Joint Detection and Tracking Processing Algorithm for ... - IEEE Xplore

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IEEE SENSORS JOURNAL, VOL. 15, NO. 11, NOVEMBER 2015

Joint Detection and Tracking Processing Algorithm for Target Tracking in Multiple Radar System Junkun Yan, Hongwei Liu, Member, IEEE, Bo Jiu, Member, IEEE, Zheng Liu, and Zheng Bao, Senior Member, IEEE Abstract— In this paper, a joint detection and tracking processing (JDTP) algorithm is proposed for target tracking in clutter using a multiple radar system. In this paper, the data association events are formed with a reasonable assumption that each radar can at most receive one measurement originated from a target. Moreover, we explore the idea of feeding the information from the tracker to the detector. In this scenario, the tracker can guide the detectors of multiple radars where to look for a target while keeping the constant false alarm rate property. From a practical point of view, the detection threshold is depressed near where a target is expected to be and elevated where it is unexpected. Simulation results demonstrate the efficiency of the proposed JDTP algorithm, in terms of the detection and the tracking performance, when compared with the existing works. Index Terms— Constant false alarm rate, multiple radar system, target detection, target tracking, feedback.

I. I NTRODUCTION A. Background and Motivation

T

ARGET tracking plays an important role for many applications in radar and sonar systems [1], such as battlefield surveillance, air defense, air traffic control, and fire control. For target tracking in clutter, a number of measurements may be obtained at each scan, and it is not known which of them, if any, is originated from the target. In past several decades, many algorithms have been developed to solve the problem of single target tracking in clutter [3]–[8]. Two simple solutions are the strongest neighbor filter (SNF) [3] and the nearest neighbor filter (NNF) [3]. In the SNF, the signal with the highest intensity among the validated measurements (in a gate) is used for track update and the others are discarded. In the NNF, the measurement closest to the predicted measurement is used. However, these methods begin to fail with either increase false alarm (FA) Manuscript received July 8, 2015; accepted July 23, 2015. Date of publication July 28, 2015; date of current version September 14, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61201285 and Grant 61271291, in part by the Program for New Century Excellent Talents in University under Grant NCET-09-0630, and in part by the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant FANEDD201156. The associate editor coordinating the review of this paper and approving it for publication was Dr. Richard T. Kouzes. (Corresponding author: Hongwei Liu.) The authors are with the National Laboratory of Radar Signal Processing, Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [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/JSEN.2015.2461435

or decrease probability of detection (PD) [3]. Instead of using only one measurement among the received ones and discarding the others, the probabilistic data association (PDA) approach [4] avoids a hard association decision by updating a track with a set of measurements and their corresponding weights. With the recent development in real application, many enhanced PDA-based estimators are proposed. A novel joint PDA (JPDA) technique for multitarget tracking is proposed in [5], which greatly reduces the system computation. A particle filter estimator in conjunction with PDA [6] is analyzed to track the maneuvering target in clutter. However, the aforementioned algorithms assume that at most one measurement can be generated by a target in each scan. For target tracking in multiple radar system (MRS), there exists the scenario that multiple detections from the same target fall within the association gate [7]. Aiming at this problem, a multiple detection probabilistic data association (MD-PDA) filter is proposed in [7] and [8] in which the combinatorial association events are formed to handle the possibility of multiple measurements from the same target. Performance evaluation results show the effectiveness, with respect to estimation accuracy, of the MD-PDA algorithm. It is worthwhile to note that, two aspects of the MD-PDA algorithm that need to be improved are: (1) The MD-PDA assumes that multiple detections of a target may be generated, while not considers the restriction that each radar can at most receive one measurement originated from a target; (2) The detector of the MD-PDA approach operates according to Neyman-Pearson (NP) criterion, and a fixed threshold is adopted. In this way, the information flows only one way: from detector to tracker. In this paper, we explore the idea of feeding the information from the tracker to the detector. In this scenario, the tracker can guide the detectors of multiple radars where to look for a target while keeping the constant false alarm rate (CFAR) property. Our previous work [2] proved that, by appropriately using the feedback information, the detector may operate in a Bayesian mode, and the detection performance may be efficiently enhanced. Integrating the above two refinements into the MD-PDA approach, a joint detection and tracking processing (JDTP) algorithm is proposed in this paper, whose block diagram can be found in Fig. 1. B. Main Contributions The major contributions of this paper are as follows: 1) We redefine the combinatorial association events in [7] with a reasonable assumption that each radar can at most

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YAN et al.: JDTP ALGORITHM FOR TARGET TRACKING IN MULTIPLE RADAR SYSTEM

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between successive frames and we refer it to tracking interval, the target is then located at coordinate. (x Tk , yTk ). at time kT0 and moves with a speed (x˙Tk , y˙Tk ). A. Target Dynamic Model The target motion is prescribed by a constant-velocity (CV) model [14]: Fig. 1.

