Adaptive simultaneous multibeam resource management for colocated MIMO radar in multiple targets tracking

Signal Processing 172 (2020) 107543

Contents lists available at ScienceDirect

Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Adaptive simultaneous multibeam resource management for colocated MIMO radar in multiple targets tracking Yang Su a, Ting Cheng a,∗, Zishu He a, Xi Li a School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China

a r t i c l e

i n f o

Article history: Received 17 September 2019 Revised 3 February 2020 Accepted 18 February 2020 Available online 19 February 2020 Keywords: Colocated MIMO radar Sub-array division Multiple targets tracking Adaptive resource management Simultaneous multibeam mode

a b s t r a c t Different from conventional phased array radar, the colocated multiple-input multiple-output (MIMO) radar can form multiple beams simultaneously through transmitting orthogonal waveforms by its subarrays. The illuminating mode of multiple beams can be changed with the sub-array number, providing greater flexibility in system resource management. In this paper, a novel resource management optimization model for the colocated MIMO radar in multiple targets tracking is proposed, where the average time and energy resource consumption of the system are minimized under the guarantee that the illuminated targets can be effectively detected. When solving the proposed optimization problem, the adaptive simultaneous multibeam resource management (ASMRM) algorithm for the colocated MIMO radar is obtained, where the sub-array number, the beam directions, the system sampling period, the transmit waveform energy and the working mode can be controlled adaptively. Simulation results demonstrate that the working parameters of the colocated MIMO radar can be changed effectively. Furthermore, in terms of the average system resource consumption, the proposed algorithm is superior to the algorithms with fixed parameters while realizing the effective multiple targets tracking. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Multiple-input multiple-output (MIMO) radar, which is a new generation of radar system, has raised growing attention recently [1–8] and is on a path from theory to practical use. Compared with phased array radar, MIMO radar achieves better performance in target localization, target identification, and can obtain higher resolution and sensitivity when detecting the slow moving target [9–11]. Generally, MIMO radar can be divided into two types, that is, colocated MIMO radar [10] as well as distributed MIMO radar [12,13]. In the distributed MIMO radar system, the transmit antennae are located far apart from each other in comparison with their distance to the target [12]. However, many actual difficulties still stop distributed MIMO radar from being applied in practice [14]. Compared with distributed MIMO radar, the colocated MIMO radar system, where the transmit and receive antennae are deployed close to each other relative to their distance to the target, can be considered as an advancement of the existing phased array radar. Hence, based on the current technical conditions, the colocated MIMO radar system has more practical value than the dis-

∗

Corresponding author. E-mail address: [email protected] (T. Cheng).

https://doi.org/10.1016/j.sigpro.2020.107543 0165-1684/© 2020 Elsevier B.V. All rights reserved.

tributed MIMO radar [15]. Compared with phased array radar, multiple orthogonal beams can be transmitted in the colocated MIMO radar simultaneously. Therefore, when the colocated MIMO radar system is applied in multiple targets tracking, the primary problem to be solved is how to allocate multiple beams among multiple targets, which is actually the resource management problem. For radar resource management, the essence is to control its working parameters adaptively. Existing researches on resource management for colocated MIMO radar mainly focus on the waveform design and power allocation. The authors in [16] propose an efficient optimization method to design a constant modulus probing signal, which can synthesize a desired beam pattern while maximally suppressing both the autocorrelation and crosscorrelation side lobes between given spacial angles. In power allocation and waveform design for target identification and classification, two waveform design problems with constraints on waveform power have been investigated in [17]. The power allocation in jamming is considered in [18], where the robust jamming power allocation strategies are proposed. Furthermore, the problem of antenna allocation in resource management has been studied in [19,20], where the optimal distribution of antennae is found by applying the relevant operators to the Cramér-Rao lower bound (CRLB). In addition, the authors in [21] propose an active jamming suppression method based on the transmit array designation for the colocated MIMO radar.

2

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543

The target tracking process is not considered in the aforementioned studies. The authors in [22–24] put forth resource management strategies based on the Minimax criterion in multiple targets tracking, whose aim is to run out of the power budget to improve the worst tracking accuracy. In [22], a simultaneous multibeam resource allocation (SMRA) strategy is proposed, where the beam number, beam directions and the transmit power of each beam are adjusted. In addition, the cases when one beam illuminates a single target and multiple targets are both taken into consideration in [22]. It is extended to the netted radar system [23]. The power allocation algorithm for the colocated MIMO radar in clutter is proposed in [24]. In [25], the power allocation in clutter with netted radar system for single-target tracking is addressed. Consider the influence of waveform selection on resource management, the authors in [26] put forward a joint beam, power and waveform selection strategy. The aforementioned studies are based on the Minimax criterion, while a resource management model considering the desired tracking accuracy is proposed in [27], where the sub-array number, the working mode and the transmit energy can be controlled adaptively. In addition, an adaptive cost function (ACF), which is related to the desired tracking accuracy, is proposed in [28–30], where the ACF is optimized by allocating the beam and power resources adaptively. Furthermore, resource management for sensor network and radar communication integration system has also begun to receive attention[31–33]. The authors in [31] propose a collaborative detection and power allocation (CDPA) scheme, where the false alarm rate and transmit power of each node can be controlled. Based on the Markov decision process (MDP), the authors in [32] propose a new resource scheduling method for target tracking in clutter in the radar network, where the selection of radar node and its resource can be optimized. For resource management in the integrated radar and communications system (IRCS), a power minimization-based joint sub-carrier assignment and power allocation (PM-JSAPA) strategy is proposed in [33], where the total radiated power of the IRCS is minimized. While existing studies have made seminal contributions to the MIMO radar resource management problem, there are still some issues to be addressed: Firstly, resource management for MIMO radar focuses on the energy resource in [22–24,26–30], where energy resource management considers the spacial distribution of the transmitted energy at each illuminating moment. Diverse energy resource management strategies are derived from different resource management problem models. However, the time resource management, which focuses on how to choose the optimal illuminating moment for the system, is not considered. Secondly, in the existing studies [22–26], in order to improve the worst tracking performance among the multiple targets, all system resource is used up. However, in practice, how to minimize the system resource consumption on the premise of ensuring effective targets tracking is a more valuable problem. Thirdly, each beam corresponds to one single target in [23,24,26,28–30]. The mechanism of how multiple beams are formed and the ability of each beam to detect multiple targets have not been taken into account. Therefore, in the resulting resource management strategies, the ability of a single wide beam to illuminate multiple targets is ignored. Based on above, an adaptive resource management optimization model for the colocated MIMO radar in multiple targets tracking is established in this paper. In the optimization model, the objective function considers comprehensively the average energy and time resources consumed by the illuminated targets, and the constraints ensure the effective tracking of the targets. In solving the optimization model, the simultaneous multibeam mode of the colocated MIMO radar is fully considered and an adaptive resource manage-

