SQPbased Power Allocation Strategy for Target Tracking in MIMO Radar Network with Widely Separated Antennas
Subject Areas : Signal Processing
Mohammad
Akhondi Darzikolaei
^{
1
}
(Department of Electrical and Computer Engineering, Babol Noshirvani University of Technology,Babol, Iran)
Mohammad Reza
KaramiMollaei
^{
2
}
(Department of Electrical and Computer Engineering, Babol Noshirvani University of Technology,Babol, Iran)
Maryam
Najimi
^{
3
}
(Department of Electrical and Computer Engineering, University of Science and Technology of Mazandaran,Behshahr, Iran)
Keywords: MIMO radar, Power allocation, SQP, Target tracking,
Abstract :
MIMO radar with widely separated antennas enhances detection and estimation resolution by utilizing the diversity of the propagation path. Each antenna of this type of radar can steer its beam independently towards any direction as an independent transmitter. However, the joint processing of signals for transmission and reception differs this radar from the multistatic radar. There are many resource optimization problems which improve the performance of MIMO radar. But power allocation is one of the most interesting resource optimization problems. The power allocation finds an optimum strategy to assign power to transmit antennas with the aim of minimizing the target tracking errors under specified transmit power constraints. In this study, the performance of power allocation for target tracking in MIMO radar with widely separated antennas is investigated. Therefore, a MIMO radar with distributed antennas is configured and a target motion model using the constant velocity (CV) method is modeled. Then Joint Cramer Rao bound (CRB) for target parameters (joint target position and velocity) estimation error is calculated. This is utilized as a power allocation problem objective function. Since the proposed power allocation problem is nonconvex. Therefore, a SQPbased power allocation algorithm is proposed to solve it. In simulation results, the performance of the proposed algorithm in various conditions such as a different number of antennas and antenna geometry configurations is examined. Results affirm the accuracy of the proposed algorithm.
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SQPbased Power Allocation Strategy for Target Tracking in MIMO Radar Network with Widely Separated Antennas
Department of Electrical and Computer Engineering, Babol Noshirvani University of Technology,Babol, Iran M.akhondi@stu.nit.ac.ir Mohammad Reza Karami Mollaei * Department of Electrical and Computer Engineering, Babol Noshirvani University of Technology,Babol, Iran Maryam Najimi Department of Electrical and Computer Engineering, University of Science and Technology of Mazandaran,Behshahr, Iran
Received: 22/Aug/2021 Revised: 06/Nov/2021 Accepted: 07/Dec/2021 
Abstract
MIMO radar with widely separated antennas enhances detection and estimation resolution by utilizing the diversity of the propagation path. Each antenna of this type of radar can steer its beam independently towards any direction as an independent transmitter. However, the joint processing of signals for transmission and reception differs this radar from the multistatic radar. There are many resource optimization problems which improve the performance of MIMO radar. But power allocation is one of the most interesting resource optimization problems. The power allocation finds an optimum strategy to assign power to transmit antennas with the aim of minimizing the target tracking errors under specified transmit power constraints. In this study, the performance of power allocation for target tracking in MIMO radar with widely separated antennas is investigated. Therefore, a MIMO radar with distributed antennas is configured and a target motion model using the constant velocity (CV) method is modeled. Then Joint Cramer Rao bound (CRB) for target parameters (joint target position and velocity) estimation error is calculated. This is utilized as a power allocation problem objective function. Since the proposed power allocation problem is nonconvex. Therefore, a SQPbased power allocation algorithm is proposed to solve it. In simulation results, the performance of the proposed algorithm in various conditions such as a different number of antennas and antenna geometry configurations is examined. Results affirm the accuracy of the proposed algorithm.
Keywords: MIMO radar; Power allocation; SQP; Target tracking.