Block diagram of the JDTP algorithm.

receive one measurement originated from a target. In this scenario, the JDTP algorithm can efficiently decrease the number of the association events; 2) Unlike our previous cognitive tracking scheme [9], [10], in which the prior information is fed back from the tracker to the transmitter, the JDTP algorithm constitutes a form of feedback from the tracker to the detectors of multiple radars. Then, the detectors no longer operate in NP mode and instead become Bayesian [11]. From a practical point of view, the Bayesian threshold is depressed near where a target is expected to be and elevated where it is unexpected [11]. Therefore, the JDTP algorithm belongs to a type of detect before track (DBT) technique, and it is different from the track before detect (TBD) algorithm [12], [13], which simultaneously detect and track targets with unthresholded data. Moreover, we calculate the false alarm rate of each Bayesian detector, such that the CFAR property can analytically be ensured. In this scenario, we can enhance the PD efficiently under the precondition that the averaged FA of the gate is fixed [2]. The rest of the paper is structured as follows: Section II formulates the system model. In Section III, the JDTP algorithm is developed. The BCRLB is derived for target tracking in clutter with an adaptive threshold in Section IV. Several numerical results are provided in Section V to verify the effectiveness of the proposed JDTP technique. Finally, the conclusion of the paper is made in Section VI. II. S YSTEM M ODEL Without loss of generality, we consider a MRS with N spatially diverse monostatic radars. These radars are labeled 1, 2, ..., N ≥ 2 with the location of the i th radar denoted by (x i , yi ).1 We assume that: (1) Each radar transmits a signal with different carrier frequency; (2) The radar station i is equipped with only a matched filter that correlates to its own transmitted signal. Thus, any radar station can receive its own signal, and the target echoes from the other signals generate near-zero outputs at the matched filter because it does not correlate with any of them. In this case, each radar operates in a monostatic way, and sends its measurement to the fusion center. We assume that a single target initially located at (x T0 , yT0 ) with an initial speed of (x˙T0 , y˙T0 ). Set T0 as the time interval 1 For brevity, throughout this paper the radar index “i” and the time index “k” will often be omitted, unless doing so causes confusion.

ξ k = Fξ k−1 + uk−1 ,

(1)

where the target state is given by ξ k = [x Tk , x˙Tk , yTk , y˙Tk ]T . [x Tk , yTk ] and [x˙Tk , y˙Tk ] denote the position and velocity of the target, respectively. F is the transition matrix 1 T0 . (2) F = I2 ⊗ 0 1 where the operator ⊗ denotes the Kronecker operator, and Im denotes an identity matrix of order m. The term uk−1 in (1) denotes the process noise, and is assumed to be zero-mean, Gaussian with a known covariance Qk−1 [14]: ⎡1 1 2⎤ T03 T ⎢ 2 0 ⎥ (3) Qk−1 = q1 I2 ⊗ ⎣ 3 ⎦ 1 2 T T0 2 0 where q1 control the amount of process noise in target dynamic model. B. Detection Model Suppose that the i th radar examines the echoes of the target within its own validation gate [2] to decide whether this target is present. Hence, a test of absence or presence of a target at location zli,k is to be performed. Hypothesis H0 is that there is no target and hence, the measured return is due simply to noise. Hypothesis H1 is that there is indeed a target

l l |H0 = exp −ai,k H0 : p ai,k

l

ai,k 1 l (4) H1 : p ai,k |H1 = exp − 1 + μi,k 1 + μi,k l where ai,k is the corresponding amplitude of the target, and μi,k is the signal to noise ratio (SNR). The usual implementation is according to the NP criterion that the PD be maximized subject to a constraint on the FA, and the resulting test can easily be shown to be a comparison of l to a fixed threshold γi,k,NP [11]. However, with the prior ai,k knowledge obtained from the previous recursion cycle in target tracking [2], the appropriate test can be written in a Bayesian mode

l H p ai,k 1 H≥1 pH0 zli,k

ηi,k,BD

(5) < l p ai,k H0 H0 pH1 zli,k

where pH0 zli,k and pH1 zli,k are the predicted prior information of the i th radar

