Target Tracking using Range and RCS Measurements in a MIMO Radar Network Bin Sun† , Xuezhi Wang‡ , Bill Moran‡ and Xiang Li† †
School of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan, 410073, P.R.China Email:
[email protected],
[email protected] ‡
Melbourne Systems Laboratory, School of Engineering, University of Melbourne, Australia. Email: {xwang,wmoran}@unimelb.edu.au
Abstract It is evidenced that for a target of non-isotropic reflectivity, a radar range measurement may associated with a large variance due to the fluctuation of target RCS. A MIMO radar is able to observe a target from multiple directions and therefore is likely to capture the target scattering pattern via widely separated receivers. This paper considers the case of tracking a target using both range and RCS measurements collected from a MIMO radar network. The bistatic RCS measurements of a target are obtained from the amplitudes of received signal strength for each of the transmitting-receiving paths. A pre-calculated look-up table is used to take target reflectivity pattern into account for measuring the amplitude of received signal strength. The latter is assumed to be proportional to the RCS of a point target. The RCS measurement model are then incorporated with the bistatic range measurements for estimating the posterior probability density of the target by using a particle filter. The numerical results show that by incorporating target RCS measurement, the root mean square error performance of the estimate can be improved and the improvement is significant if a large number of well separated antennas are used in the MIMO radar network. Keywords: MIMO radar networks, Target tracking, Particle filter, RCS model.
1
Introduction
A distributed multiple input and multiple output (MIMO) radar network consists of multiple widely separated transmitters and receivers and is capable of improving target detection performance by observing target from multiple aspects thus obtaining radar cross section (RCS) diversity gain [1, 2]. Typically, a MIMO radar measures the time delays of the waveforms transmitted by multiple transmitters and reflected off a target in different directions via a fusion center [3] and thus can obtain the bistatic ranges after multiplying by the light speed. In addition, the reflectivity pattern of a target may also be measured from a set of amplitude measurements of the received signal strength “observed” by multiple receivers [4]. The potential of a MIMO radar system for accurately localizing a target was explored and discussed in [3, 5–7]. A significant improve-
ment was shown in [8] for target tracking using the measurements of a MIMO radar over a monostatic phased array radar. A jointly tracking and classification algorithm which augments the measurement vector to include target RCS was reported in [9], where an angle-independent statistical model for RCS fluctuation is assumed. Similar RCS models were found elsewhere which are difficult to be associated with the target kinematic state. In general, the RCS measurement of a target is a function of aspect angles between the target and emitters/receivers [10]. The problem of maneuvering target tracking using kinematic measurements in conjunction with target orientation measurements was studied in [11]. It is shown that apart from the improvement in tracking accuracy the time delay for adapting to unexpected target maneuvers is also reduced. In this paper, the modeling of RCS measurement of a target in a MIMO radar environment is studied and incorporating the RCS measurements to improve target kinematic state tracking performance is considered. A recursive Bayesian particle filter is adopted to cope with the multiple bistatic range measurements as well as bistatic RCS measurements for tracking a moving target. In order to evaluate the likelihood of a target RCS measurement, a RCS amplitude look-up table for the target is precomputed based on the underlying target reflectivity pattern. A particle filter (PF) procedure for update the posterior density of the target state using both range and RCS measurements is presented. The root mean squared error (RMSE) performance of the PF using target RCS measurements is compared with the PF without using target RCS measurements via the MonteCarlo multiple runs in a constant velocity target tracking scenario. It is demonstrated in the simulations that the additional RCS measurements can help improve target tracking accuracy, in particular, when a large number of well separated antennas are involved. In Section 2, the target tracking problem is described, particularly, we present a target RCS measurement model to extract target RCS information at a given target state from a set of received signal strength values. In Section 3, A PF algorithm which incorporates both range and RCS measurements collected from a MIMO radar network to update the underlying posterior density of target is presented. The RMSE performance of proposed algorithm is compared with that of a PF algorithm
without considering the RCS measurements by means of simulation in Section 4. Finally, concluding remarks are addressed in Section 5.
