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Prior Knowledge-Based Simultaneous Multibeam Power Allocation Algorithm for Cognitive Multiple Targets Tracking in Clutter Junkun Yan, Bo Jiu, Member, IEEE, Hongwei Liu, Member, IEEE, Bo Chen, Member, IEEE, and Zheng Bao, Life Senior Member, IEEE
Abstract—In this paper, a power allocation scheme for tracking multiple targets, with radar measurements either target generated or false alarms, is developed for colocated multiple-input multipleoutput (MIMO) radar system. Such a system adopts a multibeam concept, in which multiple simultaneous transmit beams are synthesized by different probing signals from various colocated transmitters. To ensure that the limited power resource can be exploited effectively, we adjust the transmit power of each beam according to the prior knowledge predicted from the tracking recursion cycle. Specifically, the whole algorithm can be viewed as a reaction of the cognitive transmitters to the environment, in order to improve the worst case tracking performance of the multiple targets. By incorporating an information reduction factor (IRF), the Bayesian Cramér-Rao lower bound (BCRLB) gives a measure of the best achievable performance for target tracking in clutter. Hence, it is derived and utilized as an optimization criterion for the simultaneous multibeam power allocation algorithm. The optimization problem is nonconvex and is solved by the modified gradient projection (MGP) method in this paper. Simulation results show that the proposed algorithm significantly outperforms equal power allocation, in terms of the worst case tracking root mean-square error (RMSE). Index Terms—BCRLB, cognitive tracking, colocated MIMO radar, multiple target, simultaneous multibeam power allocation.
I. INTRODUCTION A. Background and Motivation
T
HE problem of multiple targets tracking (MTT) has been of great interest for various commercial and military applications [1]–[4]. For instance, application areas include ballistic missile defense, air defense, air traffic control and battlefield surveillance. Generally, MTT deals with the state estimation of multiple moving targets. Available measurements may Manuscript received August 14, 2014; revised November 07, 2014; accepted November 10, 2014. Date of publication November 20, 2014; date of current version December 18, 2014. The associate editor coordinating the review of this manuscript and approving it for publication was Porf. Sofiene Affes. This work is partially supported by the National Natural Science Foundation of China (61271291, 61201285), Program for New Century Excellent Talents in University (NCET-09-0630), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD-201156) and the Fundamental Research Funds for the Central Universities. (Corresponding author: H. Liu.) The authors are with the National Laboratory of Radar Signal Processing, 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/TSP.2014.2371774
Fig. 1. simultaneous multiple transmit beams, multiple receive beams a. Transmit beam pattern b. Receive beam pattern.
both arise from the targets, if they are detected, and from clutter. Clutter is generally considered to be a model describing false alarms [4]. Many modern avionics systems including multiple sensors as well as individual sensors are capable of tracking multiple targets with different working modes. Technically, based on the multibeam concept [5], multiple targets can be tracked by colocated multiple-input multiple-output (MIMO) radar systems [6], [7]. Herein, the expression ‘multibeam’ refers to a mode of operation where multiple simultaneous transmit beams are synthesized by different probing signals from various colocated transmitters [8]. When the transmit illumination is defocused, receive beams must be chosen to span the volume of space illuminated by the transmitter, see Fig. 1. Typically, this is done by using multiple, highly focused receive beams (collectively spanning the region of transmit illumination) [5]. Thus, the signals from multiple targets can be separated using their spatial diversity. In this working mode, the user can lower the peak power to achieve low probability of interception, while extending the integration time so as to maintain system sensitivity [5]. This technique eases radar equipment specifications, and can offer improved performance (e.g., increase Doppler resolution) [5]. However, there exists the fact that the consumption of the transmit power in the multibeam technique grows greatly with the number of targets. In order to avoid the radar transmitter consumption power surpassing the endurable ability of radar system physical equipment, the power constraint must be taken into consideration. In other words, at any illumination, the total transmit power of the multiple beams needs to be constrained. For traditional colocated MIMO radar system with a multibeam working mode, any prior information is not used for power allocation. For convenience, the fixed system power budget is uniformly allocated to multiple beams to formulate the desired beam pattern. However, in practice, prior knowledge can be