⎧ ⎪ pH0 zli,k = 1/Vi,k ⎪ ⎪ ⎨

T (6) pH1 zli,k = N zli,k ; hi,k ξ k|k−1 , Hi,k C k|k−1 Hi,k ⎪

⎪ ⎪ ⎩ = N zli,k ; zi,k|k−1 , D i,k|k−1

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with ⎧ ⎪ 2 2 ⎪ = h R Ri,k ξ k = (x Tk − x i ) + (yTk − yi ) ⎪ ⎨ i,k θi,k = h θi,k ξ k = arctan [(yTk − yi )/(x Tk − x i )] (12) ⎪ − x x ⎪ Tk i 2 ⎪ Ri,k ⎩ fi,k = h fi,k ξ k = − χi (x˙ Tk , y˙Tk ) yTk − yi corresponding to different measurement components. Hence, we have n z = 3. In (12), χi denotes the carrier wavelength of the i th radar. We assume that the measurement error wi,k is zero-mean, Gaussian with a covariance

(13) i,k = blkdiag σ R2i,k , σθ2i,k , σ 2fi,k , Fig. 2.

where σ R2i,k , σθ2i,k , and σ 2fi,k are the CRLBs on the estimation

The sketch map of the target measurement model.

In (6), Vi,k is the validation region, zi,k|k−1 and Ck|k−1 are the predicted measurement and covariance matrix defined in Section III. ηi,k,BD is a constant which ensures the Bayesian detector operates within a constant false alarm rate mode [2]. With some deviation, we should decide H1 if

l H p ai,k 1 pH1 zli,k ηi,k,BD

> = η¯ i,k,BD , (7) Vi,k p a l H0 i,k

l to a The test can easily be shown to be a comparison of ai,k location-dependent threshold ⎡ ⎤ H1 1 + μ 1 + μ η ¯ ≥ i,k i,k,BD i,k l ⎦ ai,k ln ⎣

BD < l μi,k N z ; zi,k|k−1 , D i,k|k−1 H0

mean square error (MSE) of the target’s range, bearing, Doppler information [15] ⎧ −1

2 ⎪ ⎪ σ R2i,k ∝ μi,k Bi,k ⎪ ⎨ −1 (14) σθ2i,k ∝ μi,k /Bi,N N ⎪ −1

⎪ ⎪ ⎩ σ 2 ∝ μi,k T 2 fi,k i,k In (14), Bi,N N is the null to null beam width of the receiver antenna [15]. The term Bi,k and Ti,k denotes the effective bandwidth and the effective time duration [2]. FAs are modeled as independent and uniformly distributed over the observation volume Vi,k with probability density function (PDF) 1 p υ i,k = Vi,k

i,k

= γi,k,BD .

(15)

(8)

It is clear that the threshold is depressed near where a target is expected to be and elevated where it is unexpected. In other words, the PD of the Bayesian detector is different at distinct locations. Hence, we have to calculate the PD Pdi,k and the FA Pi,fa averaged over the validation gate via the test and the threshold of (8) (see details in Appendix).

III. J OINT D ETECTION AND T RACKING P ROCESSING A LGORITHM Previously, a MD-PDA approach is proposed in reference [7] to solve the problem of single target tracking in MRS, where the number of the association events is defined as Nk =

C. Measurement Model

min(N,M k)

CiMk =

i=0

For target tracking in clutter, we may receive multiple measurements (see Fig. 2). Let m i,k denotes the number of measurements that exceed γi,k,BD , and thus the measurements in the validation gate can be denoted as m i,k j Zi,k = zi,k . (9) j =1

j zi,k

At sampling time k, each measurement has the general form hi,k ξ k + wi,k if target generated j (10) zi,k = υ i,k if false alarm. where hi,k ξ k is the measurement function, T hi,k (·) = h Ri,k (·) , h θi,k (·) , h fi,k (·) (11)

min(N,M k) i=0

Mk ! i ! (Mk − i )!

(16)

N

where Mk = i=1 m i,k . However, a common assumption for real application is that each radar can at most receive one measurement originated from the target. In this case, the number of the association events is supposed to be Nk =

N

m m C1 i,k + C0 i,k .