2
Problem Formulation
The target tracking problem considered in this work is to estimate the posterior density of a target state based on a sequence of the bistatic range and RCS measurements observed by a MIMO radar network. 2.1
Target Model
A single straight line motion target in a two-dimensional Cartesian coordinate system is considered, where a nearly constant velocity (CV) motion model is used to describe the target dynamics. Specifically, the state of the target at time k is denoted as xk = [xk , x˙ k , yk , y˙ k ]T , which contains the target position pk = [xk , yk ]T and velocity vk = [x˙ k , y˙ k ]T . The target state is assumed to evolve according to xk = Fxk−1 + νk−1 where
⎡
1 ⎢ 0 ⎢ F=⎣ 0 0
T 1 0 0
⎤ 0 0 0 0 ⎥ ⎥ 1 T ⎦ 0 1
(1)
(2)
is the state transition matrix, and T is the tracking interval, νk denotes a Gaussian process noise with zero-mean and the covariance matrix Q is given by ⎤ ⎡ 3 0 0 T /3 T 2 /2 ⎢ T 2 /2 T 0 0 ⎥ ⎥ (3) Q = q⎢ 3 2 ⎣ 0 0 T /3 T /2 ⎦ T 0 0 T 2 /2 where q denotes the power spectral density (PSD) of the process noise and indicates the process noise intensity. The model in (1) can be used to determine the transition distribution from k to k + 1 as p(xk+1 |xk ) = N (xk+1 ; Fxk , Q)
(4)
where N (·; Fxk , Q) denotes Gaussian probability density with mean Fxk and covariance matrix Q. 2.2
(1,1)
Measurement Models
We assume that a widely separated MIMO radar network consisting of M transmitters simultaneously illuminates a moving target with a set of approximately orthogonal waveforms which are received by a set of N receivers. Target range and RCS measurements are obtained by jointly processing the returned signals in a fusion center. The location of the mth transmitter t T ] , m = 1, · · · , M , and the lois denoted by ptm = [xtm , ym r cation of the nth receiver pn = [xrn , ynr ]T , n = 1, · · · , N . All these locations are assumed known a priori.
(M,N )
]T be the multiple bistatic range Let rk = [rk , · · · , rk (1,1) (M,N ) T ] the measurements vector and αk = [αk , · · · , αk multiple bistatic RCS measurements vector at time k. The augmented measurements can be expressed as Zk = [rk , αk ]T
(5)
2.2.1 Range Measurement At time-step k, each transmitter-receiver pair measures the bistatic distance to the target, and therefore the range measurement equation is given by rk = hrk (xk ) + εr ⎡ (1,1) pk − pt1 + pk − pr1 + εr ⎢ . .. =⎢ ⎣ pk −
ptM
+ pk −
prN
+
(M,N ) εr
⎤ ⎥ ⎥ ⎦
(6)
where · is the Euclidean norm, pk − ptm is the range from transmitter m to the target while pk − prn is range from the target to receiver n at time-step k, and the measurement error εr is modeled as a zero-mean Gaussian random vector with covariance R = E[εr εTr ] = σr2 IM N , where σr2 is the noise power and IM N is an identity matrix of size M N . This assumption is reasonable for noises which come from different propagation paths are independent of each other. The set of bistatic range measurements (6) are nonlinear of the state vector x with independent Gaussian error distributions. 2.2.2
RCS Measurement
At time-step k, a received signal strength measurement, denoted by ξ, for each transmitter-receiver pair is responsible for the bistatic RCS scattering from the target. So, the set of RCS measurements given by the MIMO radar are denoted by α k = hα k (xk , εα ) In general, for a target with anisotropic shape the RCS is dependent on many factors, such as the incident radiation’s frequency and polarization, the object’s electrical size and orientation with respect to the transmitters and receivers, and its constituent materials, and so on [12]. In this paper, we assume that the RCS of a target is proportional to the received signal power and therefore, it is measured by the value of received signal strength intensity. In addition, the bistatic RCS of a target is considered to be a function of the relative incident angle as well as the scattering angle with respect to the target. We also assume that the underlying target reflectivity pattern is known. Unfortunately, it is generally difficult to find an analytical expression for the RCS measurement hα k (xk , εα ). In this work, we propose the methodology of look-up tables which can be pre-calculated with knowledge of target reflectivity pattern to facilitate the measurement likelihood required for target tracking. As illustrated in Fig.1, let φtm,k denote the line of sight bearing between the line of the mth transmitter to target and the
Y
and time k. The corresponding RCS is defined as
Mk
(m,n)
αk
Imt ,k
( xmt , ymt )
⎡ ⎢ ⎢ αk = ⎢ ⎣
I
r n,k
Ren
( xnr , ynr ) X
Figure 1: 2-D Location of transmitters and receivers with respect to the target. The antennas are assumed to be in the far field of the target.