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obtained from the previous recursion cycle in target tracking, e.g., the predicted target state and the target radar cross section (RCS). If the prior knowledge can be used in a cognitive manner for the power allocation strategy, the limited power resource could be exploited effectively. Here, the expression ‘cognitive’ is meant to adaptively adjust the transmitter to perceive the environment according to the predicted prior information [9]. So far, there exist a plethora of works that aim at enhancing the utilization efficiency of the scarce radar power resource [10]–[13]. Reference [10] proposes a power allocation scheme for CS-based colocated MIMO radar systems, with the aim of rendering the sensing matrix as orthogonal as possible. In [11], the authors propose a method to minimize the total transmit power such that a predetermined localization Cramér-Rao lower bound (CRLB) is met, or to minimize the CRLB by optimizing power allocation among the transmit radars for a given total power budget. The problem of power allocation among sensor nodes for locating a static target or for tracking a dynamic target in either linear or nonlinear sensing systems has been addressed in [12]. Besides, [13] computes the Bayesian CRLB (BCRLB) [14] on the estimates of the target state and the channel state, and uses it as an optimization criterion for the antenna selection and power allocation under a complex urban environment. Generally, the above works provide us an opportunity to deal with the power management problem. However, in these works [11]–[13], the detection and tracking strategy are isolated, and two ideal detection precondition [15] are assumed: (1) the probability of detection is one; (2) the false alarm rate is zero. Under these assumptions, a measurement always contains the location information of the target. Such a favorable circumstance may not always hold true, due to false alarms and missed detections [16]. Therefore, a closed-loop cognitive framework is built herein for the case of tracking multiple, widely spread targets in colocated MIMO radar system under the clutter environment. The feedback of the closed-loop system is formed to enhance the utilization efficiency of the limited power resource in the simultaneous multibeam working mode. Unlike the previous power allocation scheme for colocated MIMO radar [17], in which the transmit power of each element is allocated appropriately, we adjust the transmit power of the multiple beams to formulate the desired beam pattern. The whole algorithm can be viewed as a reaction of the cognitive transmitter to the environment, in order to minimize a specified performance metric. Conventional metrics like overall mean square error (MSE) do not efficiently capture the estimation accuracies in multi-target scenario [18]. For example, even if the estimates of some of the targets are poor, the overall MSE (averaged over all the targets) can still be small if the estimates of the other targets are very accurate. Thus, our approach is based on the optimization of the worst tracking performance of multiple targets (we refer it to the worst case tracking performance). The BCRLB gives a measure of the achievable optimum performance and, importantly, this bound can be calculated predictively [14]. Hence, it is utilized as an optimization criterion for the simultaneous multibeam power allocation strategy.
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B. Main Contributions The major contributions of this paper are fourfold: 1) By incorporating the Bayesian detector [19] into our system model, we derive the expression of the tracking BCRLB under the clutter environment. 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. This is termed measurement origin uncertainty (MOU) [20]–[24]. Previous results show that in the calculation of the BCRLB for a given measurement, the presence of false alarms and clutter manifests itself as a scale factor under certain conditions [20], namely information reduction factor (IRF). By incorporating an IRF, we can efficiently calculate the tracking BCRLB with MOU. Different from the BCRLB derived for MTT in [25], this paper incorporates a Bayesian detector into the system model, which will yield a distinct IRF; 2) A simultaneous multibeam power allocation algorithm for MTT in clutter is proposed based on the feedback from the tracker. The BCRLB with MOU, which provides a reliable bound for the performance of target tracking in clutter [24], is utilized as an optimization criterion for dynamic power management. Implementing the resource allocation techniques into real-time systems necessitates the quick and efficient calculation of the BCRLB. To achieve this goal, we decouple the effect of the MOU and the target state uncertainty with some moderate assumptions [23]. In this scenario, the MOU can be expressed as a state-dependent IRF which can be calculated off-line. It is shown that the power optimization problem is nonconvex, and thus, a modified gradient projection (MGP) technique is proposed to find a suboptimal solution; 3) We build a close-loop cognitive MTT scheme in the presence of clutter for colocated MIMO radar system. For a number of well separated targets, MTT can be simplified as a number of single target tracking problems which can be solved independently [26]. For each target, the probabilistic data association (PDA) [27] approach is adopted to obtain an approximate estimate of the hybrid state vector (target state and channel state). Then, the predictive information obtained from the tracking recursion cycle is used to form a probabilistic understanding of the environment [28], and the colocated MIMO radar incorporates this prior knowledge into its probing strategy, thereby rendering it a closed-loop system. Note that the feedback information from the tracker may also enhance the detection probability, and thus we incorporate the Bayesian detector [19] into our system model. In this case, the close-loop cognitive tracking system constitutes a form of feedback from the tracker to the transmitter as well as the detector, and its block diagram is given in Fig. 2; 4) In order to have a better understanding of the power allocation strategy and the Bayesian detection strategy, some physical explanations are provided. The rest of the paper is structured as follows: Section II formulates the system model, in which the Bayesian detection
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where is the carrier frequency, and represents the transmit power. The term denotes the normalized complex envelope of the transmit signal. The transmit waveform of the th beam is assumed to be a narrowband signal with an effective bandwidth [29] (2) and an effective time duration Fig. 2. Block diagram of the close-loop cognitive tracking system.
