; ; ; ; >,” respectively. (a) Accumulated phase measurements in 50 scans. (b) Accumulated possible target locations mapped from (a).
+2
3
Fig. 6. Demonstration of resolving target location ambiguity in phase measurement distribution discrimination. (a) A target fluctuating in the same distribution at ambiguous locations. (b) Phase measurement distributions.
where is 2 2 identity matrix. The phase measurement model (4) is then written as
(10) where
and
are independent and (11)
(9)
where , , 2, 3 take values of integer due to the modulo . operator and As we will see later, the distribution of the phase measurement (9) is a wrapped Gaussian. To derive this distribution, we first consider the range term in (9)
It is well known that for a fluctuating target with its motion modeled as a zero-mean Gaussian distribution, the range mea, 2, 3 between the sensor and target are of surements , Rice distribution. The vector valued range of the three sensors is a multivariate distribution with correlated components and so it is impossible to obtain an exact analytical expression of this distribution. However, in view of the fact that the target “noise” is usually much smaller compared to the range itself, a Taylor expansion can be used to approximate the range measurement.
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and in (12) The weighted sum of the two distributions and variance is also a Gaussian distribution and its mean are given by (13) (14) Therefore, the range measurement can be approximated by a Gaussian distribution (15) The covariance between
and
can be calculated as
Cov
(16) Taking the correlations between the noise from three range measurements into account, the distribution of the range measurement (10) can be approximated as a multivariate Gaussian (17) where (18) (19) where the correlation coefficient between Fig. 7. Measurements of a circular fluctuating target. (a) Identical motions are performed at all ambiguous locations. (b) The measured phase measurement trajectories corresponding to target motion in (a).
For an asymptotic consideration, the high order terms of the Taylor expansion are negligible and will be ignored and the expansion of range measurement can be written as
and
are given by
(20) As a consequence, the first term in the phase measurement equation (9) is approximately given by a multivariate Gaussian distribution as (21) where
and the phase measurement (9) is (22)
(12)
where , 2, 3 and distances between the th sensor and the target.
are the true
Remark: • The covariance matrix in (19) is not full rank with one zero eigenvalue, which means that the density of range measurement has only two degrees of freedom. In fact, due to the correlation of the noise, the range measurements can be represented as points on a parameterized surface in the measurement space. • When the range is modulo , all the ambiguities in the will have the same sample target location space mean with the one at true location but different covariance matrix. Due to modulo operation, the phase measurement becomes a wrapped distribution well-known as the directional or circular
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Fig. 8. Probability density of wrapped distribution. Top: Probability density function p(x). Middle: The wrapped Gaussian distribution p (#) defined in (25). Bottom: Directional representation of p (#).
data. Directional data may be visualized as points on a surface, a hypersphere, or in two dimensions on the circumference of a circle, while the conventional data can be regarded as linear data [20]. The corresponding statistics are called directional statistics which are mainly concerned with observations which are unit vectors in the plane or in three-dimensional spaces. Thus the sample space is typically a circle or a sphere. It turns out that standard methods for analyzing linear univariate or multivariate measurement data cannot be used [21]. There are fundamental differences between linear and directional statistics, such as the calculation of circular mean of samples from the circular data. Physicists and statisticians have developed a methodology for dealing with statistics of directional data. Most recent description can be found in the book of Mardia [21], [22]. of a linear variAny given probability density function able on the line can be wrapped around the circumference of a circle of unit radius [20]. That is, the probability density funcof the wrapped variable , where tion (23) is
Fig. 9. An example of the wrapped Gaussian distribution.
For a univariate Gaussian distribution univariate Gaussian distribution denoted as fined as [20]
, the wrapped is de-
(25) When , the overlap of neighboring Gaussian wraps in (25) is negligible. In this case, it is permissible to approximate by only one, but the most meaningful wrap of it [20]
(24) (26) Fig. 8 illustrates probability density of the wrapped distribution. There are two circular distributions that should serve as appropriate substitutes for the univariate linear normal distribution. One is the wrapped Gaussian distribution, and the other the von Mises distribution [23]. As the univariate wrapped distribution can be easily extended to the multivariate context, so the wrapped Gaussian distribution is the more natural choice for the problem.
For the case
,
can be approximated as [24] (27)
Fig. 9(a) shows an example of the wrapped Gaussian distribut different bution with identical standard deviations , 2.5, 4.5, respectively. Fig. 9(b) shows the parameter
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example of the wrapped Gaussian distribution with identical pabut different standard deviations , 0.8, rameters 1.5, respectively. The concept of a univariate wrapped distribution can be extended to the multivariate case by an extension of the single sums that cover all disimple sum in (24) to a number of mensions in the feature space, i.e.
(28) is the th Euclidean basis vector. where , , with a For a multivariate Gaussian distribution , where denotes the square root small covariance of matrix , and the identity matrix, the wrapped multivariate can be approximated as Gaussian distribution
(29) According to the above analysis, the phase measurement can be approximated by a wrapped multivariate Gaussian distribution (30) The measurement noise , which is independent of target noise, can also be approximated by a zero-mean wrapped , i.e. Gaussian distribution with covariance (31) In summary, the distribution of phase measurement (9) with both small target noise and sensor noise can be approximated by a wrapped multivariate Gaussian distribution of the form (32) Discussions: 1) The covariance matrix is given by
of the phase measurement
(33)
and it becomes full rank in the presence of sensor noise. 2) For a fluctuating target, the distribution of the phase measurement (32) is of the same mean but different covariance at all ambiguous target locations. In fact, the correlation coefficients in (33) are unique for all locations in the target location space. Such uniqueness can be regarded as target location dependent signatures to be used to discriminate the true target location from the ambiguous location candidates. This uniqueness of the phase measurement distribution is justified as follows. Suppose that
Fig. 10. Illustration of the tensor field constructed by 6( ) in the target location space of interest. The plots demonstrate the uniqueness of 6 ( ) for every target locations. (a) [v ; v ] versus . (b) [v ; v ] versus . (c) [v ; v ] versus .
is the eigenvector with maximal eigenvalue of the covari. It is understood that the direction of the ance matrix
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Fig. 11. Procedure of the proposed target localization method based on ambiguous phase measurements.