positive x axis viewed at the transmitter origin. Similarly, let φrn,k be the line of sight bearing between the line of the nth receiver to target and the positive x axis. ϕk = arctan( xy˙˙ kk ) signifies the target orientation angle between target motion direction and the positive x axis. Thus, for the (m, n)th propt is given agation path, the relative incident aspect angle θm,k t t r by θm,k = |ϕk − φm,k | and the scattering aspect angle θn,k r r is calculated as θn,k = |π − (ϕk − φn,k )|. Therefore, for a t r and θn,k at location given target with the specified angles θm,k (x, y), the “ground truth” RCS of the target can be found via a pre-calculated look-up table. An example of the RCS data look-up table is given in Fig. 2, where from the given target reflectivity pattern in (a) we can calculate the look-up table for the values of received signal strength over the area of interest (b).
90
60 4 3
150
(1,1)
2
3
As aforementioned, we choose to use the Bayesian sequential Monte Carlo technique to recursively perform the nonlinear filtering since a PF tracker can address a nonlinear filtering problem without the need of an explicit form measurement likelihood [9]. The diagram of the MIMO PF tracker is illustrated in Fig.3.
Updated Importance Weights
Particle Filter Range Measurements
0
210
330
240
⎥ ⎥ ⎥ ⎦
Particle Filter Tracking Algorithm
1 180
⎤
(8) where the subscripts R and I identify the real and imagi(m,n) nary components of ζk respectively, εα is modeled as a zero-mean Gaussian random vector with covariance A = E[εα εTα ] = σα2 IM N , where σα2 is the noise power. Clearly, the bistatic RCS measurements in (8) is a nonlinear and nonGaussian function of the target state x. At this point, we should make it clear that since the measurement models in (6) and (8) are non-linear and non-Gaussian, a nonlinear filter is required for estimating the posterior density of target state p(xk |Z1:k ), where Z1:k = {Z1 , · · · , Zk }. However, there is no closed-form relationship between target state and the its RCS, the simple extended Kalman filter (EKF), which needs the first and second moments of the measurement distribution, can not be used. Hence, we adopt a nonparametric particle filter for the underlying tracking problem which is discussed in Section 3.
30
2
(7)
2 (1,1) (1,1) + ζk,I + εα .. . 2 2 (M,N ) (M,N ) (M,N ) (M,N ) ζk,R + εα + ζk,I + εα (1,1)
ζk,R + εα
RCS Measurements
5
120
.
We consider the following model for the RCS measurement in the presence of thermal noise:
( xk , yk )
Trm
(m,n) 2
= ζk
Target State Estimate
Updated New Particles
Motion model
300 270
(a)
(b)
Figure 2: Target RCS pattern in decibels. (a) Bistatic scattering pattern with transmitter illuminating the target at 0o and the receiver moving in a circle around the target; (b) MIMO scattering pattern look-up table. (m,n)
denote the amplitude of received signal strength Let ζk of a target corresponding to the (m, n)th path at location (x, y)
Figure 3: A PF-based tracker for MIMO radar tracking with range and RCS measurements. In this work, we employ a standard sequential importance resampling (SIR) PF algorithm described in [13], where target transition density (4) is used as the importance density. The resampling, which can alleviate particle degeneracy, is applied at every time step. The philosophy of the PF algorithm is to approximately represent the posterior density of target state p(xk |Z1:k ) by using a set of samples with associated weights
is given by
expressed as p(xk |Z1:k ) ≈
Np
j=1
(j) ωk δx(j) (xk k
−
(j) xk )
(9)
· exp −
(j)
