Abstract
Considering the demands of different location accuracy for multiple targets tracking, performancedriven resource allocation schemes in distributed MIMO radar system are proposed. Restricted by the tracking antenna number, location estimation meansquare error (MSE), and target priorities, an optimization problem of the minimal antenna subsets selection is modeled as a knapsack problem. Then, two operational schemes, modified fair multistart local search (MFMLS) algorithm and modified fair multistart local search with one antenna to all targets (MFMLS_OAT) algorithm, are presented and evaluated. Simulation results indicate that the proposed MFMLS and MFMLS_OAT algorithm outperform the existing algorithms. Moreover, the MFMLS algorithm can distinguish targets with different priorities, while the MFMLS_OAT algorithm can perform the tracking tasks with higher accuracy.
1. Introduction
Multiple Input Multiple Output (MIMO) radar [1] can be divided into two basic regimes of architecture, centralized MIMO radar and distributed MIMO radar. The former can obtain the waveform diversity and degree of freedom to improve the parameter estimation performance with centralized antennas, which is suitable for a point target model due to the lack of angle diversity [2, 3]. The latter offers enhanced target detection and localization capabilities by widely spaced antennas where an extended target model can be achieved [4, 5]. With the development of stealth technology, the targets may be invisibility, while the target fluctuation characteristic is difficult to control. Considering that the distributed MIMO radar has an advantage on overcoming the target radar cross section (RCS) fluctuation in complex electromagnetic environments [6] and performs better antiinterception and antidestroying character than centralized radar system, in this paper we focus on the distributed MIMO radar to improve target tracking capacity with a centralized signal fusion structure where global information can be collaborated and the allocation results can be directly transformed to each antenna by internal communication network, such as satellite link and fiber optic link. In distributed MIMO radar system, the advantages of multiple channels depend on the optimal system architecture and flexible signal design. Both system architecture and transmit parameters can be considered as system resources. Therefore, reasonable resource allocation schemes are needed for better system performance in distributed MIMO radar system [7â€“11]. The topology of the transmitters and receivers with respect to targets needs to be available for resource allocation, so the target tracking process needs to be considered [12]. Moreover, distributed MIMO radar system has advantages on multiple targets management [13]. Resource allocation schemes play an important part in multiple tasks system. Overall, researches on resource allocation problem in distributed MIMO radar system for multiple targets tracking are especially necessary.
In terms of resource allocation on system architecture, existing researches mainly focus on the intelligence of antenna selection [14, 15]. Some antennas may have greater impacts on system performance than others [16], affected by the different propagation paths, target reflection coefficients, and topologies of system with respect to targets. Reasonable antenna selection can maximize antenna utilization and minimize computational complexity and communication cost among stations. Currently, the antenna selection problem in MIMO radar system is usually modeled as a knapsack problem (KP). The Bayesian CramerRao bound (BCRB) is used as the performance metric for parameter estimation. Compared with an exhaustive search, the existing heuristic algorithm can offer considerable reduction in computational complexity. Considering the data transmission and computational complexity, a single target localization scheme based on antenna selection is proposed in [17] where two optimization models are included. The first idea achieves the minimum antenna subset out of available transmitters and receivers within a given MSE threshold. The second is to select antenna subset with a specific subset size such that the location estimation capability is maximized where a constraint on operational cost is considered. Antenna cluster method for multiple targets localization is generated in [18] where each target is just tracked by the corresponding antenna subset. Greedy multistart local search (GMLS) algorithm and fair multistart local search (FMLS) algorithm are proposed. The former offers lower computational complexity with random target sequence, while the latter offers better selection performance with a more balanced allocation of antennas. Joint schemes of antenna selection and power allocation for target detection and localization are studied in [14, 15] to improve system performance. A specific TOAbased passive localization method is presented in [19] where parametric belief propagation (BP) algorithm could be attractive under the impact of receiversâ€™ position uncertainty, due to the significantly lower computational complexity.