eigenvector represents the main direction of the ellip. The tensor fields soid corresponding to the matrix of , and versus are plotted in Fig. 10(a), (b), and (c), respectively. Clearly, the values of the tensor are different for all states and indicate the cois unique at all locations in the region. variance matrix 3) Fundamentally, the uniqueness of the covariance matrix reflects the fact that when a micromotion target is at different locations, the information collected from a sequence of phase measurements will be different everywhere in the location plane. However, such differences will tend to be smaller for locations far from the sensors where the measurement noise (31) will dominate the entire noise term. 4) Knowledge of target noise is not necessarily required and as we will see that it can be estimated from data. From (33) are identical for we can find that the diagonals of all possible locations with the same circular mean , the difference only lies in the off-diagonals for true location and ambiguities. So we can use the diagonal elements of to estimate the standard deviation of the target noise, i.e.
can be from a sequence of phase measurements established by the following steps: Step 1) Estimate sample mean and covariance from the received measurement sequence. According to [21], the circular mean and covariance are estimated as (35) (36) . where Step 2) Estimate the target noise variance using (34). possible target locations from the Step 3) Find the set of look-up table, i.e. (37) Step 4) Calculate the theoretical phase measurement distributions for each of the possible target locations via (32):
(34) (38) where
is the sample covariance of phase measurement.
B. Target Localization Procedure and an Illustrative Example As shown in Fig. 11, the procedure of the proposed method for estimating the location of a target with fluctuated motion
Step 5) Calculate KLD between each of the theoretical phase measurement distributions and the sample distribution, i.e. (39)
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Fig. 12. Case 1: ambiguity discrimination using measurement density signature.
Fig. 13. Case 2: ambiguity discrimination using measurement density signature.
where is the distribution summarized from measurements. In our case, a closed expression of (39) is given
rameters and calculated results are listed in Table I. The computed KLDs are marked on the corresponding figures. In both situations, the computed minimum KLDs pointed to the correct target locations and thus resolved target location ambiguities. We should point out that the proposed method is also for small ). number of samples (say,
(40) and the correct target location is found by selecting the one with minimum KLD, i.e. (41) To demonstrate the effectiveness of the proposed distribution discrimination method, an example of localizing a fluctuating target is given. As shown in Figs. 12 and 13, the underlying target was in two such locations that the generated phase measurements give the same set of possible target locations. Pa-
C. Statistical Results The robustness of the proposed localization method for target of micromotion is investigated via Monte Carlo statistics. Parameters used for the Monte Carlo runs are same as those listed in Table I. The true location of the underlying target is assumed to be uniformly distributed across the region of interest whose boundary is shown in Fig. 13. A successful run is counted if the target location is estimated correctly, i.e., the minimum KLD between a theoretical measurement density and the one estimated from measurements is corresponding to the ground truth in this run. In Table II, statistical values (mean and standard deviation) of normalized KLDs between the estimated distribution (from
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TABLE I RESULTS OF THE TWO SCENARIOS SHOWN IN FIGS. 12 AND 13
TABLE II NORMALIZED DISTANCES COMPARISON AVERAGED OVER 100 MONTE CARLO RUNS
samples) and theoretical distributions computed via (33) at all possible target locations are given. These values are averaged from 100 Monte Carlo runs where the ground truth target location are uniformly generated over the region of interest. For a comparison, the Euclidean distances between the estimated and all possible target locations are also provided. Note that these statistics (Table II) only count successful runs (98 out of 100). The successful rate versus the number of samples used at each run for estimating the target measurement density via (35) and (36) is illustrated in Fig. 14, which has evidenced that as the number of samples used to estimate the target measurement distribution increased, a higher probability of correctly estimating the target location will be achieved. As indicated in Table II, the KLDs for measurement at target true and false locations can be well separated and the average localization error is quite small in this example. VI. CONCLUSION In this paper, estimating the trajectory/location of a target using phase-only measurements observed by three sensors in different locations is considered, where phase ambiguity caused by phase wrapping is a challenging issue. Methods for resolving the phase ambiguities are proposed. First, the phase wrapped measurement is mapped into target location space via a look-up table. We then resolve the target location ambiguity in two cases. For targets in significant motion, the technique of stochastic filtering with data association is suggested. For targets with jitter (micromotion), a distribution discrimination method is proposed and the probability density function of the
Fig. 14. Rate of successful localization of a target of micromotion versus the number of samples used for estimating measured distribution at each run. The plot indicates that as the number of samples used for estimating target measurement density increased, the probability of successful target localization will approach to one.
phase measurement is derived. The effectiveness and robustness of these methods are demonstrated. A challenging and interesting issue is to localize multiple jittering targets using phase-only measurements. We will address this issue in the future research work. From Fisher information point of view, the accuracy of tracking/localization can be improved if more than three sensors are considered in the presence of measurement noise. The current work can be straightforwardly extended to the case of using more than three sensors. As the dimension of phase measurement increased, the overall computational complexity, such as building and searching the look-up table in the region of interest as described in Section III and computing measurement distributions via (33), is expected to rise accordingly. However, the number of ambiguous target locations from a phase measurement will be less. REFERENCES [1] A. Wenzler and S. Steinlechner, “Nonlinear processing of n-dimensional phase signals,” in Proc. IEEE Int. Symp. Circuits Syst. (ISCAS), 2002, pp. I-805–I-808. [2] C. Wang, Q. Yin, and W. Wang, “An efficient ranging method for wireless sensor networks,” presented at the 2010 IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP), Dallas, TX, Mar. 14–19, 2010. [3] W. D. Boyer, “A diplex, Doppler, phase comparison radar,” IEEE Trans. Aerosp. Navigat. Electron., vol. ANE-10, no. 1, pp. 27–33, Mar. 1963. [4] I. Urazghildiiev, R. Ragnarsson, and A. Rydberg, “High-resolution estimation of ranges using multiple-frequency CW radar,” IEEE Trans. Intell. Transport. Syst., vol. 8, no. 2, pp. 332–339, Jun. 2007. [5] M. Maroti, B. Kusy, G. Balogh, P. Volgyesi, K. Molnar, A. Nadas, S. Dora, and A. Ledeczi, “Radio interferometric positioning,” Inst. Software Integr. Syst., Vanderbilt Univ., Nashville, TN, Tech. Rep. ISIS-05-602, Nov. 2005. [6] B. Kusy, A. Ledeczi, M. Maroti, and L. Meertens, “Node-density independent localization,” presented at the 5th Int. Conf. Inf. Process. Sens. Netw. (IPSN’06), Nashville, TN, Apr. 19–21, 2006. [7] G. L. Mader, “Rapid static and kinematic global positioning system solutions using the ambiguity function technique,” J. Geophys. Res., vol. 97, no. B3, pp. 3271–3283, 1992.