ment algorithm is proposed. In the proposed algorithm, five working parameters can be controlled adaptively, including the subarray number, the beam directions, the system sampling period, the transmit waveform energy and the working mode. The main contributions of this paper are the following: 1.A novel resource management optimization model for the colocated MIMO radar in multiple targets tracking is put forth. In the proposed resource management model, the objective function considers not only the energy resource but also the time resource, namely, the average time and energy resource consumption of the system. Then, to solve the established optimization model, the adaptive simultaneous multibeam resource management (ASMRM) algorithm is proposed to minimize the total system resource consumption while ensuring effective targets tracking. 2.In our work, we consider that the multiple beams are realized by the sub-array division. The sub-array number and its influence on beam width are also taken into account, that is, the number of multiple beams and the width of each beam will be changed with the sub-array number. When the sub-array number is large, a single beam in the multiple beams is wide enough to detect multiple targets simultaneously under certain conditions. As a result, in addition to the optimization of traditional working parameters, the optimization of the sub-array number is mainly considered. The rest of this paper is organized as follows: The problem formulation is given in Section 2. In Section 3, the resource management optimization model is proposed. Then, an adaptive resource management algorithm for colocated MIMO radar based on the simultaneous multibeam mode is given in Section 4 to solve the proposed optimization model. To show the effectiveness of the proposed algorithm, several numerical simulation results are provided in Section 5. Finally, the conclusions are drawn in Section 6. 2. Problem formulation Consider a monostatic MIMO radar system whose transmit and receive arrays are colocated, as shown in Fig. 1. Assume that there are N array elements in the colocated MIMO radar. When the array is divided into K sub-arrays, each sub-array will contain L = N/K array elements. In the colocated MIMO radar system, the sub-arrays transmit multiple orthogonal signals simultaneously [16,23,34]. Obviously, the beam formed by each sub-array in the colocated MIMO radar is much wider than that in the conventional phased array radar, and the transmit gain is lower [35]. When the transmit beams are defocused, multiple narrow receive beams are formed with the digital beam-forming (DBF) technique to span the volume of space that is illuminated by the transmit beams [22], as shown in Fig. 1. The measurements of different targets are extracted from diverse narrow receive beams. Therefore, the multitarget tracking problem can be regarded as the combination of

Fig. 1. The simultaneous multiple beams of the colocated MIMO radar.

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543

multiple single-target tracking problems that can be solved independently. Through the control of sub-array number, the colocated MIMO radar can transmit different number of multiple beams simultaneously to illuminate multiple targets. Furthermore, each beam may illuminate multiple targets simultaneously if it is wide enough. Therefore, the colocated MIMO radar has larger flexibility in illuminating multiple targets and greater degree of freedom in system resource management. In the resource management of this paper, in addition to the optimization of traditional working parameters, the optimization of sub-array number is mainly considered. The reason is that the sub-array number determines the width and the gain of the transmit beam. For a given target scene and beam pointing, the beam width determines the number of targets that are illuminated by the beam. Meanwhile, the gain of the beam determines the echo signal-to-noise ratio (SNR). Assume that D targets are under tracking, the maneuverability of the target is considered here, so the interacting multiple model (IMM) filter is adopted here to realize effective target tracking [32,36–39]. Consider there are J models in the IMM filter, and the target motion of the jth model [40] is

xi( j ) (tk ) = F i( j ) (Tk−1 )xi( j ) (tk−1 ) + i( j ) (Tk−1 )wi( j ) (tk−1 ), j = 1,2, · · · , J

σb,i (tk+1 ) =

c·





3

Bw 2SNRi (K, Tsys , M, us , e )

(5)



where r tk+1 is the range resolution of transmit waveform, Bw is the round trip beam width, c is a constant, SNRi (K, Tsys , M, us , e) is the SNR of target i and it is related to the controllable working parameters, which will be given in detail later. In order to realize the multiple targets tracking, they should be detected firstly. Therefore, how to minimize the consumed system resources under the guarantee that the targets can be detected effectively is the resource management problem to be addressed. The working parameters of the system including the sub-array number, the system sampling period, the beam directions, the transmit waveform energy and the working mode are to be controlled adaptively to realize the effective resource management. 3. Optimization model of resource management for the colocated MIMO radar In this section, the resource management optimization model, which includes the objective function and the constraints, for colocated MIMO radar is established.

(1) ( j)

( j)

( j)

( j)

( j)

( j)

T

( j)

where xi (tk ) = [xi (tk ), x˙ i (tk ), x¨i (tk ), yi (tk ), y˙ i (tk ), y¨ i (tk )] denotes the range, velocity and acceleration of target i at tk .  ( j) F i Tk−1 is the transition matrix of the jth model, where Tk−1 ( j)

is the time interval between tk−1 and tk . wi (tk−1 ) denotes the process noise of the jth model, which is assumed to be zero( j) mean Gaussian distributed with a known covariance Q i (tk−1 ),

i( j ) (Tk−1 ) is the input matrix of process noise of the jth model.

After update moment tk , the proposed resource management algorithm for the colocated MIMO radar will allocate the time and energy resource of the system. For time resource allocation, the next system sampling time tk+1 = tk + Tsys should be given, where Tsys is the system sampling period. For energy resource allocation, the sub-array number K, the beam directions us , the transmit waveform energy e and the working mode M will be determined, which means energy resource e is to be consumed according to the form described by K and us on the targets in working mode M. Here, M represents the illuminated targets set. For instance, when two targets are under tracking, there will be three possible working modes, namely M ∈ {{1}, {2}, {1, 2}}. M = {1} indicates that target 1 is illuminated, M = {2} means that target 2 is illuminated, and M = {1, 2} represents both targets are illuminated. When target i is illuminated at instant tk+1 , the measurement of it is

zi (tk+1 ) = h(xi (tk+1 )) + vi (tk+1 )

(2) T

where h(xi (tk+1 )) = [ri (tk+1 ), bi (tk+1 )] , ri (tk+1 ) and bi (tk+1 ) are the range and azimuth of the target, respectively. vi (tk+1 ) denotes the measurement error, which is assumed to be zero-mean Gaussian distributed, and the measurement error covariance is



Ri (tk+1 )= J ·

σr,i2 (tk+1 ) 0



0 · JT 2 σb,i (tk+1 )