(17)

i=1

where Nk ≤ Nk . Based on the above discussion, we develop a JDTP algorithm. Moreover, we also explore the idea of feeding the information from the tracker to the detectors. In this scenario, the tracker can guide the detectors where to look for a target while keeping the CFAR property. The general steps of this cognitive tracker can be summarized as Table I.


TABLE I T HE S TEPS OF THE P ROPOSED JDTP A LGORITHM

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the MSE of any estimator cannot go below a bound

T " E ξˆ k|k (Zk ) − ξ k ξˆ k|k (Zk ) − ξ k ≥ J−1 ξ k

(26)

J ξ k is the BIM of the target state: # $ ξ J ξ k = −Eξ k ,Zk ,m i,k ξ k ln p Zk , ξ k

(27)

k

the The notation κη = η κT represents second-order partial derivative vectors. The term p Z , ξ k k is the joint PDF of Zk , ξ k . Similar to [2], J ξ k can be split into two parts (28) J ξ k = J P ξ k + JZ ξ k where J P ξ k and JZ ξ k are the Fisher information matrix (FIM) of the prior information and the data, respectively. A. Prior Information

Similar to [10], J P ξ k can be formulated as ξ J P ξ k = −Eξ k ξ k ln p ξ k k

−1 22 11 12 (29) D J ξ + D = Dk−1 − D21 k−1 k−1 k−1 k−1

For the linear Gaussian described in (1), the FIM of the case prior information J P ξ k can be calculated as [14] # $−1 J P ξ k = Qk−1 + FJ−1 ξ k−1 FT . (30) B. Data Information On the other hand, the FIM of the data can be denoted as JZ ξ k = E ξ k ln p Zk | ξ k ξTk ln p Zk | ξ k . (31) As the measurements from different radars are independent from one another, JZ ξ k can be written as follows N JZ ξ k = Ji,Z ξ k i=1

=

N

Eξ k ,Zi,k ,m i,k ξ k ln p Zi,k ξ k

i=1

×ξTk ln p Zi,k ξ k ,

(32)

where the expectation is over ξ k , Zi,k and m i,k . In (32), Ji,Z ξ k denotes the FIM of i th radar’s measurement. Since m i,k is a nonnegative random integer determining the cardinality of Zi,k , Ji,Z ξ k can be posed as Ji,Z ξ k ∞ = p m i,k Eξ k ,Zi,q,k ξ k ln p Zi,k ξ k m i,k =0

IV. BAYESIAN C RAMÉR -R AO L OWER B OUND Let ξˆ k|k (Zk ) be an estimate of ξ k , which is a function of Zk . The Bayesian Cramér-Rao inequality [16] shows that 2 For notation convenience, we convert the joint event superscript J into

representation of J = [2 · · · 136] (the length of J is N ). For example, !205 denotes the event that the second measurement of radar 1, the fifth k measurement of radar 3 is originated from the target, while none of the measurement from radar 2 is correct.

=

∞ m i,k =0

ξTk ln p Zi,k ξ k m i,k p m i,k

× Eξ k ,Zi,q,k ξ k ln p Zi,k ξ k , m i,k ξTk ln p Zi,k ξ k , m i,k &' ( % m Ji,Zi,k (ξ k ) (33)

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Fig. 4. Fig. 3.

Deployment of the target with respect to our MRS.

The target SNR with respect to different radar.

TABLE II T HE N UMBER OF THE A SSOCIATE E VENTS

In (33), p m i,k is the probability that there exist m i,k observations in the i th radar’s validation gate [17]. Then, m we can achieve Ji,Zi,k ξ k as reference [2]. As m i,k = 0 means no measurement is obtained, we can rewritten (33) as ∞ m p m i,k Ji,Zi,k ξ k Ji,Z ξ k =

(34)

m i,k =1

In this scenario, the FIM of i th radar’s measurement Ji,Z ξ k can be specified as [2] # $ T T μi,k −1 H (35) Ji,Z ξ k = Eξ k Hi,k i,k i,k where T μi,k is the information reduction factor [2]. Substituting (32), (34) and (35) into (28), we have the BIM $−1 # J ξ k = Qk−1 + FJ−1 ξ k−1 FT +