However, previous researches on antenna selection have failed to consider specific system tasks. Therefore, further studies are still necessary in this paper. Existing studies on antenna selection mainly focus on the static target. For the resource allocation it is necessary to predict target state in advance. Antenna selection problem for moving targets is considered in this paper. Antenna utilization is ignored in [16]. For the improvement of the antenna utilization and the demands of other tasks, a constraint on tracking antenna utilization is proposed in this paper. The same MSE threshold is set for all targets in [18]. For further close to practical application, location accuracy and priorities for different targets should be different. A modified cluster method based on [18] is presented to solve it. Antenna cluster method can reduce the data transmission to fusion center, but it is difficult for higher location accuracy. Therefore, an idea with every antenna to track all targets for different location accuracy is introduced.
In this paper, resource allocation schemes for the demands of different location accuracy are proposed in distributed MIMO radar system, which have not been studied before. The paper is organized as follows: The system model is introduced in Section 2, including the derivation of the BCRB. Resource allocation schemes are proposed in Section 3, including problem formulation and antenna selection algorithms. Simulation results in different scenarios are provided in Section 4. Finally, Section 5 concludes the paper.
2. System Model and Preliminaries
Consider a widely distributed MIMO radar system where only one antenna is configured for each station. There are transmitters located at and receivers located at . At state , define the time interval where is the observation interval. moving targets are located and move with velocity . A set of orthogonal waveforms are transmitted where , and is the duration time of the transmitted signal. Define the transmit power vector , the effective bandwidth vector , and the effective illumination time vector for transmitters.
For simplicity time synchronization has been satisfied between different receivers. The lowpass signal observed at receiver can be written aswhere denotes propagation channel along the path transmitter target receiver . is the time delay along path , satisfyingwhere and , respectively, denote the Euclidean distances from transmitter to target and from target to receiver . is the speed of light. , the Doppler shift due to the target velocity, can be expressed aswhere and are the angle from transmitter to target and the angle from receiver to target . models the pathloss in free space along the path where . is the carrier frequency. represents the corresponding targets reflection coefficients. For sufficiently spaced antennas each target is modeled as a collection of reflection coefficients forming the RCS model. The noise is assumed circularly symmetric, zeromean, complex Gaussian noise, spatially and temporally white with autocorrelation function . According to [20], , where is pulse repetition frequency.
Define the state vector for target . At state , the statetransition model for target is a linear motion model, represented aswhere , the statetransition matrix for uniform motion, is of the form, plant noise modeled as Gaussian noise process with covariance matrix , is of the form is the noise intensity.
Define measurable vector , where , .
The observation vector is a nonlinear function of the vector :where stands for the observation process and is the observation noise.
At high signaltonoise ratio (SNR) [21], the target location estimation MSE is close to the BCRB; the latter may be used to evaluate the location estimation performance. At state , BCRB matrix of target can be expressed aswhere denotes the Bayesian Information Matrix (BIM) of target , and the recursive BCRB is of the formwhere is the expectation; is the Fisher Information Matrix (FIM), which can be obtained by applying the chain rule , where the Jacobi matrix is ; is the FIM of , which can be derived by the conditional probability distribution function .
Define antenna selection vectors for the transmitters and for receivers. , where means abandoned and 1 means selected. The diagonal elements of BCRB matrix satisfy . An expression for localizing target with antenna selection is derived in [22]where , , , and are, respectively, expressed aswhere
3. Resource Allocation Schemes
In multiple tasks system radar systems need to deal with the problem of insufficient resources. Reasonable resource allocation schemes are important for application demands of given location MSE. Antenna selection is considered to improve resource utilization when the resources on transmit parameters are enough. The detailed allocation process can be described as follows: at one state, orthogonal signals transmitted by transmitters are reflected by the targets and observed by receivers. All the receivers directly transfer the a global signal from multiple targets to one centralized fusion center where the effective data is employed to predict target state and calculate the optimal resource allocation strategy for next state. In the end, the allocation results can be directly informed to each antenna by internal communication network for next tracking.
In this paper, targets are assumed to be moving in low attitude. All the targets are not treated equally, which are divided into general targets, suspicious targets, and hot targets. And the corresponding tracking location estimation MSE are successively reduced with MSE ranges 20^{2}~30^{2}â€‰m^{2}, 10^{2}~20^{2}â€‰m^{2}, and 0~10^{2}â€‰m^{2}. The lower the MSE threshold is, the higher the priority will be.