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[8] M. L. Skolnik, Introduction to Radar Systems, 2nd ed. New York: McGraw-Hill, 1980. [9] P. Setlur, M. Amin, and F. Ahmad, “Cramer-Rao bounds for range and motion parameter estimations using dual frequency radars,” presented at the 2007 IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP), Honolulu, HI, Apr. 15–20, 2007. [10] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero, III, R. L. Moses, and N. S. Correal, “Locating the nodes—Cooperative localization in wireless sensor networks,” IEEE Signal Process. Mag., pp. 54–69, Jul. 2005. [11] L. J. Mordell, Diophantine Equations. New York: Academic, 1969. [12] H. Cohen, Number Theory: Volume I: Tools and Diophantine Equations. New York: Springer, 2007. [13] J. Steduing, Diophantine Analysis. Boca Raton, FL: Chapman & Hall/CRC, 2005. [14] R. G. McKilliam, B. G. Quinn, I. V. L. Clarkson, and B. Moran, “Frequency estimation by phase unwrapping,” IEEE Trans. Signal Process., vol. 58, no. 6, pp. 2953–2963, 2010. [15] P. Smaragdis and P. Boufounos, “Learning source trajectories using wrapped-phase hidden Markov models,” in Proc. IEEE Workshop Appl. Signal Process. Audio Acoust., New Paltz, NY, Oct. 2005, pp. 114–117. [16] Y. Bar-Shalom and T. E. Fortmann, Tracking and Data Association. New York: Academic, 1988. [17] M. S. Arulampalam, S. Mansell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process., vol. 50, no. 2, pp. 174–188, Feb. 2002. [18] M. Orton and W. Fitzgerald, “A Bayesian approach to tracking multiple targets using sensor arrays and particle filters,” IEEE Trans. Signal Process., vol. 50, no. 2, pp. 216–223, Feb. 2002. [19] C. Kreucher, K. Kastella, and A. O. Hero, III, “Multitarget tracking using the joint multitarget probability density,” IEEE Trans. Aerosp. Electron. Syst., vol. 41, no. 4, pp. 1396–1414, Oct. 2005. [20] C. Bahlmann, “Directional features in online handwriting recognition,” Pattern Recogn., vol. 39, pp. 115–125, 2006. [21] K. V. Mardia and P. E. Jupp, Directional Statistics 2nd Edition. Chichester, U.K.: Wiley, 2000. [22] K. V. Mardia and P. E. Jupp, Statistics of Directional Data. New York: Academic, 1972. [23] N. I. Fisher, Statistical Analysis of Circular Data. Cambridge, U.K.: Cambridge Univ. Press, 1995. [24] B. W. Church and D. Shalloway, “Characterizing large correlated fluctuations of macromolecular conformations in torsion-angle space using the multivariate wrapped-Gaussian distribution,” Polymer, vol. 37, pp. 1805–1813, 1996.
Yongqiang Cheng received the B.S. and M.S. degrees in information and communication engineering from National University of Defense Technology, Changsha, China, in 2005 and 2007, respectively. He is currently pursuing the Ph.D. degree from the National University of Defense Technology. From September 2009 to November 2010, he was a visiting research student with Melbourne Systems Laboratory, University of Melbourne, Australia. His research interests lie in the areas of information geometry and statistical signal processing.
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Xuezhi Wang received the B.S. degree in avionics from Northwest Polytechnical University, Xian, China, in 1982 and the Ph.D. degree in signal and systems from the Department of Electrical and Electronic Engineering, University of Melbourne, Australia, in 2001. He has been involved in radar and sonar signal processing research and development since 1983. He is now a Research Fellow with the Melbourne Systems Laboratory, Department of Electrical and Electronic Engineering, University of Melbourne. His research interests are in stochastic signal processing, information theory, Bayesian estimation, data fusion, and Situation assessment.
Terry Caelli (SM’92–F’02) received the B.S. (Hons.) degree in mathematics and psychology and the Ph.D. degree in human and machine vision from the University of Newcastle, Australia. Current interests lie in Signal Processing, Computer Vision and Machine Learning and their applications in Health and the Environment. He is Director of National ICT Australia’s (NICTA) Health Business Area. He is a Fellow of the International Association for Pattern Recognition (IAPR). He is also a Convocation Medalist from the University of Newcastle. He has spent 15 years in North American universities and research institutes, has been a DFG Professor, Germany, Killam Professor of Science, the University of Alberta, Canada, as well as having held senior positions in Australia at the University of Melbourne and Curtin University of Technology. He returned from Canada to join NICTA in 2004. Professor Caelli has served on the editorial boards of journals including the IEEE JOURNAL ON PATTERN RECOGNITION and numerous international conference committees in both human and machine vision. See http://www.nicta. com.au/people/caellit.