N

# $ T −1 Eξ k T μi,k Hi,k i,k Hi,k

(36)

i=1

V. S IMULATION R ESULTS A radar network with N = 6 spatially diverse radars is considered. The signal effective bandwidth and effective time duration of each radar are set as Bi,k = 2 MHz and Ti,k = 1 ms, respectively. The time interval between successive frames is set as T0 = 2 s, and a sequence of 60 frames of data are utilized to support the simulation. The gate size parameter is g = 8. The target is initially located at (0, 0) km with a constant speed of (−30, 0) m/s. We simulate a scenario that a target is approaching a radar network that is deployed on the borderline, and the angular spread of the multiple radars with respect to a single target is illustrated in Fig. 3. In our simulation, the SNR is set as μ0 = 8 dB for R0 = 10 km. Hence, the target SNR at different radar can be drawn in Fig. 4. In this paper, we analyze the effects of the FA on the detection and tracking results, and consider the following two case: (1) Case 1: Pi,fa = 10−2 ; (2) Case 2: Pi,fa = 10−3 , i = 1, 2, . . . N. In both cases, the PD of different radars achieved by the NP detector and the Bayesian detector are shown in Fig.5. Compare the results in Fig. 5a with that in Fig. 5b, we see that the detection performance of the

JDTP algorithm is better than that of the MD-PDA approach in case 1. Similar results can also be achieved by Fig. 5c and Fig. 5d for case 2. Moreover, compare the results of case 1 and case 2, we can conclude that the higher the FA, the larger improvement can be achieved from the Bayesian detector. Here is an intuitive explanation: the higher the FA, the larger the number of spurious detections, and thus the superiority of the Bayesian threshold can better be expressed. In Fig.6, the detection thresholds of the NP detector and the Bayesian detector are shown for different range cells in the validation gate (Take i = 1 and k = 10 for example). The results show that the Bayesian threshold is depressed near where a target is expected to be and elevated where it is unexpected. Hence, as it shown in Fig. 5, the averaged detection probability can efficiently be improved for both Case 1 and Case 2. In Table II, the number of the associate events is given for the PDA, the MD-PDA, as well as the proposed JDTP algorithms. These results are achieved by averaging over all the frames in 400 Monte Carlo runs. Generally, for higher FA case, the number of the measurements that exceed the detection threshold of each radar will be larger, and thus the number of the events in case 1 is larger than that in case 2. The PDA approach only updates the filter with a single measurement, while the MD-PDA and the JDTP algorithms use a set of measurement instead. Therefore, the number of the association event is larger for the latter two algorithms in Table II. Moreover, according to (16) and (17), we know that: (1) For Case 1, the proposed JDTP algorithm generally yields less number of association events; (2) For Case 2, the number of the events in both methods are almost the same. In other words, if the FA is too small, the JDTP algorithm cannot reduce the number of the associate events evidently, when compared with the original MD-PDA approach. To better examine the optimality of the proposed method, Fig. 7 presents the tracking accuracy of the standard PDA,


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Fig. 6. The physical explanation of the Bayesian detection strategy. (a) Case1; (b) Case2.

Fig. 5. The PD of each radar. (a) Case1 with NP detector; (b) Case1 with Bayesian detector; (c) Case2 with NP detector; (d) Case2 with Bayesian detector.

the MD-PDA [7] and the JDTP algorithms. Herein, the position root MSEs (RMSE) at the kth tracking interval is defined as ) * N MC * 2

2 * 1

j j x Tk − x Tk + yTk − y Tk RMSEk = + N MC j =1

(37)

Fig. 7. The tracking RMSE of both approaches, the BCRLB of MD-PDA can be found in reference [18]. (a) Case 1; (b) Case 2.

where N MC is the number of the Monte Carlo trials,

j j x Tk , y Tk is the state estimate at the j th Monte Carlo trial. The results in Fig. 7 show that the tracking accuracy approaches the BCRLB as the number of measurement increases. Performance evaluation results show the