1 Introduction
The RADAR is a short form of Radio Detection And Ranging. Radar utilizes electromagnetic waves to detect, locate and measure the speed of reflected objects. It transmits the electromagnetic waves into space and receives the echo signals [12]. In recent years, radar network systems such as multistatic radars and multiinput multioutput (MIMO) radars have become an attractive and improtant problem [3]. Spatial diversity [4], waveform diversity [5], and multiplexing gain [6] over common monostatic radar [7] are some positive characteristics of networked radar systems. This structure of radars helps to raise tracking accuracy in multiple target tracking scenarios with radar, sonar, and video sensors [8]. A MIMO radar is a kind of radar structure that uses a combination of antennas as transmitter and receiver and each of them emits its own waveform apart from others [9]. Widely separated antennas and collocated antennas are two famous categories for MIMO radar. In collocated MIMO radar, the antennas are close to each other. The antennas in MIMO radar with widely separated antennas are far from each other. In other words, the transmit and receive antennas are located in a wide area. Therefore, the target is seen from different angles by antennas. In this type of radars, each receiver should receive all signals from all transmitters and then emit them to the central processor. This means that each receiver does not process or make a decision individually and sends its signals to the central processor to process all signals. This feature is the main difference between multistatic and MIMO radar with widely separated antennas. Power allocation is usually performed in radar networks to find the best strategy to assign power among various transmit antennas, aiming at minimizing the estimation error under specified transmit power constraints or its converse [10]. Power allocation is an important section of military operations in a hostile environment to obtain a low probability of interception [11]. Power allocation in radar networks is studied in last researches. For example, in [12] power allocation in widely separated multiinput multioutput (MIMO) radar for rangeonly target tracking is evaluated. This is performed by maximizing the Bayesian Fisher information matrix (BFIM). BFIM is derived for the predetermined signal model and then the problem is modeled as one cooperative game. [13] Shows the commercial application of power allocation in radar networks. In fact, it puts a cognitive radar network in an urban environment and this network tracks cars and vehicles. The power allocation problem for radar networks in a cooperative gametheoretic structure is considered in [14] to enhance the low probability of intercept (LPI) performance. In addition, by considering transmit power constraint and minimum signal to noise and interference ratio (SINR) for each radar, a cooperative Nash Bargaining power allocation game based on LPI is expressed. The radar network in [15] consists of unmodulated continuous wave (UCW) radars. This network utilizes a power allocation algorithm for Doppleronly target tracking. This algorithm minimizes the mean square error of target state estimation with a power budget constraint. [16] Investigates power allocation for radar networks to increase the performance of low probability of intercept. Two power allocation strategies are stated in this reference. One is for optimizing transmit power allocation with predetermined mutual information (MI) threshold and another is for finding optimal power allocation with minimum mean square error (MMSE) threshold. In [17], power allocation and target assignment in radar networks is mentioned to improve LPI performance. The target geometrical method is used to fuse information for measuring target localization by different radars. Authors in [18] propose joint antenna selection and power allocation for localization in distributed MIMO radar networks. The sensor management is performed by solving a constrained problem which is expressed to minimize the error in estimating target position, while it is constrained by transmitter number and power budget. [19] Talks about sensor selection in radar networks and for target tracking, the number of radars are selected. This problem is just for one target tracking. In this reference, sensor selection is performed by information theory. Joint transmitter and receiver selection in target tracking in distributed MIMO radar is described in [20]. Due to resource restriction in radars, it is necessary to select some radars in the MIMO radar network at each time and also keep the system performance in the best condition. So, lower bound PCRLB is u as an optimization criterion for an optimization problem in this literature. In [21], a joint power allocation and sensor selection algorithm for multitarget tracking in an LPI radar network with N monostatic radars is introduced. This algorithm can minimize the total transmitted power of a radar network based on predefined mutual information threshold between reflected signal and target impulse computed predictively with the estimation of the target state is needed for estimation of target parameters. [22] Describes the joint beam selection and power allocation strategies for multitarget tracking in collocated MIMO radar. Each radar works based on a multibeam working mode, in which multiple simultaneous transmit beams are synthesized. This strategy applies an optimization technique to control the limited beam and power resource of each radar to obtain accurate target state estimation. Therefore, Bayesian Cramer Rao Lower Bound is extracted, normalized, and utilizes as the optimization criterion. To increase the system performance and resource utilization of widely separated MIMO radar, a joint resource allocation for velocity estimation problem in multitarget tracking is proposed in [23]. The authors chose one target as a key target and then examined their strategy by this target. They considered a Mean Square Estimation of velocity estimation of the key target as a minimization problem criterion. With limited resources and requirements for velocity estimation for targets, a joint optimization model with the selection of numbers of receivers and transmitters and allocation of transmit power and signal time is introduced. The Authors in [24] claims that since transmitters in MIMO radars with widely separated antennas emit waveforms with different powers and bandwidths, therefore these two parameters are limited. In this reference, they offer power allocation, bandwidth allocation, and joint power and bandwidth allocation problems. They compute Cramer Rao for target localization accuracy and utilize it as optimization criteria. In [25], a solution for joint beam and power scheduling in the netted Collocated MIMO radar systems for distributed multitarget tracking is suggested. An adaptive sensor scheduling integrated with power and bandwidth allocation is presented for centralized multiple target tracking in the netted collocated MIMO radar in [26].