Bill Moran (M’95) received the B.S. (Hons.) degree in mathematics from the University of Birmingham in 1965 and the Ph.D. degree in mathematics from the University of Sheffield, London, in 1968. He is a professor of electrical engineering with the University of Melbourne, Australia, where he is the Research Director of Defence Science Institute and Technical Director of Melbourne Systems Laboratory. Previously, he was a professor of mathematics with the University of Adelaide and Flinders University. He also serves as a Consultant to the Australian Department of Defence through the Defence Science and Technology Organisation. His research interests are in signal processing, particularly with radar applications, waveform design and radar theory, and sensor management. He also works in various areas of mathematics, including harmonic analysis and number theory and has published widely in these areas. Dr. Moran has been a member of the London Mathematical Society since 1967, of the Australian Mathematical Society since 1976, and of the American Mathematical Society since 1978.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 8, AUGUST 2011
Tracking and Localizing Moving Targets in the Presence of Phase Measurement Ambiguities Yongqiang Cheng, Xuezhi Wang, Terry Caelli, Fellow, IEEE, and Bill Moran, Member, IEEE
Abstract—When tracking a target using phase-only signal returns, range ambiguities due to the modulo 2 in measured signal phases are a major issue. Standard approaches to such problems would typically involve the use of Diophantine equations. In this paper, a simpler and robust solution is examined which uses look-up tables defined between the phase measurement and target location spaces to determine the phase measurement mapping. We show, first, that when the target motion is significant between data sampling intervals the location ambiguity can be resolved over time via known target-in-cluster tracking techniques. This method determines the optimal allocation of location with respect to phase measurements relative to the quantization of look up table values. Second, when the target is undergoing micromotions (jitter) which results in the same collection of candidate locations from phase measurements over time, the location ambiguity can be resolved using a novel phase distribution discrimination method. In this method a probability density function of the ambiguous phase-only measurement is derived that takes both sensor noise and target motion distributions into account based on directional statistics. Optimal locations are inferred from such distributions. Examples are given to demonstrate the effectiveness of these proposed methods. Index Terms—Data association-based filtering, distributed sensor localization, phase-only measurement, range ambiguity, target tracking.
I. INTRODUCTION
ANY measurement methods in physics and engineering result in a set of ambiguous phase signals from which the unknown quantity (for example, a distance or angle) has to be calculated by a combination of phase measurements [1]. While phase measurements provide higher accuracy than time
M
Manuscript received November 17, 2010; revised March 10, 2011; accepted April 26, 2011. Date of publication May 10, 2011; date of current version July 13, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ta-Hsin Li. This work was supported in part by the Melbourne Systems Laboratory, Department of Electrical and Electronic Engineering, the University of Melbourne and by the Queensland Research Laboratory, National ICT Australia (NICTA). Y. Cheng is with the School of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan, China (e-mail: [email protected]). X. Wang and B. Moran are with the Department of Electrical and Electronic Engineering, University of Melbourne, Australia (e-mail: [email protected]; [email protected]). T. Caelli is with the Victoria Research Laboratory, National ICT Australia. He is also with the Department of Computer Science and Software Engineering, University of Melbourne, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2011.2152399
delay measurements, the problem of integer ambiguity of the unknown quantity is inevitably introduced [2]. Specially, in the ranging/positioning applications of sensor networks, the ranging techniques comprised of two or more frequencies known as dual-frequency or multiple-frequency continuous waves (MFCW) [3], [4] are often employed to increase the maximum unambiguous detection range. An example of such applications is the Radio Interferometric Positioning System (RIPS) [5], [6], which utilizes frequency interference by measuring the relative phase offset of the interference signal in multiple frequencies to obtain a combination of distance differences among four motes. Similar approaches have been applied to the Global Positioning System (GPS) for high resolution positioning in differential GPS [7]. However, it is impossible to extend the maximum unambiguous detection range while, at the same time, improving ranging accuracy [8], [9]. More complex techniques such as stagger frequency differences may be adopted to deal with such limitations but these approaches present enormous practical challenges in terms of computational and constructional cost, sensing ability, and accuracy when applied to the resource-limited wireless sensors [10]. Single frequency continuous waves greatly facilitate the hardware configuration of wireless sensors where high frequency continuous wave returns from a target are often available at different locations. When they are used to measure the distances between the sensors and target, multiple values can be identified as a result of the received signal phase shifts modulo . The measurement model involving integer ambiguities of target locations can be ascribed to a class of Diophantine Equations [11]. A Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems usually have fewer equations than unknown variables and involve finding the integers that satisfy all equations [12], [13]. The problem is also known as the ambiguity due to phase wrapping in the signal processing community [14]. There are several approaches to resolve the Diophantine Equations caused by the underlying measurement model of sensor networks. For example, Wenzler [1] proposed a nonlinear processing approach based on the nonius principle in order to achieve an optimized determination of the unknown quantity from ambiguous measurements. Using this method all possible solutions to the Diophantine equations can be found. However, in the context of tracking or localizing a target, solutions to the Diophantine equations cannot always remove the ambiguity of target locations. Other approaches to solving ambiguous phase measurement is phase unwrapping such as [14] where the frequency and phase of unknown signal are found by applying lattice theory and the
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wrapping number of the phase is estimated by a least square estimator, and measurement density modeling methods for identifying source trajectories as discussed in [15]. This latter technique has similarities to the proposed approach for moving targets. Nevertheless, the way that we considered to solve the phase ambiguity problem is quite different from the cited works. In particular, for jittering target we consider it as a discrimination problem to identify the true target location. The main focus of this paper is to deal with the ambiguities caused by phase measurements and to elucidate how to identify and remove these ambiguities in tracking and localization context. Specifically, we combine the above approaches in terms of creating mappings between target location and phase measurement spaces so that the nonlinear and indeterminate Diophantine problem reduces to the acquisition of a finite set of possible target locations over a region of interest. We then present two tracking/localization methods which are able to resolve the ambiguity of target location in the context of: 1) tracking a target with significant motion; 2) localizing a target of random vibration or micromotion (jitter) which is described by a probability distribution. In the former case we show that the target location ambiguities can be resolved via the techniques of multitarget tracking in clutter. In the latter case a probability density discrimination method is proposed to resolve the ambiguities in target location space. The analytical phase measurement distribution for a fluctuating target is derived and an inference method for prediction target locations is developed. Following the Introduction, the range ambiguity problem caused by measuring signal phases in the context of target tracking is described in Section II. The look-up table which maps phase measurements into target location space is introduced in Section III. In Section IV, we demonstrate how the location ambiguities due to phase-only measurements can be resolved for moving target tracking. The approach for localizing a target of jitter described by a distribution is given in Section V, where the derivation of phase measurement distribution, the localization procedure and technical justification of the proposed method are presented. Finally, conclusions are given in Section VI. II. PROBLEM Consider the problem of tracking a target using phase-only measurements as shown in Fig. 1. A continuous wave of single frequency , which can be modeled as (1) is transmitted by each of the three sensors. The received return signal at the sensor from the target with range is represented by (2) where the signal phase is measured by the sensor, and is the speed of propagation. Denoted by , the target state at time will typically consist of position and velocity components, although only target location is considered in this work.