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effectiveness, with respect to estimation accuracy, of the proposed JDTP algorithm. The standard PDA tends to distribute the weight among the target originated from measurement, while the MD-PDA and the JDTP algorithms assign weights to multiple measurement originated from a target. Therefore, the PDA filter may achieve worse estimation accuracy compared with those two algorithms. According to (16) and (17), we see that, for higher FA case, the number of the association events is greatly reduced. Thus, the tracking accuracy can better be improved when compared with the MD-PDA approach. However, for lower FA case, the tracking accuracy is slightly improved due only to the adoption of a Bayesian detector (the number of the association events is almost the same, see Table II). VI. C ONCLUSION Based on the assumption that each radar can at most receive one measurement originated from the target, a JDTP algorithm is proposed in this paper for single target tracking in MRS. The basis of our tracking technique is to adjust the detection threshold of each radar according to the prior tracking information, with the aim of enhancing the detection performance under a CFAR constraint. Simulation results show that the JDTP algorithm can not only reduce the number of association events, but also improve the detection and tracking performance, when compared with the MD-PDA approach. In the future, we will extend the proposed JDTP algorithm to the distributed MIMO radar case, and verify its effectiveness using real data. A PPENDIX D ERIVATION OF Pdi,k AND Pi,fa It is apparent that the PD of the Bayesian detector is different at distinct locations, and thus we have to calculate the PD averaged over the validation gate via the test and the threshold of (8) ,

Pd zli,k pH1 zli,k dzli,k Pdi,k = Vi,k ⎛ ⎡ ⎤⎞ , 1 + μi,k η¯ i,k,BD 1 ⎦⎠ = exp ⎝− ln ⎣

μi,k Vi,k N zli,k ; zi,k|k−1 , D i,k|k−1

× N zli,k ; zi,k|k−1 , D i,k|k−1 dzli,k =

− 1 1 + μi,k η¯ i,k,BD μi,k , #

$ς × dzli,k N zli,k ; zi,k|k−1 , D i,k|k−1 Vi,k

− μ 1

⎡ · ⎣

√ ς

⎤ς−1

⎦ (2π) i,k , D i,k|k−1 − ς2 l ·ς N zi,k ; zi,k|k−1, √ dzli,k ς Vi,k ⎡ ⎤ς−1 √ − μ 1 ς ς ⎦ · ς− 2 i,k · ⎣ = 1 + μi,k η¯ i,k,BD (2π)nz i,k =

1 + μi,k η¯ i,k,BD

i,k

nz

(38)

In (38), Pd zli,k is the PD of location zli,k , and 1 + μi,k ς= . μi,k

(39)

Similarly, we calculate the average FA ,

Pi,fa = Pfa zli,k pH0 zli,k dzli,k Vi,k ⎛ , 1 + μi,k = exp ⎝ − × μi,k Vi,k ⎡ ⎤⎞ 1 + μi,k η¯ i,k,BD 1 ⎦⎠ ln ⎣

dzli,k l V i,k N zi,k ; zi,k|k−1 , D i,k|k−1 =

−ς 1 1 + μi,k η¯ i,k,BD Vi,k ⎤ς−1 ⎡ √ ς ς ⎦ · ⎣ · ς− 2 (2π)nz i,k

(40)

Constraining the FA Pi,fa to be an arbitrary small value, we can find η¯ i,k,BD as well as Pdi,k through (38)-(40). R EFERENCES [1] S. S. Blackman, Multiple-Target Tracking with Radar Applications. Norwood, MA, USA: Artech House, 1986. [2] J. Yan, B. Jiu, H. Liu, B. Chen, and Z. Bao, “Prior knowledge-based simultaneous multibeam power allocation algorithm for cognitive multiple targets tracking in clutter,” IEEE Trans. Signal Process., vol. 63, no. 2, pp. 512–527, Jan. 2015. [3] M. E. Liggins, D. L. Hall, and J. Llinas, Eds., Handbook of Multisensor Data Fusion: Theory and Practice. Boca Raton, FL, USA: CRC Press, 2009. [4] Y. Bar-Shalom and X. R. Li, Estimation and Tracking: Principles, Techniques and Software. Boston, MA, USA: Artech House, 1993. [5] A. M. Aziz, “A joint possibilistic data association technique for tracking multiple targets in a cluttered environment,” Inf. Sci., vol. 280, pp. 239–260, Oct. 2014. [6] Y. Bar-Shalom, F. Daum, and J. Huang, “The probabilistic data association filter,” IEEE Control Syst. Mag., vol. 29, no. 6, pp. 82–100, Dec. 2009. [7] B. K. Habtemariam, R. Tharmarasa, T. Kirubarajan, D. Grimmett, and C. Wakayama, “Multiple detection probabilistic data association filter for multistatic target tracking,” in Proc. 14th IEEE Int. Conf. Inf. Fusion, Chicago, IL, USA, Jul. 2011, pp. 1–6. [8] B. Habtemariam, R. Tharmarasa, T. Thayaparan, M. Mallick, and T. Kirubarajan, “A multiple-detection joint probabilistic data association filter,” IEEE J. Sel. Topics Signal Process., vol. 7, no. 3, pp. 461–471, Jun. 2013. [9] J. Yan, H. Liu, B. Jiu, and Z. Bao, “Power allocation algorithm for target tracking in unmodulated continuous wave radar network,” IEEE Sensors J., vol. 15, no. 2, pp. 1098–1108, Feb. 2015. [10] J. Yan, H. Liu, B. Jiu, B. Chen, Z. Liu, and Z. Bao, “Simultaneous multibeam resource allocation scheme for multiple target tracking,” IEEE Trans. Signal Process., vol. 63, no. 12, pp. 3110–3122, Jun. 2015. [11] P. Willett, R. Niu, and Y. Bar-Shalom, “Integration of Bayes detection with target tracking,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 17–29, Jan. 2001. [12] B. K. Habtemariam, R. Tharmarasa, and T. Kirubarajan, “PHD filter based track-before-detect for MIMO radars,” Signal Process., vol. 92, no. 3, pp. 667–678, Mar. 2012. [13] F. Papi, V. Kyovtorov, R. Giuliani, and F. Oliveri, “Bernoulli filter for track-before-detect using MIMO radar,” IEEE Signal Process. Lett., vol. 21, no. 9, pp. 1145–1149, Sep. 2014. [14] B. Ristic, S. Arulampalam, and N. Gordon, Beyond the Kalman Filter: Particle Filters for Tracking Applications. Norwell, MA, USA: Artech House, 2004.