where Np is the total number of particles, and ωk is the weight (j) of particle xk . As the number of particles becomes large, the approximation becomes accurate. We summarize the proposed PF algorithm in Algorithm 1 below. Algorithm 1 Framework of Particle filtering • Step 1 Initialization: At time k = 0, draw Np samples to form the initial (j) Np with identical weight 1/Np to particles set {x0 }j=1 approximate the initial distribution of the target state, p(x0 ) • Step 2 Importance sampling: At time k = 1, 2, · · · ,
1 2πσr2
m=1 n=1 2 (m,n) −(pk −ptm +pk −prn ) rk
(13)
2σr2 (m,n)
is similar to a Rician distribution, The distribution for αk except there no square root in (8). The resulting likelihood is of the form M N (m,n) (m,n) 2 + ζk αk 1 exp − p(αk |xk ) = 2σα2 2σα2 m=1 n=1 ⎡ ⎤ (m,n) (m,n) 2 αk ζk ⎦ ·I0 ⎣ (14) σα2 (m,n)
(j) N
p using the dynam– Predict new particles {xk }j=1 ic target motion model in (1)
)
(10)
and, normalize them as below (j)
ω ˜ (j) ωk = Npk
j=1
(j)
ω ˜k
(11) 4.1
• Step 3 Resampling: Np (j) Generate a new particle set {xk , 1/Np }j=1 by resampling procedure with replacement Np times from (j) (j) Np (j) (j) (j) , where Pr(xk = xk ) = ωk {xk , ωk }j=1 • Step 4 Output: Np State estimation: ˆxk = j=1
In this section, we compare the tracking performance of the proposed PF algorithm with that of a PF which does not use the target RCS measurement observed by the underlying MIMO radar. The tracking performance is measured using the RMSE averaged over multiple Monte Carlo runs. Parameters Case 1
4
x 10
Tx Radar Rx Radar
1
0.5
0.5
0
−0.5
1 (j) N p xk
Case 2
4
x 10 Tx Radar Rx Radar
1
Y axis
(j)
Simulation results
Y axis
(j)
ω ˜ k ∝ ωk−1 p(Zk |xk
2 is obtained from a pre-calculated look-up tawhere ζk ble for a given target state at time k with known positions of the mth transmitter and the nth receiver, and I0 is a zero-order modified Bessel function of the first kind.
4
– Calculate importance weights by (12) (j)
M N
p(rk |xk ) =
0
−0.5
−1
−1 −1
• Step 5 k = k + 1, go to Step 2.
−0.5
0
0.5
1
X axis
−1
−0.5
x 10
(a) Tx Radar Rx Radar
(12)
According to (6) and the independency of measurements, the likelihood function of the multiple bistatic range measurements
4
x 10
Tx Radar Rx Radar
1
0.5
Y axis
0.5
0
−0.5
0
−0.5
−1
−1 −1
−0.5
0
X axis
p(Zk |xk ) = p(rk |xk ) · p(αk |xk )
1
Case 4
4
x 10
1
Y axis
In order to recursively calculate the posterior density of target state, we need to compute three distributions, namely the initial state distribution p(x0 ) at time 0, the state transition distribution in (4) and the likelihood function p(Zk |xk ). The latter depends on the measurement model. In this work, we assume that the RCS measurements are independent from range measurements, which implies that
0.5
(b)
Case 3
4
x 10
0
X axis
4
(c)
0.5
1
−1 4
x 10
−0.5
0
X axis
0.5
1 4
x 10
(d)
Figure 4: MIMO radar configurations and the target trajectory. The black circle represents the starting point.
4
x 10
Case 1
Tx Radar Rx Radar
1
2.4
Position RMSE(m)
10
0
−0.5
2.3
10
2.2
10
2.1
10
−1 −1
−0.5
0
0.5
1
X axis
5
4
10
x 10
(a) 3 × 3 Radar configuration
20
25
30
(b) Position RMSE
Case 1
1
1 0.9
0.9 1
0.8
0.8
0.7
0.6
0.6
0.4
0.5
0.2
0.4
0 400
0.3 200
400
Normalized Likelihood
Normalized Likelihood
1
0.8
0.8
0.7
0.6
0.6
0.4
0.5
0.2
0.4 0.3
0 400
0.2
200
200
0
0
−200
Y axis
400
0.1
−400
−200 −400
Y axis
X axis
0.2 0.1
0
−200
−200 −400
200
0
(a) Without RCS
−400
X axis
(b) With RCS
Figure 7: Normalized likelihood function by 6×6 MIMO radar system in Case 1. Target is located at (0, 0). Case 1
where x ˆk and yˆk are the position estimates made at time k and Nmc is the total number of Monte Carlo simulations.