(3)
model is introduced. In Section III, the Bayesian information matrix (BIM), whose inverse yields the BCRLB, is derived for target tracking in clutter with an adaptive threshold. Then, some tricks are adopted to ensure that the BCRLB can be calculated quickly and efficiently. The simultaneous multibeam power allocation algorithm is proposed in Section IV. The basis of the allocation scheme is introduced in Section IV-A. Section IV-B derives the criterion for the allocation strategy. The resulting nonconvex optimization problem is solved by MGP method in Section IV-C. In Section IV-D, we divide the MTT problem into a number of single target tracking problems, and utilize the PDA filter to achieve the hybrid state estimation of each target in a cognitive manner. Several numerical results are provided in Section V to verify the effectiveness of the proposed cognitive tracking technique. Finally, the conclusion of the paper is made in Section VI.
The baseband representation of the received signal from the th target is an attenuated version of the transmit signal, which is delayed by and frequency shifted by
(4) The attenuation represents the variation in the signal strength due to path loss effects ( is the range between the th target and the radar). In this paper, the target reflectivity is modeled as a random parameter ( and are the real and imaginary parts of ), and needs to be estimated at every instant. is a zero-mean, complex white Gaussian noise. B. Target Motion Model The target motion is prescribed by a constant-velocity (CV) model [30]
II. SYSTEM MODEL Consider a colocated MIMO radar system, arbitrarily located at coordinate . A set of independent point targets is aswith an sumed. The th target is initially located at initial speed of . Set as the time interval between successive frames and we refer it to tracking interval, the th , ) at time and target is then located at coordinate ( moves with a speed , . The radar system is to track the locations and has available estimates of some other unknown parameters of these targets. This system adopts a multibeam concept in which multiple simultaneous and independent beams are synthesized by different probing signals from various colocated transmitters, with the th beam tracks the th target only. To begin with, in order to simplify the problem, we give some moderate assumptions: 1) The number of targets is known a priori (this information can be obtained from the radar search mode); 2) The target is widely distributed in the surveillance region. A. Signal Model Assuming that the th beam transmits a signal th tracking interval with the following waveform1:
[29]
at the
(1) 1For brevity, throughout this paper the target/beam index “ ” and the time index “ ” will often be omitted, unless doing so causes confusion.
(5) where the target state is given by with a dimension of . and denote is the position and velocity of the th target, respectively. the transition matrix of the target state (6) denotes an identity where is the Kronecker operator, and matrix of order . The term in (5) denotes the process noise, and is assumed to be zero-mean Gaussian with a known covariance [31] (7) where denotes the level of the process noise for the th target. The transition model of the th target’s channel is assumed to be a first order Markovian process [13], and is described by the following equation: (8) where the noise is white Gaussian with a known co. The term represents the variance channel state vector. Now, we form an extended state vector for the th target by concatenating the target state vector and the
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channel state vector into a single vector of dimension , . The state transition which can be defined as equation for is then
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where , , , and are the CRLBs on the estimation MSEs of the th target’s range, bearing, Doppler and RCS [29], [32] information
(9) where
is the overall transition matrix (16)
(10) is the corresponding Gaussian process noise with The term zero mean and covariance , denotes the block diagonal mawhere the operator trix. Hereafter, when we say state vector, we refer to the extended state vector formed by concatenating the target state and the channel state. C. Measurement Model We analyze the general case in which there exists MOU, with measurements either target generated or false alarms. In this case we can receive multiple measurements for each target. Let denotes the number of measurements ( is the dimension of each measurement) available for the th target. These measurements can all be false alarms, or there is one true detection (with probability ) and false alarms, or there might even be no observations . For notation simplicity, we define the measurement vector of the th target to be (11) At sampling time , each measurement of the th target, has the general form if target generated if false alarm
is the signal to noise ratio (SNR), and is the In (16), null to null beam width of the receiver antenna [32]. Here, we remind the reader that attenuation parameter and transmit power are defined in (4) and (1), respectively. False alarms are modeled as independent and uniformly distributed over the observation volume with probability density function (PDF) (17) and the distribution of the number of false alarms is Poisson with mean [33] (18) is the spatial density of the false alarms, and its exwhere pression is defined in Section IV-D. With the probability of detection denoted as , the probability that there are observations is given by:
(12) (19)
where
with
(13)
where the indicator function
is defined as (20)
The probability that one measurement is target generated given that there are measurements is
(14) corresponding to different measurement components. Hence, we have . In (14), denotes the carrier wavelength of the th beam, is a zero vector of length with the th element to be one. We assume that the measurement error is zero-mean, Gaussian with a covariance (15)
The conditional PDF of
, given
and
(21) , is then [22]
(22)
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where is PDF of the true measurement2 [see (23) at the bottom of the page].