[15] H. L. Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory. New York, NY, USA: Wiley, 2002. [16] R. Niu, P. Willett, and Y. Bar-Shalom, “Matrix CRLB scaling due to measurements of uncertain origin,” IEEE Trans. Signal Process., vol. 49, no. 7, pp. 1325–1335, Jul. 2001. [17] X. Zhang, P. Willett, and Y. Bar-Shalom, “Dynamic Cramer-Rao bound for target tracking in clutter,” IEEE Trans. Aerosp. Electron. Syst., vol. 41, no. 4, pp. 1154–1167, Oct. 2005. [18] B. K. Habtemariam, R. Tharmarasa, M. Mallic, and T. Kirubarajan, “Performance comparison of a multiple-detection probabilistic data association filter with PCRLB,” in Proc. IEEE Int. Conf. Control Autom. Inf. Sci., Ho Chi Minh City, Vietnam, Nov. 2012, pp. 18–23.

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Bo Jiu (M’13) received the B.S., M.S., and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, China, in 2003, 2006, and 2009, respectively. He is currently an Associate Professor with the National Laboratory of Radar Signal Processing, Xidian University. His research interests are radar signal processing, radar automatic target recognition, radar imaging, and cognitive radar.

Junkun Yan was born in Sichuan, China, in 1987. He received the B.S. and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, China, in 2009 and 2015, respectively. He is currently doing post-doctoral research with the National Laboratory of Radar Signal Processing, Xidian University. His research interests include adaptive signal processing, target tracking, and cognitive radar.

Zheng Liu was born in 1964. He received the B.S., M.S. and Ph.D. degrees in 1985, 1991, and 2000, respectively. He is currently a Professor, the Doctoral Director, and the Vice Director of the National Laboratory of Radar Signal Processing with Xidian University, Xi’an, China. His research interests include the theory and system design of radar signal processing, radar precision guiding technology, and multisensor data fusion.

Hongwei Liu (M’04) received the M.Eng. and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, China, in 1995 and 1999, respectively. From 2001 to 2002, he was a Visiting Scholar with the Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA. He is currently a Professor with the National Laboratory of Radar Signal Processing, Xidian University, and the Director of the National Laboratory of Radar Signal Processing. His research interests are radar signal processing, radar automatic target recognition, adaptive signal processing, and cognitive radar.

Zheng Bao (M’80–SM’90) was born in Jiangsu, China. He is currently a Professor with Xidian University and the Chairman of the Academic Board of the National Key Laboratory of Radar Signal Processing. He has authored or co-authored six books and published over 300 papers. His research fields include space-time adaptive processing radar imaging, automatic target recognition, and over-the-horizon radar signal processing. He is a member of the Chinese Academy of Sciences.