(3)

where J is the Jacobian transform matrix from polar coordinates to 2 (t 2 Cartesian coordinates, σr,i k+1 ) and σb,i (tk+1 ) are the variances of the range measurement error  and  the bearing   measurement error of target i. Specifically, σr,i tk+1 and σb,i tk+1 can be calculated as follows:

σr,i (tk+1 ) = 

r (tk+1 ) SNRi (K, Tsys , M, us , e )

(4)

3.1. Constraint conditions In multiple targets tracking, the targets whose states are to be updated should be illuminated and detected by the system. Specifically, the illumination means that targets are covered by the transmit beam. For instance, when the working mode is M = 1, target 1 should be in the coverage of the transmit beam. Therefore, for each target in the working mode M, it should be illuminated by the transmit beam, such that:

us,k −

φ (K ) 2

≤ u pre,i ≤ us,k +

φ (K ) 2

, ∀i ∈ M, ∃k ∈ {1, 2, · · · , K } (6)

where us,k is the kth element in us , representing the transmit beam direction of the kth sub-array. upre,i represents the predicted azimuth of target i, and φ (K) represents the beam width that is related to the sub-array number [41], that is:

φ (K ) =

1.76K N

(7)

Note that for brevity, throughout this paper, the time index tk+1 will often be omitted, unless doing so causes confusion. According to (6), each target in the working mode M should be illuminated by at least one transmit beam. Furthermore, in order to detect the illuminated target effectively, the detection probability of the target must exceed the given probability threshold, that is:

P di (K, Tsys , M, us , e ) ≥ P dth , ∀i ∈ M

(8)

where Pdth is the detection probability threshold. Pdi (K, Tsys , M, us , e) represents the detection probability of target i under the combination (K, Tsys , M, us , e), and it is related to the SNR. For example, for the target whose radar cross section (RCS) obeys the Swerling I distribution, its detection probability can be calculated as follows: 1 1+SN Ri K,Tsys ,M,us ,e

P di (K, Tsys , M, us , e ) = Pf(a

(

))

, ∀i ∈ M

(9)

3.2. Objective function Colocated MIMO radar has advantages in multiple targets tracking due to its characteristics of simultaneous multibeam and wide transmit beam. Therefore, the aim of resource management for

4

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543

colocated MIMO radar is to maximize the utilization of system resource. Specifically, the minimum system resource that meets system detection performance requirements is consumed to accomplish as many targets tracking tasks as possible. In this paper, the consumed system resource includes both energy and time resources. The energy consumption is determined by the transmit energy e. The time consumption is determined by the system sampling period Tsys , where more time resource is consumed with small sampling period. The weighted sum of normalized energy and time resource consumption is used to describe the synthetical system resource consumption. The total number of targets detected by the system at each moment is used to describe the multiple targets tracking ability, therefore, how many targets share the resource is also considered. We hope more targets to share the resource to increase the utilization of the consumed resource. Therefore, the average resource consumption is formed as the objective function:

C (K, Tsys , M, us , e ) =

αψ {e(us )} + βψ



1



Tsys (K )

NI (M )

(10)

where NI (M) represents the number of targets being illuminated, which is determined by the working mode M. It is worthwhile noting that even if a target is illuminated by multiple sub-arrays, the target still appears only once in the set M. Because the energy resource and the time resource have different units, a normalization function ψ { · } is introduced in (10), which is defined as ψ{xi } = xi /xmax . The coefficients α and β in (10) represent the emphasis of the system on the energy resource and the time resource respectively, which meet α +β =1. When α is larger than β , it means that the colocated MIMO radar emphasizes more on the energy consumption; otherwise, more attention is paid on the time resource of the system. In summary, combine the objective function in (10) and the constraints in (6) and (8), the colocated MIMO radar resource management optimization model is obtained as follows:

{1, 2, · · · , 2i−1 , · · · , N}, (i = 1, 2, · · · , log2 N + 1 ) and the set size of Kset is denoted as NK . In practice, the system sampling period usually has a maximum value Tmax and a minimum value Tmin . Therefore, the possible system sampling period set Tset can be obtained by discretization of [Tmin , Tmax ], where the set size of Tset is denoted as NT . When the number of targets is D, the possible working mode set Mset has NM = 2D − 1 elements. As to the waveform energies, consider there are different waveforms with totally Ne kinds of energies in the preset waveform library, so the possible transmit waveform energy set eset has Ne elements in it. The proposed ASMRM algorithm in multiple targets tracking for colocated MIMO radar includes the following steps: Step 1: For each possible sub-array number K j , ( j = 1, 2, · · · , NK ) in Kset , namely K j = Kset ( j ), calculate the beam width φ (Kj ). According to the adaptive sampling period method based on the prediction error covariance threshold [42], the sampling period Ti (tk , φ (K j )), i = 1, 2, · · · , D of each target i, i = 1, 2, · · · , D is obtained, where Ti (tk , φ (Kj )) is the maximum value in the sampling period set that satisfies the following constraints:

σr,pre (tk(i) , Ti (tk , φ (K j ))) ≤ rth σb,pre (tk(i) , Ti (tk , φ (K j ))) ≤ bth (K j )

(12)

where σ r,pre (tk(i) , Ti (tk , φ (Kj ))) and σ b,pre (tk(i) , Ti (tk , φ (Kj ))) represent the standard deviation of range prediction error and the standard deviation of azimuth prediction error, respectively. They can be predicted from the IMM filter. In (12), rth is the threshold in the range, bth (Kj ) that is related to the sub-array number represents the threshold in the azimuth [43]. Step 2: According to the previous update time tk(i) and the sampling period Ti (tk , φ (Kj )) of each target, tk+1(i ) ∼ = tk(i ) + Ti (tk , φ (K j )) is obtained. The next sampling time of the system is tk+1 = tk+1(imin ) = min tk+1(i ) , therefore, the sampling period of the i=1,2,··· ,D

Through solving the resource management optimization model in (11), an adaptive resource management algorithm for the colocated MIMO radar can be obtained.

system is Tsys (K j ) = tk+1 − tk . Target imin is called as the main target. Select the working modes from the working mode set Mset that contain the main target, and denote them as Mm , m = 1, 2, · · · , N j . Step 3: Under the parameters combination ( (K, Tsys ) j , Mm ), j = 1, 2, · · · , NK , m = 1, 2, · · · , N j , form the main virtual target and normal virtual targets. A)Main virtual target formation: Subtract the predicted azimuth upre,i of the remaining target in Mm from the predicted azimuth u pre,imin of the main target, and the absolute value of the difference is denoted as |u pre |imin ,i , i ∈ {1, 2, · · · , imin − 1, imin + 1, · · · , D}. Regard target i, i ∈ {1, 2, · · · , imin − 1, imin + 1, · · · , D} that satisfies (13) and target imin as the main virtual target, and denote the illumination direction of the main virtual target as uK j ,1 = u pre,imin .