ensures the Bayesian detector operates within a constant false alarm rate mode. With some deviation, we should decide if
D. Detection Model We assume that a radar examines the echoes given in (4) within a rectangular validation gate to decide whether the target is present. Hence, a test of absence or presence of a target at location is to be performed. Hypothesis is that there is and hence, the measured return is due no target at location simply to noise. Hypothesis is that there is indeed a target at location and thus, the return is due to a combination of noise and signal energy. Therefore, we write
(24) where is the corresponding amplitude (magnitude-square output of a matched filter, with a Swerling I target fluctuation model implicit) of the th target. The usual implementation is according to the Neyman-Pearson criterion [34] that the probability of detection be maximized subject to a constraint on the false alarm rate, and the resulting test can easily be shown to be a comparison of to a fixed threshold [35] (25) However, with the prior knowledge obtained from the previous recursion cycle in target tracking, the appropriate test can be written in a Bayesian mode
(28)
The test can easily be shown to be a comparison of location-dependent threshold
to a
(29) 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 detection probability of the Bayesian detector adopted in this paper is different at distinct locations. Hence, we have to calculate the probability of detection and the false alarm rate averaged over the validation gate via the test and the threshold of (29) (see details in Appendix A). The general state space model and Bayesian detection model of the th target is given by (9), (12) and (73), respectively. At the th tracking interval, the parameter to be estimated is the hybrid target state . However, the inaccuracy of the target dynamic model and the measurement model may lead to the uncertainty of the estimated results. In the next section, the analytical expression of the BIM with MOU, which provides a bound on the performance of the estimators of , will be given. Organizing the transmit parameter in order to control the target state estimation error bounds (determined from the sequence of BIMs), which forms the basis of the simultaneous multibeam power allocation technique, will then be introduced in Section IV.
(26) III. BAYESIAN FISHER INFORMATION MATRIX WITH MOU where tion
and
are the predicted prior informa-
(27) and are the predicted measurement and covariis a constant which ance matrix defined in Section IV-D. 2A standard notation PDF of variable with mean
. is used in this paper to denote the Gaussian and covariance matrix .
be an estimate of , which is a function of the Let measurement vector . The Bayesian Cramér-Rao inequality [20], [22] shows that the MSE of any estimator cannot go below a bound
(30) where the expectation is over BIM of the th target’s state:
,
and
.
is the (31)
(23)
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The notation represents the second-order partial derivative vectors. The in (31) is the joint PDF of , which can be factorized as: (32) is the joint conditional PDF, and is the where PDF of the target state. Reference [14] provided an elegant method of computing the BIM recursively without manipulating the large matrices at each time index
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Since is a nonnegative random integer determining the cardinality of , can be posed as [15] (38), shown at the bottom of the page. As means no measurement is obtained for the th target, we can rewritten (38) as (39) is detailed in the Appendix B. SubThe derivation of stituting (82) into (39), the FIM of the data is then
(33) and are the Fisher information matrix where (FIM) of the prior information and the data, respectively. A. Prior Information According to [14],
can be formulated as (40)
(34) , For the linear Gaussian case described in (9), the terms and are deterministic and the expectation operator can be dropped out [30]
Reference [21] and [22] show that the off-diagonal elements of , which involve integration of odd-symmetric matrix terms, are all zero, and thus the information reduction matrix (41)
(35) Substituting (35) into (34) and using the matrix inversion can lemma as [30], the FIM of the prior information be calculated as
is diagonal. Then, we have the BIM
(42)
(36) B. Data Information On the other hand, the FIM of the data can be denoted as
depends on the transmit power We note that the BIM through the detection probability and the measurement covariance . C. Calculation of From (82), we have (43) at the bottom of the next page, where the notation
(37)
(44)
(38)
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Defining at the bottom of the page, with
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, we can write (43) as (45)
integration techniques [36] are generally used for this calculacan be obtained by tion. In this case, the term substituting (19), (49) and (50) into (41), and the BIM is
(46) In the calculation of , numerical integration is necessary. For reasons of computational load, it is reasonable to restrict our measurements to a validation gate [22] (47) is the th component of the th measurement, and where is the standard deviations of the corresponding measurement component given in (16); is a constant controlling the size of validation region. In (45), the gate volume is then [21] (48) Substituting (48) into (45) and using the similar simplification given in [21] (replace with , where , for and ), we have (49) where [see (50) at the bottom of the page]. A closed form solution to (50) is not known, and thus Monte Carlo or numerical
(51) The scalar the IRF [22].