By investigating the above references, we understand that power allocation for target tracking in MIMO radar with widely separated antennas is very essential and the performance of this problem should be improved. With our reviews, there are some challenges in the power allocation problem for target tracking in MIMO radar with widely separated antennas which are not investigated in other papers and we consider them in this paper.
1. In this paper, for the calculation of target tracking errors, joint target velocity and position estimation is used to improve tracking performance. Although someone works on this issue for MIMO radar, this is not used in power allocation problems in MIMO radar with widely separated antennas. For example, in [25] and [7], joint estimation is considered but certain power is determined for the transmit power and power allocation strategy is not performed. And also, in [25], it does not exactly specify its MIMO radar structure (distributed or collocated). In other researches about power allocation for target tracking in MIMO radar with widely separated antennas, joint estimation is not worked. For instance, in [12], rangeonly estimation for target tracking in MIMO radar with widely separated antennas is performed and it is not considered velocity. Whereas, in [23], velocity estimation is utilized to compute target tracking for power allocation problem in distributed MIMO radar. These are just some papers which we investigated and we concluded that joint estimation for target tracking in power allocation problem for MIMO radar with widely separated antennas is not used yet. Although it may be performed for collocated MIMO radar. Thus , joint target velocity and position estimation is the first novel idea for our power allocation strategy in MIMO radar with widely separated antennas.
2. Using random mathematical statistics for target RCS is another unique characteristic of this paper. Because in other researches, or a deterministic model for target RCS is considered [7, 24, 18, 22, 23], or if they supposed a random target RCS, they neglected it in their next calculations for simplicity [20, 21]. Therefore, none of the previous researches did contribute a power allocation strategy for target tracking in MIMO radar with widely separated antennas by considering random complex Gaussian target RCS. They usually neglected it or put a deterministic number instead of variance of Gaussian distribution. But in this paper, in all calculations, random complex Gaussian with random variance in different transmitreceive paths is considered. Note that assuming random RCS is necessary for MIMO radar with widely separated antennas. Because each antenna sees the target in a specified angle and target reflections in different transmitreceive paths are different with respect to each other.
The main contributions of this paper are as follows:
1. The system model for MIMO radar with widely separated antennas is introduced. Then CV model is considered for target motion. Considering a complex Gaussian random model for target RCS and using this feature in the next calculations, Cramer Rao bound for target parameters estimation error, is one of the prominent aspects of this paper.
2. Maximum likelihood (ML) estimation for unknown target parameters which were target position and velocity is calculated and then joint CRB for target position and velocity estimation is computed. Output of joint CRB is considered as an objective function for power allocation problem.
3. A power allocation strategy is formed. In fact, joint CRB for target position and velocity estimation function (target tracking errors) subject to some constraints such as limitation in total transmit power and transmit power of each transmit antenna is the power allocation problem of this paper. Our goal is to minimize tracking errors by using the mentioned constraints.