4. Adaptive resource management algorithm for colocated MIMO radar based on simultaneous multibeam mode

|u pre |imin ,i ≤

minC (K, Tsys , M, us , e )

s.t.

us,k −

φ (K ) 2

≤ u pre,i ≤ us,k +

P di (K, Tsys , M, us , e ) ≥ P dth

φ (K ) 2

, ∀i ∈ M, ∃k ∈ {1, 2, · · · , K } (11)

From the proposed resource management optimization model in (11), it can be seen that five working parameters of the colocated MIMO radar can be controlled, including the sub-array number, the system sampling period, the working mode, the beam directions and the transmit waveform energy. Suppose at tk , after filtering the state information is {tk(i ) , xˆi (tk(i ) ), P i (tk(i ) )}, where tk(i) is the last update time for target i, xˆi (tk(i ) ) is the updated state estimation of target i at tk(i) and Pi (tk(i) ) is the corresponding estimation error covariance. Since target i might not be in the illuminated targets set, it is obvious that tk(i) ≤ tk . Right now, the update moment tk+1 = tk + (Tsys )opt and the corresponding optimal working parameters (Kopt , Mopt , (us )opt , eopt ) at tk+1 need to be determined. When the total number of array elements in the colocated MIMO radar is N, which is always the integer power of 2, the possible sub-array number set can be formed as Kset =

φ (K j ) 2

(13)

B)Normal virtual targets formation: Extract the targets in Mm that are not included in the main virtual target, store their predicted directions into the vector sort usort pre in the descending order, and the size of u pre is denoted as Nsort . Start from the first element in usort , find the first one pre that is less than u pre,imin − φ (K j )/2 , whose index is denoted as o. sort sort Therefore, two sub-vectors [usort pre (1 ), u pre (2 ), · · · , u pre (o − 1 )] and sort (o + 1 ), · · · , usort (N [usort ( o ) , u ) ] are obtained. sort pre pre pre sort sort 1) For the sub-vector [usort pre (o), u pre (o + 1 ), · · · , u pre (Nsort )], find the first usort ( v ) from o + 1 to N that satisfies (14): sort pre sort |usort pre (o) − u pre (v )| > φ (K j ), v = o + 1, o + 2, · · · , Nsort

(14)

Therefore, all targets whose predicted azimuth angles are among  sort sort sort [usort usort pre (o), u pre (o + 1 ), · · · , u pre (v − 1 )], namely pre (v ), u pre (o) , are regarded as a virtual target, and the illumination direction of

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543

it is

1)

uK j ,2 = usort pre (o) −

φ (K j )

(15)

2

sort sort Then, remove [usort from pre (o), u pre (o + 1 ), · · · , u pre (v − 1 )] sort sort the sub-vectors [usort to get pre (o), u pre (o + 1 ), · · · , u pre (Nsort )] sort sort [usort pre (v ), u pre (v + 1 ), · · · , u pre (Nsort )]. The remaining targets are divided into virtual targets in the same way and the corresponding illumination directions are determined. sort sort 2) For the sub-vector [usort pre (1 ), u pre (2 ), · · · , u pre (o − 1 )], find the first usort pre (h ) from o − 2 to 1 that satisfies (16):

|

usort pre

(h ) −

usort pre

( o − 1 )| > φ ( K j ), h = o − 2, o − 3, · · · , 2, 1 

sort usort pre (o − 1 ), u pre (h ) , are regarded as a virtual target, and the illumination direction of it is

uK j ,3 = usort pre (o − 1 ) +

φ (K j ) 2

(17)

sort sort Then, remove [usort from pre (h − 1 ), u pre (h − 2 ), · · · , u pre (o − 1 )] sort sort the sub-vectors [usort to get pre (1 ), u pre (2 ), · · · , u pre (o − 1 )] sort sort [usort pre (1 ), u pre (2 ), · · · , u pre (h )]. The remaining targets are divided into virtual targets in the same way and the corresponding illumination directions are determined. Finally, NVT (Kj , Mm ) virtual targets are formed and their illumination directions are denoted as uK j ,q , q = 1, 2, · · · , NV T (K j , Mm ), they are stored in the vector uVT (Kj , Mm ). Step 4: Combine each possible transmit waveform energy e p , p = 1, 2, · · · , Ne with the obtained ( (K, Tsys ) j , Mm ), j = 1, 2, · · · , NK , m = 1, 2, · · · , N j , the combinations ( (K, Tsys ) j , Mm , e p ), j = 1, 2, · · · , NK , m = 1, 2, · · · , N j , p = 1, 2, · · · , Ne are obtained. In the multibeam mode, the SNR of target i can be calculated as follows:

N 3 e p ( π ηA )

σi λ2 ni ( ( ))4 N0 2

SN Ri =

( 4π )

3

K 2j Ri tk−+1

(18)

where ni represents the number of beams that illuminate target i, λ is the wavelength, σ i is the RCS of target i, ηA is the aperture efficiency, N0 is the power spectrum density of radar receiver noise. Ri (tk−+1 ) is the predicted range of target i at tk+1 , where the symbol · (tk−+1 ) denotes the predicted value at instant tk+1 . If the threshold of detection probability is Pdth , the threshold of SNR can be calculated according to (9) as follows:

SN Rth =

ln Pf a −1 ln P dth

(19)

The minimum beam number needed for target i to reach Pdth is calculated as below:

 (4π )3 K j 2 (Ri (tk+1 ) )4 N0 , i ∈ Mm , (ni )th = SNRth · N 3 e p (π ηA )2 σi λ2 j = 1, 2, · · · , NK , p = 1, 2, · · · , Ne

(20)

where the symbol · denotes rounding up operation. Therefore, the minimum beam number required by the illuminated virtual target q is:

numq = max( (ni )th ), i = 1, 2, · · · , Nq , q = 1, 2, · · · , NV T (K j , Mm ) (21) where Nq is the number of real targets that are contained in the virtual target q. Calculate the number of remaining beams: NV T (K j ,Mm )

Kle f t = K j −

 q=1

numq

Kleft

(22)

<

0,

the

parameters

combination

( (K, Tsys ) j , Mm , uK j , e p ) is not feasible;

2) When Kleft ≥ 0, the remaining Kleft beams are allocated to virtual targets according to (23) and (24):