500 True location 3×3: Without RCS 3×3: With RCS
400 300
4.2
15 Scan Number
Figure 6: Position RMSE for target state estimates by 3 × 3 MIMO radar system Case 1
The MIMO radar data is processed in a fusion center with sampling interval T = 2 s and total 30 sampling intervals are considered. The PSD value in (3) is q = 0.1. The standard deviation of range measurement noise is σr = 1000 m, and the standard deviation of the received signal strength noise is σα = 0.1. All simulation results are averaged over 100 Monte Carlo runs and at each run 500 particles are generated. The RMSE for the estimated target position at k is calculated by mc 1 N
[xk (i) − x ˆk (i)]2 + [yk (i) − yˆk (i)]2 , (16) Nmc i=1
PF with RCS PF without RCS
2.5
10
0.5
Y axis
The MIMO radar is assumed to consist of M = 6 transmitters and N = 6 receivers. Four scenarios for the MIMO radar antenna configurations are considered as shown in Fig.4 to demonstrate the influence of antenna distribution on the tracking performance. The distances from origin to all antennas are assumed to be identical 10 km. Signals received from all propagation paths are assumed to be synchronized. = Target initial state is assumed to be x0 [3000, −100, −3000, 100]T with a covariance matrix ⎤ ⎡ 0 0 5002 0 ⎢ 0 0 0 ⎥ 22 ⎥ (15) P0 = ⎢ 2 ⎣ 0 0 ⎦ 0 500 0 0 0 22
6×6: Without RCS 6×6: With RCS
200
Simulation Results Y axis
100 Case 1
Case 2 PF with RCS PF without RCS
PF with RCS PF without RCS
0 −100
Position RMSE(m)
Position RMSE(m)
−200
2
10
−300 −400 −500 −500 −400 −300
2
10
−200 −100
0
100
200
300
400
500
X axis 5
10
15 Scan Number
20
25
30
5
10
(a)
20
25
30
Figure 8: Contour of normalized likelihood function by 6 × 6 MIMO radar system in Case 1. The contour level is 0.75.
(b) Case 4
Case 3
PF with RCS PF without RCS
PF with RCS PF without RCS
2.4
Position RMSE(m)
10
Position RMSE(m)
15 Scan Number
2.3
10
2.2
10
2
10
2.1
10
5
10
15 Scan Number
(c)
20
25
30
5
10
15 Scan Number
20
25
30
(d)
Figure 5: Position RMSEs for target state estimates by 6 × 6 MIMO radar system
The RMSEs of the estimated target position versus time index under four different antenna configurations are shown in Fig.5. Clearly, under all the four antenna configurations the tracking performance of the PF with RCS measurements is generally better than that of the PF without RCS measurements. The simulation is repeated for a MIMO radar consisting of M = 3 transmitters and N = 3 receivers and results are shown in Fig. 6. It is observed from Fig. 5(a) and Fig. 6(b) that the position RMSE increases if the number of antennas used decrease. A comparison of normalized likelihoods for target location at origin under the configuration of Fig. 4(a) is shown in both Fig.7 and Fig. 8. The result indicates that the uncertainty of
measurement is reduced by incorporating target RCS measurements. As shown in Fig. 8, the uncertainty of likelihood can also be reduced by increase the number of antennas.
5
Conclusion
In this paper we study the problem of tracking a moving target using bistatic range measurements as well as multiple RCS measurements received by a distributed MIMO radar. To incorporate target RCS information in the tracking, we use a precalculated look-up table to compute the ground truth amplitude of received signal strength from a particular transmittingreceiving path such that the RCS measurement model which is a function of target state and the amplitude of received signal strength can be established. A sequential particle filter is then implemented to cope with both the bistatic range and RCS measurements for updating the posterior density of target state. The beneficial of incorporating target RCS measurements from a MIMO radar network for target tracking is demonstrated using a numerical example. It worth mentioning that the proposed bistatic RCS measurement model has taken into account the underlying target reflectivity pattern. It turns out that when tracking multiple targets of different types, using target RCS measurement based on the proposed model can assist for data association and target classification in addition to improve target kinematic state estimation accuracy.
Acknowledgements The authors thank Mark Morelande for his insightful advice. This work is done while Bin Sun is visiting University of Melbourne, and the authors are supported by the China Scholarship Council.
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