4. For solving the previous section problem, since it is nonconvex problem, SQP^{1} based power allocation algorithm is proposed. This algorithm is formed based on the SQP algorithm and it can allot optimal power to each transmit antenna to satisfy the constraints in the problem. The rest of the paper is structured as follows:
The system model is mentioned in Section 2. Section3 exhibits the ML estimation calculations. Joint CRB for target parameters is computed in section4. Section 5 forms a power allocation problem to minimize target tracking error by considering total power limitation. And also, a proposed SQPbased algorithm is presented in this part to solve this problem. Simulation results are shown in section 6 and finally in part 7, concluding remarks are addressed.
2 System Model
Consider a MIMO radar with widely separated antennas with M transmitters and N receivers. Denote the location of mth transmitter in (), where and the coordinates of nth receiver in (), where. Target is in initial location () with initial velocity of (). A set of low pass equivalent orthogonal waveforms,, is transmitted. (. period, effective bandwidth and transmit power of th transmit waveform are shown as ,, . Target RCS corresponding to th path is modeled as a zeromean complex Gaussian random variable. Where is the variance of th path and it is known. Fig.1 shows the structure and location of antennas in MIMO radar with widely separated antennas with respect to the target.
Fig. 1 configuration of a MIMO radar with widely separated antennas
We suppose the below assumption to simplify our problem.
1. (Noise of th path with a zeromean complex Gaussian random variable and variance of ) and in different paths are mutually independent.
2. Transmit waveforms are orthogonal.
(1)
This orthogonality also remains for time delays and Doppler shifts and [26]:
(2)
3. Set without loss of generality.
4. The antennas are adequately separated [27]. Therefore, each path provides an independent observation of the target and is independent for different and paths.
The time delay of th channel in th time slot is:
(3)
Where,
(4)
In the above equations, is light velocity. is distance from target and th transmitter and is distance from a target and th receiver.
With these assumptions, the received signal from th transmit antenna at th receive antenna at time is given by:
(5)
In the above equation, represents a zeromean complex Gaussian noise with the variance of . Power variations due to path loss is shown as . Where is the carrier frequency.
Doppler frequency in path and time is given by:
(6)
is wavelength.
21 Target Dynamic Model
Target tracking in a MIMO Radar with widely separated antennas is the favorable problem of this paper. The target motion model is the constant velocity (CV). This model is as below[7]:
Target state vector is described as . This parameter is considered as unknown parameter in this paper and it should be estimated. The noise is a zeromean Gaussian matrix as . Where is state transition matrix and is the covariance matrix [27]:
 (7) 
And
 (8) 
Where denotes sample intervals and is the density of process noise.
In next section, we compute ML estimation of to use it in calculation of joint CRB.
3 Maximum Likelihood Estimation
ML estimation of the unknown parameter () [28] can be achieved by testing of the likelihood ratio for two hypothesis pair, is corresponding to target existence hypothesis modeled in (5) and corresponding to noise only hypothesis:
 (9) 
Where denotes observation signal in th receiver corresponding to th transmitter.
Loglikelihood ratio based on (10) is as:
 (10) 
Where and it does not depend to .
According to problem assumptions, noise and reflection coefficients (RCS) are independent, the joint time delay and Doppler likelihood ratio term are obtained by:
 (11) 
Where collects all observed signals from all antennas set and it is introduced as:
 (12) 
In the following part, joint Cramer Rao bound of target parameters is derived.
4 Joint Cramer Rao Bound
In this section, Bayesian Fisher Information and joint CRB for target position and target velocity under our problem assumptions are provided.
Bayesian Fisher Information provides a lower bound on the target tracking Mean Square Error (MSE) for the estimation of target state. The Cramer Rao bound represents a lower bound on the accuracy of state estimates. If be the estimation of target state, the CramerRao bound is stated as [7]:
 (13) 
Where is expectation operator and is the Bayesian Information Matrix (BIM).
According to [29], Bayesian Information Matrix for unknown parameter vector is as:
 (14) 
Where is Fisher Information Matrix (FIM).
The first step in CRB calculation is computing FIM which is a matrix corresponding to the second derivatives of joint loglikelihood [26]:
 (15) 
Where denotes the expected value operator.