⎢ ⎢ ⎢ ⎢ numq numq = numq + ⎢Kle f t  ⎣ NV T (K j ,Mm )−1  ω=1

numω

⎥ ⎥ ⎥ ⎥ ⎥ ⎦,

q = 1, 2, · · · , NV T (K j , Mm ) − 1

(16)

Therefore, all targets whose predicted azimuth angles are sort sort among [usort namely pre (h − 1 ), u pre (h − 2 ), · · · , u pre (o − 1 )],



When

5

(23)

NV T (K j ,Mm )−1

numNV T (K j ,Mm ) =K j −



numq

(24)

q=1

where the symbol  · denotes rounding down operation. For the feasible combinations, according to numq , q = 1, 2, · · · , NV T (K j , Mm ) and uVT (Kj , Mm ), the beam directions vector us (Kj , Mm ) is formed. Specifically, if there are numq beams that point to the direction of uK j ,q , element uK j ,q will be stored in us (Kj , Mm ) for numq times. Save ( (K, Tsys ) j , Mm , us (K j , Mm ), e p ), j = 1, 2, · · · , NK , m = 1, 2, · · · , N j , p = 1, 2, · · · , Ne as the feasible working parameters combinations of the system. Step 5: Calculate the values of objective function of all the feasible working parameters combinations. The working parameters combination with the minimum value of the objective function is selected as the optimal working parameters combination of the system, which is denoted as (Kopt , (Tsys )opt , Mopt , (us )opt , eopt ). Thus, the update time of the system tk+1 = tk + (Tsys )opt can be further obtained, and the state estimation result of the target included in Mopt can be updated at tk+1 to finally obtain the updated state information {xˆi (tk+1(i ) ), P i (tk+1(i ) )}, i ∈ Mopt . Step 6: Repeat Steps 1–5 above until the tracking process ends. The process of the ASMRM algorithm is summarized in Algorithms 1 and 2. In addition, the computational complexity of the ASMRM algorithm is also evaluated, that is,  O NK DNT + NK 2D−1 D + NK 2D−1 Ne + JD , where J is the number of motion models in the IMM filter. 5. Numerical simulation results To illustrate the effectiveness of the proposed algorithm, some numerical simulations are presented in this section. 5.1. Design of trajectories and radar resource Assume that there are three targets in the scenario, where target 1 moves from 0s, target 2 and target 3 begin to appear after 20s, and the ending time of simulation is 150s. Fig. 2 shows the trajectories of them, where target 1 maneuvers during the time period of 70–90s, target 2 moves with constant velocity of [120, 0]m/s and target 3 moves with constant velocity of [40, 40]m/s . All of them are assumed to be Swerling I targets with average RCS of 1 m2 . In the IMM filter, the motion model includes constant velocity model and constant acceleration model, where initial probability of each model is 0.5. The transition probability of the same model is 0.95, and that of different models is 0.05. The colocated MIMO radar has a linear array with totally 1024 elements, namely N = 1024. The distance between adjacent elements is half wavelength and the working frequency is 10GHz. λ = 0.0545m, ηA =0.5, α = 0.7, β = 0.3, Pf a = 10−6 , Pdth = 0.9. The possible working mode set is Mset = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}, and the meaning of element in Mset is shown in Table 1. The possible transmit waveform energy set is eset =

6

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543

Algorithm 1: Virtual targets formation algorithm

1 2 3 4

Input: ( (K, Tsys ) j , Mm ), imin , φ (K j ), D Output: NV T (K j , Mm ), uV T (K j , Mm ) Initialize v = 0, h = 0, ξ = 0, ς = 0; Save target imin as a member of the main virtual target; for i = 1, · · · , imin − 1, imin + 1, · · · , D do Calculate the absolute value of the difference |u pre |imin ,i in (13);

5 6 7 8 9 10

11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26

φ (K )

Â~if |u pre |imin ,i ≤ 2 j then Save target i as a member of the main virtual target; end end Acquire the main virtual target; Acquire usort pre , and start from the first element in it to find the first one that is less than u pre,imin − φ (K j )/2 , whose index is denoted as o; Let v = o + 1, ξ = o, h = o − 2, ς = o − 1; while v ≤ Nsort do sort if |usort pre (ξ ) − u pre (v )| > φ (K j ) then Targets whose predicted azimuths are among sort sort [usort pre (ξ ), u pre (ξ + 1 ), · · · , u pre (v − 1 )] are regarded as a virtual target; ξ = v; end v = v + 1; end while h ≥ 1 do sort if |usort pre (h ) − u pre (ς )| > φ (K j ) then Targets whose predicted azimuths are among sort sort [usort pre (h − 1 ), u pre (h − 2 ), · · · , u pre (ς )] are regarded as a virtual target; ς = h; end h = h − 1; end Acquire NV T (K j , Mm ) virtual targets and the vector uV T (K j , Mm ).

Algorithm 2: ASMRM algorithm for colocated MIMO radar

1 2 3 4 5

6 7

8

9 10 11

12 13 14 15 16 17 18 19 20

Input: {tk(i ) , xˆi (tk(i ) ), P i (tk(i ) )}, tk , D, Kset , Tset Output: tk+1 = tk + (Tsys )opt , (Kopt , Mopt , (us )opt , eopt ), {xˆi (tk+1(i) ), Pi (tk+1(i) )}, i ∈ Mopt Initialize j = 1, m = 1, p = 1; while j ≤ NK do Acquire K j = Kset ( j ), and calculate φ (K j ) in (7); Â~for i = 1, · · · , D do Obtain Ti (tk , φ (K j )) of target i, which is the maximum value in Tset that satisfies (12); end Acquire tk+1 = tk+1(imin ) = min tk+1(i ) and

 

i=1,2,··· ,D



 

Tsys K j = tk+1 − tk , where tk+1(i ) ∼ = tk(i ) + Ti tk , φ K j ; Call target imin as the main target, select working modes from Mset that contain target imin , and denote them as Mm , m = 1, · · · , N j ; for m = 1, · · · , N j do Form virtual targets by employing Algorithm 1; Acquire NV T (K j , Mm ) virtual targets and the vector uV T (K j , Mm ); for p = 1, · · · , Ne do Calculate numq by solving (18)-(24); Acquire the beam directions vector us (K j , Mm ); end end j = j + 1; end Acquire all feasible working parameters combinations; Acquire and output the final solution (Kopt , (Tsys )opt , Mopt , (us )opt , eopt ), tk+1 = tk + (Tsys )opt , {xˆi (tk+1(i) ), Pi (tk+1(i) )}, i ∈ Mopt .