Fig. 1. Tracking a target (x; y ) with three distributed continuous wave phaseonly sensors located at ( ; ), i = 1, 2, 3. The distances between two adjacent circles correspond to a single wavelength.
The target dynamics are assumed to follow a Markov process subject to a random fluctuation , i.e. (3) repwhere is the system transition (dynamical) model and resents process noise, which can often be usefully approximated . by a Gaussian distribution with zero mean and covariance The measurement of the system at time is modeled as the noisy phase vector of return signals transmitted by the three active sensors, i.e. (4) where
,
,
1, 2, 3, and
(5) describes the relationship between the measurewhere and the target state with the th component repment resents the phase of the return signal transmitted from and reis the ceived at the th sensor, and measurement noise, which is a zero mean wrapped Gaussian . , , 2, 3 denotes distribution with covariance the location of three sensors and mod is the modulo operation. Without loss of generality, the frequencies of the three sensors in this paper are assumed to be the same. One of the most critical problems in tracking and localization applications in such a measurement system is the ambiguity of the measurement due to the modulo of , which results in a collection of possible target locations. III. LOOK-UP TABLE Equation (5) indicates that for a given target state, a unique vector valued phase measurement can be found while a single phase measurement may correspond to a finite set of target locations in the region of interest. In this paper, a mapping between and phase measurement space target location space
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Fig. 3. Target location ambiguities. A phase measurement 8 generated by a target at location A corresponds to a set of possible target locations.
Fig. 2. (a) Look-up table, where the value of a phase measurement vector is represented by the color/gray scale of the “resolution grid.” (b) The phase measurement space that corresponds to the target location space in (a).
for a given “resolution grid” in the region of interest is proposed in look-up table form. An example of the phase measurement look-up table is given in Fig. 2(a), where at each target location “cell” the corresponding vector valued phase measurement is represented in the color/gray scale of the cell. Clearly, a phase measurement (vector) can be uniquely identified by its color/gray scale. Multiple locations may be found from the look-up table for a given phase measurement (color). The average resolution of the measurement corresponding to a look-up table is related to the size of grid in the target location space, i.e. (6) is the grid value. where Remarks: 1) The look-up table is defined by the mapping via the in. As shown in Fig. 2(a), verse function of (5), i.e., a particular vector valued phase measurement is mapped into a set of ambiguous locations of identical colors in the look-up table. Around each of these locations, the “color distribution” may not be the same as that of other locations in the set. This indicates that if the underlying target mo-
tion is significant (rather than just jittering), the resulting set of ambiguous locations will not all follow the target trajectory but “jump” to the locations of identical colors with the target true location. 2) The finer the grid of the look-up table is, the higher the target localization accuracy will be. However, this will not alter the ambiguity of target location problem. corre3) In general, a vector valued phase measurement sponds to a set of possible target locations and they can be identified from the look-up table. For example, in Fig. 3 corresponds to a phase a target located at . Multiple locations at which the target measurement yields the same phase measurement are found by using the look-up table so providing an alternative solution to the phase Diophantine approach.
IV. MOVING TARGET TRACKING When the underlying target motion (with known dynamics) is significant, large variations in the possible target location measurements occur from the observed phase sequence over time. In this case the target trajectory can be estimated via target tracking in clutter techniques such as the Probabilistic Data Association Filter [16]. The principle of measurement ambiguity removal via data association is illustrated in Fig. 4, where the of the target at time is vector valued phase measurement , mapped into a set of possible target locations via the look-up table of , while the measurement at time results in another set of possible target locations , and so on. Only the trajectory connected by the true target locations over time will be consistent with the underlying target kinematic dynamics. An example of tracking a target of constant acceleration is given below. In this case, the location of the target at is described by (7)
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Fig. 4. Illustration of target location ambiguity removal via data association.
where the initial state of the target is , the velocity and acceleration of the target are and , respectively. In this example the standard deviation of the system process noise in (7) is and the standard deviations of measurement noise in (5) rad. is A sequence of phase measurements in 50 scans is mapped into the target location space via the look-up table as illustrated in Fig. 5, where the phase measurements of the first five scans ,” respectively. are marked in order with symbols “ The underlying target trajectory can be clearly identified from the mapped location measurements over time. Fundamentally, in the target location space the distributions of target measurements and “clutter” measurements are different and can be differentiated via stochastic filtering techniques, such as particle filters [17] which is used to deal with nonlinear non-Gaussian like the underlying problems. Many other nonlinear filters are available in the literature like [18], [19]. Therefore, once phase data is mapped into target location space in terms of virtual location data, conventional multitarget tracking techniques can be used to resolve the phase measurement ambiguity and estimate the target trajectories. V. LOCALIZATION OF MICROMOTION TARGETS When the underlying target is jittering (micromotion), the ambiguous target locations mapped from a sequence of phase measurements can no longer be resolved via a target tracking technique because the yield set of possible target locations essentially have no significantly delineated change over time.
In this case phase measurement ambiguities can also be resolved using a different approach. First, we demonstrate the situation using a target of motion type described by a Gaussian distribution. Fig. 6(a) shows three possible target locations which have the same phase measurement. When the target motion at these locations follows identical Gaussian distributions, as illustrated in Fig. 6(b), the phase measurement distributions are different and distinguishable. Another example is illustrated in Fig. 7, where a target has identical microcircular motions at all ambiguous locations in the target location space and the resulting phase measurement trajectories in phase measurement space can be clearly identified. Inspired by the above observations we derive a phase measurement noise distribution by taking target “noise” into account, which serves as “ground truth” in the target localization process to compare with the one summarized from the received phase measurement sequence. The target location ambiguity problem can then be resolved using one of the density based distance measures such as the Kullback–Leibler divergence (KLD) by identifying the location of minimum density distance to its “ground truth” from the location candidate collection. A. Phase Measurement Distribution be the target micromotion distribution Let such that the underlying target state at is given by (8)
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Fig. 5. Demonstration of resolving phase measurement ambiguity via filtering technique with data association in the target location domain, where in (b) the possible target locations due to the first five phase measurements are marked with symbols “ ; ; ; ; >,” respectively. (a) Accumulated phase measurements in 50 scans. (b) Accumulated possible target locations mapped from (a).