By considering (10) which is the function of and , a new parameter as below is described:
 (16) 
Based on Chain rule, a new FIM is introduced as:
 (17) 
First, by using (17) and, we calculate :
[1] sequential quadratic programming
 (18) 
 (19) 
Therefore, the above matrix parameters are defined as:
 (20) 
 (21) 
 (22) 
 (23) 
 (24) 
As shown in the above equations, the parameters of ، ،،،، are determined by target position and velocity and is light velocity.
If is divided into matrix blocks as:
 (25) 
A,B,D are matrices and is zero matrix.
is also obtained as:
 (26) 
Where it is a matrix.
is represented as:
 (27) 
Where , , ,and are matrices. In fact, includes all secondorder derivatives terms with respect to for all and in time slot . , contains secondorder derivatives with respect to and for all and in time slot . contains secondorder derivatives with respect to for all and in time slot .
Therefore,
 (28) 
Where is an identity matrix, is Hadmard product operator, and is Kronecker product operator.
 (29) 
In fact, is an matrix.
 (30) 
 (31) 
Where , and depend on receive waveform characteristics and they are defined as:
 (32) 
 (33)  
 (34) 
Therefore, By considering the righthand matrix as which is the function of target position and velocity in each path and in different frames (although this matrix is not the function of transmit power), (36) can be reform as:
 (35) 
In fact, is a matrix that is the function of transmit power.
Note that CRB for unknown parameter estimation is achieved by the diagonal elements of the inverse of FIM :
 (37) 
In above equations ,, and respectively show the variance of target position in axis of and and also target velocity in axis of and in th frame.In next section, our power allocation problem is formed.
5 Power Allocation Problem
In this section, we present power allocation strategy for a MIMO radar with widely separated antennas. In this problem and in the formulation of objective function, we consider a random RCS for target. This is rarely done in other papers and for simplicity, they consider deterministic reflection coefficients for target in MIMO radar with widely separated antennas. So this assumption make our main part of our objective function (equation (37)) complex.
51 Problem Formulation
In definition of our power allocation problem, we consider trace of CRB matrix as target tracking error and it is shown as:
 (38) 
 (39) 
 (40) 
 (41) 
Where is a transmit power vector, is the normalization matrix and is considered as the target tracking error.
Note that in equation (15), the first term (lefthand term) is approximately constant, therefore the second part, , is utilized for the calculation of the Another notable point is about diagonal elements of , since the first and third diagonal elements are corresponding to the target position and second and fourth diagonal elements are corresponding to target velocity and their scales are different, so it is important to change them with no loss of generality to have the same scale. Fisher information matrix.Therefore we introduce as:
 (42) 
Where is identity matrix and is the Kronecker product operator.
In the above equations, is a Fourier transform of . Therefore, (36) is obtained.
 (43) 


(36)


Therefore, the power allocation problem for target tracking in MIMO Radar with widely separated antennas by considering random radar crosssection (RCS) is defined as:
In the above problem, the first constraint represents that total transmit power is less than a predetermined value,. In fact, it says that this value does not exceed , because MIMO radar tries to use the least power for target tracking and it is not intercepted by other radars. The second constraint shows that each antenna also has power limitation.
52 Problem Solving
The problem (44) is nonlinear and it is nonconvex, So we propose the sequential quadratic programming (SQP) algorithm to solve it.
521 SQP Method
Sequential quadratic programming methods are famous to be affective for solving a series of related nonlinear optimization problems because of favorable hot and warm start properties—a solution for one problem is a good estimate of the solution of the next.
SQP applies Newton’s method (or quasiNewton methods) to directly solve the KKT conditions for the original problem. As a result, the accompanying subproblem turns out to be the minimization of a quadratic approximation to the Lagrangian function subject to a linear approximation to the constraints. Therefore, this kind of process is also known as a projected Lagrangian, or the NewtonLagrange, approach. By its nature, this method produces both primal and dual (Lagrange multiplier) solutions.The procedure of SQP for constrained nonlinear problem as below [28]:
 (44) 
 (45) 
 (46) 
Where, is an objective function, and represent inequality and equality function and and are twice continuously differentiable. shows the desirable variable matrix and it is limited by lower and upper bound and .