Table 1 Description of each element in the working mode set. Mode indices

Working modes

Specific meanings

1 2 3 4 5 6 7

{1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

target 1 is tracked target 2 is tracked target 3 is tracked targets 1&2 are tracked simultaneously targets 1&3 are tracked simultaneously targets 2&3 are tracked simultaneously targets 1&2&3 are tracked simultaneously

{1.35, 2.25, 4.05, 5.85, 11.7, 23.4}J, the possible system sampling period set is Tset = {0.3, 0.5, 0.8, 1, 1.5, 1.8, 2, 2.5, 3, 3.5, 4, 4.5}s and the Kset = {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024} is the possible sub-array number set. 5.2. Adaptive control of working parameters

Fig. 2. The trajectories of the three targets

Adaptive resource management of the colocated MIMO radar is realized by the algorithm proposed in this paper. The working mode of the colocated MIMO radar is presented in Fig. 3, it can be seen that the working mode changes adaptively during the time period of 20–150s, and the working mode is equal to 7 most of the time. The reason is that the colocated MIMO radar tends to illuminate multiple targets at the same time due to the simultaneous multibeam mode. The sub-array number of the colocated MIMO radar is shown in Fig. 4, it can be seen that the sub-array number is selected adaptively, and the width of transmit beam is changing accordingly. Fig. 5 presents the combination result of virtual targets, namely Virtual-Target mode, and the meaning of the

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543

7

Table 2 Description of each Virtual-Target mode. Virtual-Target mode

Specific meaning

0 1 2 3 4 5

target 1 forms a virtual target target 1, target 2, target 3 form three virtual targets targets 1 and 2, target 3 form two virtual targets targets 1 and 3, target 2 form two virtual targets targets 2 and 3, target 1 form two virtual targets targets 1,2 and 3 form one virtual target

Fig. 3. The working mode of the colocated MIMO radar

Fig. 6. The transmit waveform energy of the colocated MIMO radar.

Fig. 4. The sub-array number of the colocated MIMO radar.

Virtual-Target mode is shown in Table 2. It is worthwhile noting that not all virtual targets are illuminated. Combine Figs. 3–5 and Table 2, the directions of multiple beams and the real targets contained in the virtual targets can be obtained. For instance, at 40s, the working mode is 7 from Fig. 3, the sub-array number is 2 from Fig. 4, and the Virtual-Target mode is 4 from Fig. 5, therefore, we can know that the virtual target that contains target 1 and the virtual target that contains targets 2 and 3 are illuminated by different beams from two sub-arrays, respectively. In addition, at 65s, the working mode is 7, the sub-array number is 8 and the VirtualTarget mode is 5, it means the virtual target, which includes targets 1, 2 and 3, is illuminated by the eight beams with the same direction. Fig. 6 depicts the selection result of the transmit waveform energy. According to Fig. 6, the transmit waveform energy increases after 20s. The reason is that multiple targets will be tracked by the colocated MIMO radar after 20s, more energy resource is required to obtain enough SNR, which helps to achieve effective tracking of multiple targets. In addition, the transmit waveform energy allocation on each target is shown in Fig. 7, where the horizontal axis is the time index, and the vertical axis indicates the target. Each colored rectangle represents the normalized transmit waveform energy on each target, which is defined as i renergy ,k =

Fig. 5. The Virtual-Target mode.

eik etotal,k

(25)

where eik is the transmit waveform energy allocated to target i at instant tk , etotal,k is the total transmit waveform energy of the sysi tem at tk . renergy is also the proportion of beams that point to ,k target i. According to Fig. 7, before 20s, all the energy is allocated to target 1. However, once target 2 and target 3 appear, the energy is allocated to multiple targets adaptively.

8

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543

Fig. 9. The RMSEs of the three targets. Fig. 7. Normalized transmit waveform energy.

5.3. Performance analysis and comparison

Fig. 8. The sampling periods of the three targets.

In order to further investigate the performance of the proposed algorithm, the numerical results of 500 Monte Carlo trials are also given. The sampling periods of the three targets are given in Fig. 8. According to Fig. 8, for target 1, the sampling period of it decreases during the time period of 20–25s. The reason is that when targets 2 and 3 appear at 20s, the system tends to illuminate targets 2 and 3 frequently to enable them to enter stable tracking quickly. At the same time, the optimal working mode selected by the system is 7, that is, targets 1, 2 and 3 are tracked simultaneously, so target 1 is also detected frequently. In addition, the sampling period of target 1 also decreases during the time period of 70–90s. The reason is that target 1 maneuvers during this time period, the radar tends to illuminate target 1 more frequently, which makes the sampling period of target 1 decrease, to ensure the normal tracking of target 1. In order to investigate the overall tracking performance, the root mean square error (RMSE) of each target is shown in Fig. 9. For target 1, its RMSE curve has a peak during the time period of 70-90s, because target 1 maneuvers during this time period. In general, the RMSEs of the three targets are convergent. Therefore, the proposed algorithm can ensure the normal tracking of multiple targets indeed.

To check the effects of different working parameters on the performance, in terms of the average resource consumption, the proposed algorithm is compared with four other algorithms, as shown in Fig. 10. Specifically, the four algorithms used for comparison include the algorithm with fixed sub-array number K = 1, the algorithm with fixed sub-array number K = 2, the multibeam algorithm with one beam corresponding to one target and the algorithm with fixed system sampling period Tsys = T¯ , where T¯ is the average value of the selected system sampling period of the proposed algorithm during the whole tracking process. It can be seen that the average resource consumption of the proposed algorithm is less than the other four algorithms. Furthermore, the improvement is largest in Fig. 10(a), where the sub-array number is fixed at 1, which corresponds to the phased array radar case. Therefore, the joint adaptive adjustment of the sub-array number, system sampling period, transmit waveform energy, beam directions and working mode can effectively reduce the average resource consumption of the system. To further investigate the influence of different coefficient combinations (α , β ) on the performance, the average values of the objective function from different algorithms are compared. The performance comparison results under different (α , β ) combinations are shown in Table 3, where the performance measure C¯ represents the average value of the objective function:

C¯ =

ξn NMC 1  1   n C tk NMC ξn n=1

(26)

k=1

where NMC is the number of Monte Carlo trials, ξ n is the tracking update times in the nth Monte Carlo run, C (tkn ) is the value of objective function at tkn . In Table 3, η represents the improvement ratio of the proposed algorithm compared with other algorithms in terms of C¯ :

η=

C¯other − C¯proposed × 100% C¯other

(27)