+2
3
Fig. 6. Demonstration of resolving target location ambiguity in phase measurement distribution discrimination. (a) A target fluctuating in the same distribution at ambiguous locations. (b) Phase measurement distributions.
where is 2 2 identity matrix. The phase measurement model (4) is then written as
(10) where
and
are independent and (11)
(9)
where , , 2, 3 take values of integer due to the modulo . operator and As we will see later, the distribution of the phase measurement (9) is a wrapped Gaussian. To derive this distribution, we first consider the range term in (9)
It is well known that for a fluctuating target with its motion modeled as a zero-mean Gaussian distribution, the range mea, 2, 3 between the sensor and target are of surements , Rice distribution. The vector valued range of the three sensors is a multivariate distribution with correlated components and so it is impossible to obtain an exact analytical expression of this distribution. However, in view of the fact that the target “noise” is usually much smaller compared to the range itself, a Taylor expansion can be used to approximate the range measurement.
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and in (12) The weighted sum of the two distributions and variance is also a Gaussian distribution and its mean are given by (13) (14) Therefore, the range measurement can be approximated by a Gaussian distribution (15) The covariance between
and
can be calculated as
Cov
(16) Taking the correlations between the noise from three range measurements into account, the distribution of the range measurement (10) can be approximated as a multivariate Gaussian (17) where (18) (19) where the correlation coefficient between Fig. 7. Measurements of a circular fluctuating target. (a) Identical motions are performed at all ambiguous locations. (b) The measured phase measurement trajectories corresponding to target motion in (a).
For an asymptotic consideration, the high order terms of the Taylor expansion are negligible and will be ignored and the expansion of range measurement can be written as
and
are given by
(20) As a consequence, the first term in the phase measurement equation (9) is approximately given by a multivariate Gaussian distribution as (21) where
and the phase measurement (9) is (22)
(12)
where , 2, 3 and distances between the th sensor and the target.
are the true
Remark: • The covariance matrix in (19) is not full rank with one zero eigenvalue, which means that the density of range measurement has only two degrees of freedom. In fact, due to the correlation of the noise, the range measurements can be represented as points on a parameterized surface in the measurement space. • When the range is modulo , all the ambiguities in the will have the same sample target location space mean with the one at true location but different covariance matrix. Due to modulo operation, the phase measurement becomes a wrapped distribution well-known as the directional or circular
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Fig. 8. Probability density of wrapped distribution. Top: Probability density function p(x). Middle: The wrapped Gaussian distribution p (#) defined in (25). Bottom: Directional representation of p (#).
data. Directional data may be visualized as points on a surface, a hypersphere, or in two dimensions on the circumference of a circle, while the conventional data can be regarded as linear data [20]. The corresponding statistics are called directional statistics which are mainly concerned with observations which are unit vectors in the plane or in three-dimensional spaces. Thus the sample space is typically a circle or a sphere. It turns out that standard methods for analyzing linear univariate or multivariate measurement data cannot be used [21]. There are fundamental differences between linear and directional statistics, such as the calculation of circular mean of samples from the circular data. Physicists and statisticians have developed a methodology for dealing with statistics of directional data. Most recent description can be found in the book of Mardia [21], [22]. of a linear variAny given probability density function able on the line can be wrapped around the circumference of a circle of unit radius [20]. That is, the probability density funcof the wrapped variable , where tion (23) is
Fig. 9. An example of the wrapped Gaussian distribution.
For a univariate Gaussian distribution univariate Gaussian distribution denoted as fined as [20]
, the wrapped is de-
(25) When , the overlap of neighboring Gaussian wraps in (25) is negligible. In this case, it is permissible to approximate by only one, but the most meaningful wrap of it [20]
(24) (26) Fig. 8 illustrates probability density of the wrapped distribution. There are two circular distributions that should serve as appropriate substitutes for the univariate linear normal distribution. One is the wrapped Gaussian distribution, and the other the von Mises distribution [23]. As the univariate wrapped distribution can be easily extended to the multivariate context, so the wrapped Gaussian distribution is the more natural choice for the problem.
For the case
,
can be approximated as [24] (27)
Fig. 9(a) shows an example of the wrapped Gaussian distribut different bution with identical standard deviations , 2.5, 4.5, respectively. Fig. 9(b) shows the parameter
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example of the wrapped Gaussian distribution with identical pabut different standard deviations , 0.8, rameters 1.5, respectively. The concept of a univariate wrapped distribution can be extended to the multivariate case by an extension of the single sums that cover all disimple sum in (24) to a number of mensions in the feature space, i.e.
(28) is the th Euclidean basis vector. where , , with a For a multivariate Gaussian distribution , where denotes the square root small covariance of matrix , and the identity matrix, the wrapped multivariate can be approximated as Gaussian distribution
(29) According to the above analysis, the phase measurement can be approximated by a wrapped multivariate Gaussian distribution (30) The measurement noise , which is independent of target noise, can also be approximated by a zero-mean wrapped , i.e. Gaussian distribution with covariance (31) In summary, the distribution of phase measurement (9) with both small target noise and sensor noise can be approximated by a wrapped multivariate Gaussian distribution of the form (32) Discussions: 1) The covariance matrix is given by
of the phase measurement
(33)
and it becomes full rank in the presence of sensor noise. 2) For a fluctuating target, the distribution of the phase measurement (32) is of the same mean but different covariance at all ambiguous target locations. In fact, the correlation coefficients in (33) are unique for all locations in the target location space. Such uniqueness can be regarded as target location dependent signatures to be used to discriminate the true target location from the ambiguous location candidates. This uniqueness of the phase measurement distribution is justified as follows. Suppose that
Fig. 10. Illustration of the tensor field constructed by 6( ) in the target location space of interest. The plots demonstrate the uniqueness of 6 ( ) for every target locations. (a) [v ; v ] versus . (b) [v ; v ] versus . (c) [v ; v ] versus .
is the eigenvector with maximal eigenvalue of the covari. It is understood that the direction of the ance matrix
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Fig. 11. Procedure of the proposed target localization method based on ambiguous phase measurements.