is
D. Decoupling the Effect of
and
However, according to (51), the IRF depend and on the target state through the detection probability the measurement covariance , and this makes things far more complicated. Indeed, it may then not be able to express as a constant IRF. The Monte Carlo integration that must then be performed involves jointly sampling target state evolutions and measurement sequences and is computationally intensive [21]. However, the BCRLB, which is used for real-time resource management, must be evaluated online as the tracking evolves. Hence, the aim of this section is to decouple the effect of the MOU (quantified by the ) and the target state uncertainty (given by ). In this scenario, the
can be determined off-line, allowing
(43)
(45)
(50)
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straightforward and computationally inexpensive calculation of . Because of our assumption that measurements are only available at discrete times , similar to [23], we may achieve the IRF by using the average SNR between sampling times and (52) where
is the average range [23] (53)
The expression of
is given in (14), and (54)
Similarly, the average target RCS is approximated as
IV. SIMULTANEOUS MULTIBEAM POWER ALLOCATION STRATEGY A. Basis of the Technique Mathematically, the power allocation strategy for the colocated MIMO radar system can be formulated as a problem of optimizing a certain system level utility function subject to some power constraints. The BIM [20], [22], derived in the previous section, bounds the error variance of the unbiased estimates of the unknown target state obtained from the radar measurements with MOU, and thus can be utilized as the optimization criterion for the power allocation strategy. At each time index, we can analyze a set of (candidate), and in each case determine a predictive BIM, from which we obtain the performance bounds. We are then in a position to design and control the transmit parameter in order to achieve the best performance. The general resource allocation strategy can be detailed as follows. B. Predictive BCRLB and the Performance Metric
(55) and Replacing , the BIM of the
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with the averaged range and RCS th target can be reformulated as
The crucial feature of the power allocation algorithm is that it must be predictive. The predictive BCRLB then gives us the ability to make decisions in advance based on current knowledge. Given the updated BIM of the th target at time index and a candidate , we can now calculate the predictive BIM at time index by using the Riccati-like recursion given in (57):
(58) (56) can be achieved by (73). is the IRF where and ), which is a function of (between sampling times the average target state. In this scenario, we can determine the relationship between and off-line, prior to resource allocation [23]. As a result, the BIM can be efficiently determined and utilized for real-time, online power management. In practice, the expected value in (56) may be evaluated using Monte Carlo techniques [30]. For real time application, it is advantageous to approximate the BIM as follows
In all, the prediction BIM can be calculated by the utilization of the prior information of the target motion, as well as the candidate . Then, the predictive BCRLB of the th target is defined as the inverse of the BIM: (59) denote the lower The diagonal elements of bound on the variances of the estimation of the th target’s hybrid state. To accomplish the assignment of the fixed power budget, it is sufficient for us to utilize the worst case tracking BCRLB as a criterion for the simultaneous multibeam resource allocation strategy3
(57) (60) where denotes the predicted state vector of the th target for the case of zero process noise [30]. Besides, some further approximations can also be made for some specified circumstances (e.g., when the false alarm is low), but it is beyond the scope of this paper. For more details, the reader may refer to [20]–[24] for an in-depth description.
where the (60) characterizes the worst case tracking accuracy at time index . 3One can also use the trace of the weighted sum of the position and velocity lower bounds as the performance metric, while the whole algorithm can also be implemented with some minor modifications.