Given an iterate (, where and are the Lagrange multiplier estimates for the equality and inequality constraints, respectively, consider the following QP subproblem as a direct extension of QP(:
Minimize:  (47) 
Subject to:  (48) 
=0,  (49) 
 (50) 
(51) is named as QP( and .
Note that in above equations, for simplicity the constraint is not considered and also note that the KKT
Conditions for QP( need that, in addition to primal feasibility, Lagrange multipliers , be found like that:
 (51)  
=0  (52)  
=0  (53) 
With and unrestricted in sign. Clearly, if , then and and yields KKT conditions to original problem. Otherwise, we should set as before, increment by 1and repeat the procedure.
522 Proposed SQPbased Power Allocation Algorithm
As mentioned before, Problem (44) subject to (4546) is nonconvex and it should be solved by nonconvex suitable algorithms. In this section, we proposed SQPbased power allocation algorithm for our desired problem. In our scenario, problem (44) is our objective function and is our desired variable . In fact, (44) is in (47) and P in our problem is in (47). By considering , we can claim that in (48) is equal to in our problem.
There is not constraint (49) is our problem and and .
Therefore, our proposed algorithm is based on previous section and it is proposed as:
++
 (54) 
+ 
(55) 
Algorithm1: proposed SQPbased power allocation algorithm  
Initialization step Set the number of decision variables (Transmit Power ()), determine lower and upper bound of ( Consider as for simplicity. Set (transmit power vector) Choose an initial primal/dual point with And a positive definite matrix . Choose the relative tolerance (). Set step size (, is chosen to ensure the decrease in the objective function. Let and go to main step.
Main step 1. Solve the quadratic subproblem QP(, with replaced by , to obtain a direction along with a set of Lagrange multipliers (,). 2. if , then satisfies the KKT conditions for problem (4446), stop. 3. if the previous section conditions are not satisfied, choose a new , find . Update to . Replace and go to step 1.
Our proposed SQPbased power allocation algorithm flowchart is shown in Fig.2.
Fig. 2 the proposed SQPbased power allocation algorithm flowchart 6 Simulation Results In this part, we consider some scenarios to investigate our proposed SQPbased power allocation algorithm. All simulations are executed in MATLAB. In this section, two various geometrical antenna deployment configurations are considered to illustrate the effect of antenna deployment on target tracking performance. These two configurations are shown in Fig. 3. Consider a MIMO radar with and. Each antenna has 10 km distance from origin. Total power ( is . Carrier frequency is 1. Target RCS variance, , is random and it is different for any transmitreceive path (for simplicity, it is usually considered one in other papers). The power spectral density and . Target primary location and velocity in direction of and axis is equal to [500 1000 50 30]. Set (watt) and .
Fig. 3 MIMO radar antenna geometry deployment configurations ( The evaluation of proposed SQPbased power allocation algorithm is performed in these 2 Cases. Then this proposed algorithm is compared with uniform power allocation and Random power allocation and PSObased algorithm. The values of the proposed SQPbased algorithm are shown in Table 1. Table 1: proposed SQPbased parameters Fig.4 shows the transmit power of each transmit antenna for two cases (Case1, Case2).Note that all results of this section are achieved by the proposed SQPbased power allocation algorithm)
Fig. 4 Transmit power of each transmit antenna (%) for two cases with respect to timeslot
By attention to Fig.4, we can realize that by closing the target to the transmitter, more power is assigned to the transmitter. Therefore, in Case1, the transmitter 1 and 4, in Case2, the transmitter 1and 2, take more power to perform better target tracking performance. Fig.5 exhibits that the proposed SQPbased power allocation algorithm has better performance than other strategies (Random and uniform power allocation) and PSObased algorithm. it shows that the CRB of target tracking error is the least when the proposed SQPbased algorithm is used to perform the power allocation strategy. This priority is seen in all cases (Case1 (Fig.5 (a)), Case2 (Fig.5 (b))). Therefore, we can conclude that the proposed strategy has the best performance. 