eigenvector represents the main direction of the ellip. The tensor fields soid corresponding to the matrix of , and versus are plotted in Fig. 10(a), (b), and (c), respectively. Clearly, the values of the tensor are different for all states and indicate the cois unique at all locations in the region. variance matrix 3) Fundamentally, the uniqueness of the covariance matrix reflects the fact that when a micromotion target is at different locations, the information collected from a sequence of phase measurements will be different everywhere in the location plane. However, such differences will tend to be smaller for locations far from the sensors where the measurement noise (31) will dominate the entire noise term. 4) Knowledge of target noise is not necessarily required and as we will see that it can be estimated from data. From (33) are identical for we can find that the diagonals of all possible locations with the same circular mean , the difference only lies in the off-diagonals for true location and ambiguities. So we can use the diagonal elements of to estimate the standard deviation of the target noise, i.e.
can be from a sequence of phase measurements established by the following steps: Step 1) Estimate sample mean and covariance from the received measurement sequence. According to [21], the circular mean and covariance are estimated as (35) (36) . where Step 2) Estimate the target noise variance using (34). possible target locations from the Step 3) Find the set of look-up table, i.e. (37) Step 4) Calculate the theoretical phase measurement distributions for each of the possible target locations via (32):
(34) (38) where
is the sample covariance of phase measurement.
B. Target Localization Procedure and an Illustrative Example As shown in Fig. 11, the procedure of the proposed method for estimating the location of a target with fluctuated motion
Step 5) Calculate KLD between each of the theoretical phase measurement distributions and the sample distribution, i.e. (39)
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Fig. 12. Case 1: ambiguity discrimination using measurement density signature.
Fig. 13. Case 2: ambiguity discrimination using measurement density signature.
where is the distribution summarized from measurements. In our case, a closed expression of (39) is given
rameters and calculated results are listed in Table I. The computed KLDs are marked on the corresponding figures. In both situations, the computed minimum KLDs pointed to the correct target locations and thus resolved target location ambiguities. We should point out that the proposed method is also for small ). number of samples (say,
(40) and the correct target location is found by selecting the one with minimum KLD, i.e. (41) To demonstrate the effectiveness of the proposed distribution discrimination method, an example of localizing a fluctuating target is given. As shown in Figs. 12 and 13, the underlying target was in two such locations that the generated phase measurements give the same set of possible target locations. Pa-
C. Statistical Results The robustness of the proposed localization method for target of micromotion is investigated via Monte Carlo statistics. Parameters used for the Monte Carlo runs are same as those listed in Table I. The true location of the underlying target is assumed to be uniformly distributed across the region of interest whose boundary is shown in Fig. 13. A successful run is counted if the target location is estimated correctly, i.e., the minimum KLD between a theoretical measurement density and the one estimated from measurements is corresponding to the ground truth in this run. In Table II, statistical values (mean and standard deviation) of normalized KLDs between the estimated distribution (from
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TABLE I RESULTS OF THE TWO SCENARIOS SHOWN IN FIGS. 12 AND 13
TABLE II NORMALIZED DISTANCES COMPARISON AVERAGED OVER 100 MONTE CARLO RUNS
samples) and theoretical distributions computed via (33) at all possible target locations are given. These values are averaged from 100 Monte Carlo runs where the ground truth target location are uniformly generated over the region of interest. For a comparison, the Euclidean distances between the estimated and all possible target locations are also provided. Note that these statistics (Table II) only count successful runs (98 out of 100). The successful rate versus the number of samples used at each run for estimating the target measurement density via (35) and (36) is illustrated in Fig. 14, which has evidenced that as the number of samples used to estimate the target measurement distribution increased, a higher probability of correctly estimating the target location will be achieved. As indicated in Table II, the KLDs for measurement at target true and false locations can be well separated and the average localization error is quite small in this example. VI. CONCLUSION In this paper, estimating the trajectory/location of a target using phase-only measurements observed by three sensors in different locations is considered, where phase ambiguity caused by phase wrapping is a challenging issue. Methods for resolving the phase ambiguities are proposed. First, the phase wrapped measurement is mapped into target location space via a look-up table. We then resolve the target location ambiguity in two cases. For targets in significant motion, the technique of stochastic filtering with data association is suggested. For targets with jitter (micromotion), a distribution discrimination method is proposed and the probability density function of the
Fig. 14. Rate of successful localization of a target of micromotion versus the number of samples used for estimating measured distribution at each run. The plot indicates that as the number of samples used for estimating target measurement density increased, the probability of successful target localization will approach to one.
phase measurement is derived. The effectiveness and robustness of these methods are demonstrated. A challenging and interesting issue is to localize multiple jittering targets using phase-only measurements. We will address this issue in the future research work. From Fisher information point of view, the accuracy of tracking/localization can be improved if more than three sensors are considered in the presence of measurement noise. The current work can be straightforwardly extended to the case of using more than three sensors. As the dimension of phase measurement increased, the overall computational complexity, such as building and searching the look-up table in the region of interest as described in Section III and computing measurement distributions via (33), is expected to rise accordingly. However, the number of ambiguous target locations from a phase measurement will be less. REFERENCES [1] A. Wenzler and S. Steinlechner, “Nonlinear processing of n-dimensional phase signals,” in Proc. IEEE Int. Symp. Circuits Syst. (ISCAS), 2002, pp. I-805–I-808. [2] C. Wang, Q. Yin, and W. Wang, “An efficient ranging method for wireless sensor networks,” presented at the 2010 IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP), Dallas, TX, Mar. 14–19, 2010. [3] W. D. Boyer, “A diplex, Doppler, phase comparison radar,” IEEE Trans. Aerosp. Navigat. Electron., vol. ANE-10, no. 1, pp. 27–33, Mar. 1963. [4] I. Urazghildiiev, R. Ragnarsson, and A. Rydberg, “High-resolution estimation of ranges using multiple-frequency CW radar,” IEEE Trans. Intell. Transport. Syst., vol. 8, no. 2, pp. 332–339, Jun. 2007. [5] M. Maroti, B. Kusy, G. Balogh, P. Volgyesi, K. Molnar, A. Nadas, S. Dora, and A. Ledeczi, “Radio interferometric positioning,” Inst. Software Integr. Syst., Vanderbilt Univ., Nashville, TN, Tech. Rep. ISIS-05-602, Nov. 2005. [6] B. Kusy, A. Ledeczi, M. Maroti, and L. Meertens, “Node-density independent localization,” presented at the 5th Int. Conf. Inf. Process. Sens. Netw. (IPSN’06), Nashville, TN, Apr. 19–21, 2006. [7] G. L. Mader, “Rapid static and kinematic global positioning system solutions using the ambiguity function technique,” J. Geophys. Res., vol. 97, no. B3, pp. 3271–3283, 1992.