3.1. Problem Formulation
In multiple targets tracking system, the key targets need to be preferred for the system demands. Meanwhile, the location accuracy for others needs to be improved. Under the constraint of tracking antenna number and location estimation MSE, a uniform problem formulation for antenna selection may be expressed aswhere denotes the selected active antenna number. is given location MSE threshold for target . are the targets with high priorities, which will be ensured beforehand. are the targets with low priorities, which will be ensured after. is the antenna utilization for targets tracking.
3.2. Antenna Selection Algorithms
Different resource allocation schemes are considered to process different tracking accuracy requirements. Two resource allocation schemes, modified fair multistart local search (MFMLS) algorithm for high location MSE and modified fair multistart local search with one antenna to all targets (MFMLS_OAT) algorithm for low location MSE, are, respectively, proposed.
3.2.1. MFMLS Algorithm for High Location Estimation MSE
Studies on antenna selection in [18] indicate that the minimal antenna subset with a given high MSE threshold could be obtained with antenna cluster method, where GMLS and FMLS algorithm are proposed. The optimized target is random by GMLS algorithm, though it may realize a prior tracking requirement for some target. A more balanced allocation is generated by FMLS algorithm, but it ignores the different targets priorities processing. In this case, MFMLS algorithm inspired by GMLS and FMLS algorithm is proposed where the targets in the same and different priorities are considered.
Assume that there are general targets and suspicious targets in multiple targets tracking system with high MSE thresholds; the latter are considered as the key targets in this section. Under the constraint of tracking antenna number, the radar system needs to ensure the demand for suspicious targets where an antenna cluster method is employed. At one measurement, every antenna can be selected one time at most for just tracking one target. Therefore, antenna selection vectors, and just effective for target , are introduced in this section. And problem (15) can be rewritten aswhere denotes the optimal total antenna number.
In Algorithm 1, the specific MFMLS algorithm is proposed. A radar subset is initially generated by selecting one transmitter and one receiver for every target. Meanwhile, they are discarded from the original antenna sets . At each iteration, either one transmitter or receiver from the remaining antennas is added to the active subset such that the trace of BCRB matrix is closer to the given MSE, until the location MSE is met , or tracking antennas use up . According to the target priorities, an idea of target classification processing is inspired by GMLS algorithm. And problem (16) can be separated into two steps of optimization. Firstly, allow the radar system to select enough antennas to ensure tracking tasks of suspicious targets . Then, select from the remaining ones to improve the tracking accuracy for general targets . An idea of a balanced allocation for targets with the same priority is inspired by FMLS algorithm. An accuracy distance is defined as . At each iteration, the target with large accuracy distance is preferred. Restricted by tracking antenna number and other factors, the radar systems may fail to achieve all of the tracking tasks. When the tracking tasks of suspicious targets are done, MFMLS algorithm continues to select the minimal antenna subsets for other tracking tasks. When the active antenna subset is more than one, more targets number and higher location accuracy are considered in turn.

3.2.2. MFMLS_OAT Algorithm for Low Location Estimation MSE
The lower the location estimation MSE, the more the antennas that the system will need. Though MFMLS algorithm can meet the demands of different target priorities, it may fail to reach the lower MSE thresholds. MFMLS_OAT algorithm based on MFMLS algorithm considering the target priority is proposed to improve the antenna utilization and ensure the tracking accuracy for the key targets. Instead of antenna cluster method, MFMLS_OAT algorithm employs every antenna to track all the targets where all the receiversâ€™ data are integrated to a fusion center for centralization processing.
Assume there are suspicious targets and hot targets with low MSE thresholds, and the latter are considered as the key targets in this section where one antenna can be utilized multiple times to track multiple targets. Antenna selection vectors and are introduced in this section. And problem (15) can be rewritten aswhere denotes the optimal antenna subset size.
In Algorithm 2, the proposed MFMLS_OAT algorithm is presented to improve the resource utilization for problem (17). Similar to MFMLS algorithm, the MFMLS_OAT algorithm allows the radar system to ensure the demands of hot targets firstly and then improve the tracking accuracy for suspicious targets with the remaining resources. For the targets with same priority, they are equally treated. At each iteration, one antenna is selected to improve the location accuracy for the target with large accuracy distance. Eventually, an optimal antenna subset with the minimal antenna number and the highest location accuracy, is obtained.