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TABLE I THE MGP ALGORITHM
in clutter [27]. Instead of using only one measurement among the received ones and discarding the others, the PDA approach [27], which uses all of the validated measurements with different weights (probabilities), is adopted in this paper. The general steps of the PDA filter with the power allocation strategy can be listed as follows: Step 0) Let , , for the th target, initiate , , where ; Step 1) Predict the target state, measurement and the covariance matrix: (62)
Step 2) Constraining the false alarm rate to be an arbitrary small value, we can obtain as well as through (73)–(75) for , with the utilization of the feedback information and . Then, we collect the measurements in the validation gate that exceed as (given in (11)). Here, the spatial clutter density of the false alarms in the validation region is [15] C. Criterion Minimization for a Predetermined Power Budget According to (58), we know that the target tracking accuracy is related to several parameters. The adaptable parameter considered in this paper is the transmit power of the mul, the aim tiple beams. For a fixed system power budget of our work is to adaptively allocate the limited power resource which can result in the minimization of the predicted worst case BCRLB, subject to the power limit of each beam. The resulting optimization problem is then
(61) where , and the transmit power for the th beam is constrained by a maximum value and a min. imum value The optimization problem in (61) is nonconvex, nonlinear, due to the IRF . According to [11] and [23], a will result with an increase in either the small increase in IRF or the Fisher information of the data without MOU , and that indicates that any decrease in the power will result in a higher BCRLB. Using this property, we propose a MGP method for the optimization problem given in (61), which is illustrated in Table I. Herein, is the suboptimal solution to the power vector, which follows the total power constraint. D. Target State Estimation In tracking multiple targets with and , data association—deciding which of the received multiple measurements to use to update each track, is crucial. For a number of well separated targets, the problem of MTT can be simplified as a number of single target tracking problems which can be solved independently. In past several decades, many algorithms have been developed to solve the problem of single target tracking
(63) total number of cells in the validation gate; where Step 3) Calculate the innovation covariance (for the correct measurement) (64) where can be obtained by substituting into (52); Step 4) State update: the conditional mean of the state can be written as (65) where is the updated state conditioned on the event that the th validated measurement is correct (66) is the corresponding innovation, and is the gain is the association probability [27] [27]. In (65), (67) is the event that the th validated measurement where is correct, and represents the event that none of the measurements is correct; Step 5) Covariance update: the covariance associated with the updated state is (68) where the covariance of the state updated with the correct measurement is [27] (69)
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TABLE II THE PARAMETERS OF EACH TARGET
Fig. 4. Deployments of multiple targets with respect to our colocated MIMO radar. Fig. 3. Two RCS models: and . of the th target at each time index. a. The first RCS model . RCS model
is the RCS b. The second
and the spread of the innovations term is [27] (70)
The angular spread of these targets with respect to our radar system is given in Fig. 4.
with (71) Step 6) Implement the power allocation algorithm given in Table I, and send the suboptimal allocation result to the colocated MIMO radar system; Step 7) Let , and go to Step 1). V. SIMULATION RESULTS In order to better reveal the effects of several factors on the power allocation results, we consider two RCS models , which are shown in Fig. 3, and two dynamic models 1) 2)
successive frames is set as , and a sequence of 40 frames of data are utilized to support the simulation. The false alarm rate is set as . The lower and upper bounds , of the power of the th beam are set as , respectively. The number of targets is set as , and the parameters of each target are given in Table II.
:
, :
A. Case 1:
and
This case supports the evaluation of the resource allocation strategy with the target RCS and the level of the process noise factored out. Hence, the allocation of power resource compensate only for the deployment of the targets. To examine the superiority of the proposed power allocation method, the worst case tracking root MSEs (RMSE) with the corresponding BCRLBs are presented for the first RCS model in Fig. 5. Here, the position RMSE is defined as
; others
Combining different models of target RCS and target motion, we investigate the following three cases. In each case, our colocated MIMO radar system is located at (127.5,3) km, the signal effective bandwidth and effective time duration of the th beam are set as and , respectively. The carrier frequency of each beam is set as 1 GHz, and thus the carrier wavelength is . The time interval between
(72) is the number of Monte Carlo trials, where is the state estimate of the th target at the th trial. ” and “ ” In Fig. 5, curves labelled “ show the tracking RMSE achieved with uniform power allocation and suboptimal power allocation, respectively. The results imply that the tracking accuracy approaches the BCRLB
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Fig. 5. The worst case tracking BCRLB and RMSE in case1.
Fig. 7. The simultaneous multibeam power allocation results in case1.
Fig. 8. The worst case tracking BCRLB and RMSE in case2.
to the targets that are farther to the radar (see Table II), so that larger gain in terms of the worst case tracking accuracy will be achieved. B. Case 2:
Fig. 6. The tracking BCRLB of different targets in case1. a. Equal power allocation b. Suboptimal power allocation.