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[8] M. L. Skolnik, Introduction to Radar Systems, 2nd ed. New York: McGraw-Hill, 1980. [9] P. Setlur, M. Amin, and F. Ahmad, “Cramer-Rao bounds for range and motion parameter estimations using dual frequency radars,” presented at the 2007 IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP), Honolulu, HI, Apr. 15–20, 2007. [10] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero, III, R. L. Moses, and N. S. Correal, “Locating the nodes—Cooperative localization in wireless sensor networks,” IEEE Signal Process. Mag., pp. 54–69, Jul. 2005. [11] L. J. Mordell, Diophantine Equations. New York: Academic, 1969. [12] H. Cohen, Number Theory: Volume I: Tools and Diophantine Equations. New York: Springer, 2007. [13] J. Steduing, Diophantine Analysis. Boca Raton, FL: Chapman & Hall/CRC, 2005. [14] R. G. McKilliam, B. G. Quinn, I. V. L. Clarkson, and B. Moran, “Frequency estimation by phase unwrapping,” IEEE Trans. Signal Process., vol. 58, no. 6, pp. 2953–2963, 2010. [15] P. Smaragdis and P. Boufounos, “Learning source trajectories using wrapped-phase hidden Markov models,” in Proc. IEEE Workshop Appl. Signal Process. Audio Acoust., New Paltz, NY, Oct. 2005, pp. 114–117. [16] Y. Bar-Shalom and T. E. Fortmann, Tracking and Data Association. New York: Academic, 1988. [17] M. S. Arulampalam, S. Mansell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process., vol. 50, no. 2, pp. 174–188, Feb. 2002. [18] M. Orton and W. Fitzgerald, “A Bayesian approach to tracking multiple targets using sensor arrays and particle filters,” IEEE Trans. Signal Process., vol. 50, no. 2, pp. 216–223, Feb. 2002. [19] C. Kreucher, K. Kastella, and A. O. Hero, III, “Multitarget tracking using the joint multitarget probability density,” IEEE Trans. Aerosp. Electron. Syst., vol. 41, no. 4, pp. 1396–1414, Oct. 2005. [20] C. Bahlmann, “Directional features in online handwriting recognition,” Pattern Recogn., vol. 39, pp. 115–125, 2006. [21] K. V. Mardia and P. E. Jupp, Directional Statistics 2nd Edition. Chichester, U.K.: Wiley, 2000. [22] K. V. Mardia and P. E. Jupp, Statistics of Directional Data. New York: Academic, 1972. [23] N. I. Fisher, Statistical Analysis of Circular Data. Cambridge, U.K.: Cambridge Univ. Press, 1995. [24] B. W. Church and D. Shalloway, “Characterizing large correlated fluctuations of macromolecular conformations in torsion-angle space using the multivariate wrapped-Gaussian distribution,” Polymer, vol. 37, pp. 1805–1813, 1996.
Yongqiang Cheng received the B.S. and M.S. degrees in information and communication engineering from National University of Defense Technology, Changsha, China, in 2005 and 2007, respectively. He is currently pursuing the Ph.D. degree from the National University of Defense Technology. From September 2009 to November 2010, he was a visiting research student with Melbourne Systems Laboratory, University of Melbourne, Australia. His research interests lie in the areas of information geometry and statistical signal processing.
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Xuezhi Wang received the B.S. degree in avionics from Northwest Polytechnical University, Xian, China, in 1982 and the Ph.D. degree in signal and systems from the Department of Electrical and Electronic Engineering, University of Melbourne, Australia, in 2001. He has been involved in radar and sonar signal processing research and development since 1983. He is now a Research Fellow with the Melbourne Systems Laboratory, Department of Electrical and Electronic Engineering, University of Melbourne. His research interests are in stochastic signal processing, information theory, Bayesian estimation, data fusion, and Situation assessment.
Terry Caelli (SM’92–F’02) received the B.S. (Hons.) degree in mathematics and psychology and the Ph.D. degree in human and machine vision from the University of Newcastle, Australia. Current interests lie in Signal Processing, Computer Vision and Machine Learning and their applications in Health and the Environment. He is Director of National ICT Australia’s (NICTA) Health Business Area. He is a Fellow of the International Association for Pattern Recognition (IAPR). He is also a Convocation Medalist from the University of Newcastle. He has spent 15 years in North American universities and research institutes, has been a DFG Professor, Germany, Killam Professor of Science, the University of Alberta, Canada, as well as having held senior positions in Australia at the University of Melbourne and Curtin University of Technology. He returned from Canada to join NICTA in 2004. Professor Caelli has served on the editorial boards of journals including the IEEE JOURNAL ON PATTERN RECOGNITION and numerous international conference committees in both human and machine vision. See http://www.nicta. com.au/people/caellit.
Bill Moran (M’95) received the B.S. (Hons.) degree in mathematics from the University of Birmingham in 1965 and the Ph.D. degree in mathematics from the University of Sheffield, London, in 1968. He is a professor of electrical engineering with the University of Melbourne, Australia, where he is the Research Director of Defence Science Institute and Technical Director of Melbourne Systems Laboratory. Previously, he was a professor of mathematics with the University of Adelaide and Flinders University. He also serves as a Consultant to the Australian Department of Defence through the Defence Science and Technology Organisation. His research interests are in signal processing, particularly with radar applications, waveform design and radar theory, and sensor management. He also works in various areas of mathematics, including harmonic analysis and number theory and has published widely in these areas. Dr. Moran has been a member of the London Mathematical Society since 1967, of the Australian Mathematical Society since 1976, and of the American Mathematical Society since 1978.