It can be seen that under all (α , β ) combinations, the average resource consumption of the proposed algorithm is less than that of other algorithms. Specifically, when compared with the algorithm with fixed sub-array number and the algorithm with fixed system sampling period, the improvement ratio η of the proposed algorithm becomes more obvious with the increase of β . The reason is that in this case the system sampling period of the proposed al-

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543

9

Fig. 10. Average resource consumption comparisons in four cases. (a) Comparison between proposed algorithm and algorithm with fixed sub-array number K=1, (b) Comparison between proposed algorithm and algorithm with fixed sub-array number K=2, (c) Comparison between proposed algorithm and algorithm with fixed sampling period, (d) Comparison between proposed algorithm and multibeam algorithm with one beam corresponding to one target. Table 3 Average resource consumption comparison of different algorithms. Coefficient combinations

( α , β ) = ( 0.3, 0.7 )

( α , β ) = ( 0.5, 0.5 )

( α , β ) = ( 0.7, 0.3 )

Algorithms

C¯

η

Proposed adaptive resource management algorithm Conventional algorithm with fixed sub-array number K=1 Conventional algorithm with fixed sub-array number K=2 Conventional algorithm with fixed system sampling period Multibeam algorithm with one beam corresponding to one target Proposed adaptive resource management algorithm Conventional algorithm with fixed sub-array number K=1 Conventional algorithm with fixed sub-array number K=2 Conventional algorithm with fixed system sampling period Multibeam algorithm with one beam corresponding to one target Proposed adaptive resource management algorithm Conventional algorithm with fixed sub-array number K=1 Conventional algorithm with fixed sub-array number K=2 Conventional algorithm with fixed system sampling period Multibeam algorithm with one beam corresponding to one target

0.1576 0.7028 0.3885 0.3223 0.1646 0.1319 0.5184 0.2904 0.2515 0.1378 0.0959 0.3336 0.1921 0.1707 0.1050

– 77.58% 59.43% 51.10% 4.25% – 74.56% 54.58% 47.55% 4.28% – 71.25% 50.08% 43.82% 8.67%

10

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543

of the proposed algorithm to the algorithm with fixed sub-array number, the algorithm with fixed system sampling period and the multibeam algorithm with one beam corresponding to one target. Compared with the algorithms with corresponding fixed parameters, the proposed algorithm can realize the effective multiple targets tracking with less average system resource consumption. Future work will concentrate on the reduction of computational complexity for resource management in colocated MIMO radar. Declaration of Competing Interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled. CRediT authorship contribution statement Fig. 11. Comparison between proposed algorithm and algorithm in [27].

gorithm is larger, namely, the improvement of the proposed algorithm in average resource consumption is mainly due to the improvement of its system sampling period. When compared with the multibeam algorithm with one beam corresponding to one target, the improvement ratio η of the proposed algorithm becomes more obvious with the increase of α . The reason is that in this case the transmit waveform energy of the proposed algorithm is smaller, that is, the average resource consumption improvement of the proposed algorithm is mainly due to the improvement of its transmit waveform energy. Furthermore, to show the superiority of the proposed strategy to the existing method, the proposed algorithm is compared with the algorithm in [27] in terms of the average resource consumption, as shown in Fig. 11. It can be seen that the average resource consumption of the proposed algorithm is less than the one of the existing algorithm. The reason is that in the algorithm in [27], multiple beams point to the same direction at each moment, and time resource management is not considered, which can be seen as a special case of the proposed algorithm. In addition, an account of the average computation time of each iteration of the proposed algorithm is also performed, where the average computation time is defined as

t¯comp =

NMC 1  1 n t NMC ξn tot

(28)

n=1

n is the total computation time of the nth Monte Carlo where ttot trial. The simulation is based on MATLAB R2016a in a computer with 2.9 GHz CPU and 8 GB RAM, and the test result of average computation time t¯comp = 0.043s.

6. Conclusions For the colocated MIMO radar in multiple targets tracking, how to minimize the resource consumption on the basis of ensuring the normal tracking has important practical value. In this paper, an adaptive resource management method for the colocated MIMO radar based on the simultaneous multibeam mode is proposed. The sub-array number, the beam directions, the system sampling period, the transmit waveform energy and the working mode can be controlled adaptively, which realizes the allocation of the time and energy resources. The numerical simulation results demonstrate the effectiveness of the proposed algorithm and the superiority

Yang Su: Conceptualization, Writing - original draft, Writing review & editing, Methodology, Software. Ting Cheng: Conceptualization, Methodology, Writing - review & editing, Resources, Supervision. Zishu He: Conceptualization, Supervision, Methodology, Resources, Writing - review & editing. Xi Li: Investigation, Validation, Data curation, Writing - review & editing. Acknowledgment This work was supported in part by Basic Research Operation Foundation for Central University, (Grant No. ZYGX2016J039). References [1] D. Dorsch, H. Rauhut, Refined analysis of sparse mimo radar, J. Fourier Anal. Appl. 23 (3) (2015) 1–45. [2] G. Cui, H. Li, M. Rangaswamy, Mimo radar waveform design with constant modulus and similarity constraints, IEEE Trans. Signal Process. 62 (2) (2014) 343–353. [3] W.Q. Wang, Mimo sar imaging: potential and challenges, Aerosp. Electron. Syst. Mag. IEEE 28 (8) (2013) 18–23. [4] A.A. Gorji, R. Tharmarasa, W.D. Blair, T. Kirubarajan, Multiple unresolved target localization and tracking using colocated mimo radars, IEEE Trans. Aerosp. Electron.Syst. 48 (3) (2012) 2498–2517. [5] Y. Yao, A.P. Petropulu, H.V. Poor, Power allocation for cs-based colocated mimo radar systems, Sensor Array & Multichannel Signal Processing Workshop, 2012. [6] D.J. Rabideau, P. Parker, Ubiquitous mimo multifunction digital array radar, in: Conference on Signals, Systems & Computers, 2003. [7] J. Li, P. Stoica, MIMO Radar Signal Processing, 2009. [8] E. Fishler, A. Haimovich, R.S. Blum, D. Chizhik, L.J. Cimini, R.A. Valenzuela, Mimo radar: an idea whose time has come, in: IEEE Radar Conference, 2004. [9] J. Li, P. Stoica, Mimo radar diversity means superiority, 2006. [10] J. Li, P. Stoica, Mimo radar with colocated antennas, IEEE Signal Process Mag 24 (5) (2007) 106–114. [11] E. Fishler, A. Haimovich, R.S. Blum, L.J.C. Jr, D. Chizhik, R.A. Valenzuela, Spatial diversity in radars-models and detection performance, IEEE Trans. Signal Process. 54 (3) (2006) 823–838. [12] A.M. Haimovich, R.S. Blum, L.J. Cimini, Mimo radar with widely separated antennas, IEEE Signal Process. Mag. 25 (1) (2007) 116–129. [13] S.H. Zhou, H.W. Liu, Signal fusion-based target detection algorithm for spatial diversity radar, IET Radar Sonar Navig. 5 (3) (2011) 204–214. [14] X. Tang, J. Tang, T. Bo, Z. Gao, B. Xin, J. Du, A new electronic reconnaissance technology for mimo radar, in: IEEE Cie International Conference on Radar, 2012. [15] L. Xu, J. Li, P. Stoica, Target detection and parameter estimation for mimo radar systems, IEEE Transactions on Aerospace & Electronic Systems 44 (3) (2009) 927–939. [16] Y.C. Wang, W. Xu, H. Liu, Z.Q. Luo, On the design of constant modulus probing signals for mimo radar, IEEE Trans. Signal Process. 60 (8) (2012) 4432–4438. [17] Y. Yang, R.S. Blum, Mimo radar waveform design based on mutual information and minimum mean-square error estimation, IEEE Trans. Aerosp. Electron. Syst. 43 (1) (2007) 330–343. [18] L. Wang, L. Wang, Y. Zeng, M. Wang, Jamming power allocation strategy for mimo radar based on MMSE and mutual information, Iet Radar Sonar Navig. 11 (7) (2017) 1081–1089.