as the number of measurements increases. These results also indicate that the cognitive tacking method dilutes the error variation through the adequate distribution of the transmit power. The process of adaptively adjust the transmitter as well as the detector to perceive the environment through the feedback may improve the tracking accuracy about 40% in this case. Fig. 6 presents the estimation performance of different targets. The results show that the power allocation strategy makes the tracking BCRLBs of different target closer to each other. In order to disclose the effects of the deployment of the targets on the allocation results, Fig. 7 depicts the power allocation results averaged over 100 Monte Carlo trials. In Fig. 7, more power is allocated to beam 1, as increasing the transmit power of the beams pointing at the targets that are farther to the radar will have a stronger impact on the worst case tracking performance (see Fig. 6(a), more power is allocated to target 1). Thus, we know that the power will added to the beams corresponding
and
We then expand our simulation with the consideration of the losses due to target RCS. The worst case tracking performance is evaluated in Fig. 8 for case 2, which proves that the utilization of the prior information for power allocation can also greatly improve the tracking performance. In Fig. 9, the tracking BCRLBs of each target are given for case 2, according to which we know that the proposed power allocation strategy can improve the worst tracking performance of the multiple targets evidently. It also brings the tracking performance of different target closer to each other. The corresponding allocation results are presented in Fig. 10, according to which we know that the targets with weaker path conditions require higher transmit power, compared with those with better propagation paths. By using the RCS model , beam 2 is distributed with more power, compared with the results given in Fig. 7. However, target 1 is still the target with worst tracking performance, and thus more power is distributed to beam 1. The above results imply that the allocation of the limited power resource is determined by the following two factors: target reflectivity and target range. C. Case 3:
and
In this part, we analyze the effects of the accuracy of the target dynamic model on the allocation results. The worst case tracking performance is given in Fig. 11 for case 3, which shows
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Fig. 9. The tracking BCRLB of different targets in case2. a. Equal power allocation b. Suboptimal power allocation.
Fig. 12. The tracking BCRLB of different targets in case 3. a. Equal power allocation b. Suboptimal power allocation.
Fig. 10. The simultaneous multibeam power allocation results in case2.
Fig. 13. The simultaneous multibeam power allocation results in case3.
Fig. 11. The worst case tracking BCRLB and RMSE in case3.
The results imply that the targets with larger tracking BCRLB for equal power allocation are allocated more power resource (similar conclusions can be drawn from Figs. 9 and 6). The relationship between the accuracy of the dynamic model and the allocation of power resource is highlighted in Fig. 13. In this case, more power is allocated to target 3, compared with the allocation results of case 1, as it has a less accurate dynamic model. Thus, we know that the power allocation strategy tends to assign more power to the targets that are farther to the radar, have weaker propagation paths and have less accurate dynamic models. D. Physical Interpretations
the superiority of the proposed simultaneous multibeam power allocation algorithm. The effect of the power allocation strategy on the tracking BCRLBs of different targets are illustrated in Fig. 12 for Case 3.
In order to have an insight into the simultaneous multibeam power allocation strategy and the Bayesian detection strategy, Figs. 14 and 15 provide us with some physical interpretations. Taking the tracking interval for each case as example,
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Fig. 15. The physical explanation of the Bayesian detection strategy in case3.
Fig. 14. The physical explanation of the simultaneous multibeam power allocation strategy in each case. The corresponding power allocation results are depicted in the subfigures. a. Equal power allocation in case1 b. Suboptimal power allocation in case1 c. Equal power allocation in case2 d. Suboptimal power allocation in case2 e. Equal power allocation in case3 f. Suboptimal power allocation in case3.
uncertainty area so that the physical interpretations can be presented clearly. The ellipses of same color denote the prior track information of the corresponding target, and are relates to the dynamic model of the targets. Therefore, the intersection area actually denotes the tracking accuracy of each target. In case 1, the intersection area of target 1, which implies the tracking accuracy of this target, is relatively larger with equal power allocation (all of the targets are with the same dynamic model, while the measurement error of target 1 is larger). Hence, more power is assigned to target 1, in order to minimize the worst case tracking BCRLB. In case 2, the sector area of target 2 with equal power allocation is larger than that in case1, as the reflectivity of target 2 is smaller in this case. Thus, beam 2 is distributed with more power, compared with the results given in Fig. 7. Similar conclusion can be drawn from the allocation results of target 1 in case 1 and case 2. As it shown in (36), inaccurate dynamic model will result in imprecise prior information, and thus target 3 in case 3 has a larger intersection area with equal power allocation. In this scenario, more power is distributed to target 3 in case 3, compared with the allocation results in case 1. Overall speaking, the results in Fig. 14 show that the area of the sector may be controlled by adjusting the transmit power, and the power allocation strategies assign more power resource to those targets who have larger intersection areas with equal power allocation. The intersection areas of multiple targets are almost same in size (see Figs. 6(b), 9(b), and Fig. 12(b) after power allocation. In other words, we average the tracking error of the multiple targets with the power allocation scheme. For real application, the proposed algorithm may improve the capability of the colocated MIMO radar system to track long range target considerably. In Fig. 15, the detection thresholds of the NP detector and the Bayesian detector are shown for different range cells in the validation gate for case 3. 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. 16, the averaged probability of detection can efficiently be improved for a fixed false alarm rate . VI. CONCLUSION
the sector areas of different colors in Fig. 14 approximate the spatial variance distribution of different targets’ measurement error, where and are two constants used to amplify the
A colocated MIMO radar system has the ability of dealing with multiple beam information, so it has more advantages over traditional radar systems. However, the simultaneous
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have been reported in literature. Fortunately, according to the derivation of this paper, we know that the proposed optimization scheme can be directly implemented with the adoption of those tracking techniques, once the corresponding tracking BCRLBs are calculated and utilized as the criterion.
Fig. 16. The probability of detection of distinct target with different detector in case 3.
multibeam working mode will increase the power consumption greatly. To ensure that the limited power resource can be exploited effectively, the basis of our cognitive tracking technique is to adjust the transmit power of the multiple beams according to the prior tracking information fed back from the tracker. Simulation results demonstrate the superiority of the proposed cognitive tracking strategy. The results indicate that the targets, which are farther to the radar, have weaker propagation paths and have less accurate dynamic models, are allocated more resource. It is worth to note that several new techniques that outperform PDA such as JPDA [27], MCMC based techniques [37], random finite set based techniques [38]
APPENDIX A AND DERIVATION OF It is apparent that the detection probability of the Bayesian detector adopted in this paper is different at distinct locations, and thus we have to calculate the probability of detection averaged over the validation gate via the test and the threshold of (29) [see (73) at the bottom of the page]. Herein, we have abused the notation somewhat: is the detection probability and is the transmit power. In (73), is the probability of de, and tection at location (74) Similarly, we calculate the average false alarm rate [see (75) at the bottom of the page]. Constraining the false alarm rate to as well as be an arbitrary small value, we can find through (73)–(75). APPENDIX B DERIVATION OF According to (38), the term
can be represented as (76)
(73)
(75)
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With (22), it can easily be shown that
(77) is the Jacobian matrix of the nonlinear function evaluated at the true state . With some mathematical derivation, (77) can be rewritten as
(78) where
(79) and
(80) Substituting (78) into (76), the FIM tained
can be ob-
(81) where
is a
matrix
(82)
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Bo Jiu (M’13) was born in Henan Province, China, 1982. He received the B.S., M.S., and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, China, in July 2003, March 2006, and June 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, cognitive radar, radar automatic target recognition and radar imaging.
Hongwei Liu (M’04) received the M.S. and Ph.D. degrees, both in Electronic Engineering, from Xidian University, Xi’an, China, in 1995 and 1999, respectively. He worked at the National Laboratory of Radar Signal Processing, Xidian University, after that. From 2001 to 2002, he was a visiting scholar at the Department of Electrical and Computer Engineering, Duke University, Durham, NC. He is currently a Professor at the National Laboratory of Radar Signal Processing, Xidian University. His research interests are radar automatic target recognition, radar signal processing, and adaptive signal processing.
Bo Chen (M’13) received the B.S. and Ph.D. degrees in electrical engineering from Xidian University, Xian, China, 2003 and 2008. His Ph.D. thesis received the honorable mention for National Excellent Doctoral Dissertation of P.R. China in 2010. From 2008 to 2013, He is a research scientist with the Department of Electrical and Computer Engineering, Duke University. From 2013, he has been selected to young thousand talent program by Chinese Central government. He is currently a professor at National Lab of Radar Signal Processing, Xidian University. His research interests include statistical machine learning, statistical signal processing and radar automatic target recognition.
Zheng Bao (M’80–LSM’90) was born in Jiangsu, China. Currently, he is a professor with Xidian University and the chairman of the academic board of the National Key Lab of Radar signal Processing. He has authored or co-authored 6 books and published over 300 papers. Now, his research fields include space-time adaptive processing (STAP), radar imaging (SAR/ISAR), automatic target recognition (ATR) and over-the-horizon radar (OTHR) signal processing. Professor Bao is a member of the Chinese
Junkun Yan was born in Sichuan Province, China, 1987. He received the B.S. degree in electronic engineering from Xidian University, Xi’an, China, in Jul. 2009. He is currently working toward the Ph.D. degree with the National Laboratory of Radar Signal Processing, Xidian University. His research interests include adaptive signal processing, target tracking, and cognitive radar. Academy of Sciences.