Y. Su, T. Cheng and Z. He et al. / Signal Processing 172 (2020) 107543 [19] A.A. Gorji, T. Kirubarajan, R. Tharmarasa, Antenna allocation for mimo radars with collocated antennas, in: International Conference on Information Fusion, 2012. [20] A.A. Gorji, R. Tharmarasa, W.D. Blair, T. Kirubarajan, Optimal antenna allocation in mimo radars with collocated antennas, IEEE Trans. Aerosp. Electron.Syst. 50 (1) (2012) 542–558. [21] J. Zhang, Y. Zhao, S. Zhao, L. Zhang, L. Nan, Active jamming suppression based on transmitting array designation for colocated multiple-input multiple-output radar, IET Radar Sonar Navig. 10 (3) (2016) 500–505. [22] J. Yan, H. Liu, J. Bo, C. Bo, L. Zheng, B. Zheng, Simultaneous multibeam resource allocation scheme for multiple target tracking, IEEE Trans. Signal Process. 63 (12) (2015) 3110–3122. [23] J. Yan, H. Liu, W. Pu, S. Zhou, B. Zheng, Joint beam selection and power allocation for multiple target tracking in netted colocated mimo radar system, IEEE Trans. Signal Process. 64 (24) (2016) 6417–6427. [24] J. Yan, J. Bo, H. Liu, C. Bo, B. Zheng, Prior knowledge-based simultaneous multibeam power allocation algorithm for cognitive multiple targets tracking in clutter, IEEE Trans. Signal Process. 63 (2) (2014) 512–527. [25] J. Yan, H. Liu, B. Zheng, Power allocation scheme for target tracking in clutter with multiple radar system, Signal Process. 144 (2017) 453–458. [26] H. Zhang, J. Xie, J. Shi, T. Fei, J. Ge, Z. Zhang, Joint beam and waveform selection for the mimo radar target tracking, Signal Process. 156 (2019) 31–40. [27] T. Cheng, S. Li, J. Zhang, Adaptive resource management in multiple targets tracking for co-located multiple input multiple output radar, IET Radar Sonar Navig. 12 (9) (2018) 1038–1045. [28] Y. Yuan, W. Yi, L. Kong, A complete power allocation framework for multiple target tracking with the purpose of minimizing the transmit power, in: International Conference on Information Fusion, 2018. [29] Y. Yuan, W. Yi, L. Kong, X. Yang, M. Ge, An adaptive resource allocation strategy for multiple target tracking with different performance requirements, in: IEEE Radar Conference, 2018. [30] Y. Yuan, W. Yi, T. Kirubarajan, L. Kong, Scaled accuracy based power allocation for multi-target tracking with colocated mimo radars, Signal Process. 158 (2019) 227–240.

11

[31] J. Yan, W. Pu, S. Zhou, H. Liu, Z. Bao, Collaborative detection and power allocation framework for target tracking in multiple radar system, Inf. Fusion 55 (2020) 173–183, doi:10.1016/j.inffus.2019.08.010. [32] Z. Zhang, Y. Tian, A novel resource scheduling method of netted radars based on Markov decision process during target tracking in clutter, Eurasip J. Adv. Signal Process. 2016(1) 16. [33] C. Shi, F. Wang, S. Salous, J. Zhou, Joint subcarrier assignment and power allocation strategy for integrated radar and communications system based on power minimization, IEEE Sens. J. 19 (23) (2019) 11167–11179, doi:10.1109/ JSEN.2019.2935760. [34] J. Li, L. Xu, P. Stoica, K.W. Forsythe, D.W. Bliss, Range compression and waveform optimization for mimo radar: a Cramerrao bound based study, IEEE Trans. Signal Process. 56 (1) (2008) 218–232. [35] Z. He, J. Li, H. Liu, Q. He, MIMO radar, 2017. [36] E. Mazor, A. Averbuch, Y. Bar-Shalom, J. Dayan, Interacting multiple model methods in target tracking: a survey, IEEE Trans. Aerosp. Electron.Syst. 34 (1) (2002) 103–123. [37] E. Daeipour, Y. Bar-Shalom, IMM tracking of maneuvering targets in the presence of glint, Aerosp. Electron. Syst. IEEE Trans. 34 (3) (1998) 996–1003. [38] E. Daeipour, Y. Bar-Shalom, An interacting multiple model approach for target tracking with glint noise, Aerosp. Electron. Syst. IEEE Trans. 31 (2) (1995) 706–715. [39] X.R. Li, Y. Bar-Shalom, Performance prediction of the interacting multiple model algorithm, IEEE Trans. Aerosp. Electron.Syst. 29 (3) (1993) 755–771. [40] Y. He, J. Xiu, X. Guan, Practical Application of Radar Data Processing, 2016. [41] G. Zhang, Phased Array Radar System, 1994. [42] G.A. Watson, W.D. Blair, Tracking performance of a phased array radar with revisit time controlled using the IMM algorithm, in: Radar Conference, 1994. [43] Research on Resource Management Technology for Phased Array Radar and its Network in Tracking System. Ph.D. thesis, National University of Defense Technology 2004