# Radar Selection Method Based on an Improved Information Filter in the LPI Radar Network

Authors: Zhenkai Zhang; Bing Zhang; Zhibin Xie; Yi Yang
Journal: International Journal of Antennas and Propagation
 (2018)

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## Abstract

In order to save the radar resources and obtain the better low probability intercept ability in the network, a novel radar selection method for target tracking based on improved interacting multiple model information filtering (IMM-IF) is presented. Firstly, the relationship model between radar resource and tracking accuracy is built, and the IMM-IF method is presented. Then, the information gain of every radar is predicted according to the IMM-IF, and the radars with larger information gain are selected to track target. Finally, the weight parameters for the tracking fusion are designed after the error covariance prediction of every working radar, in order to improve the IMM-IF. Simulation results show that the proposed algorithm not only saves much more radar resources than other methods but also has excellent tracking accuracy.


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## Body

## 1. Introduction

Multiple radar systems have been shown to offer significant advantages over traditional monostatic radars [1]. When it comes to the reasonable configuration of multiple radar systems, the concept of resource scheduling becomes very important and has been drawing more and more attention in recent years.As we know, low probability intercept (LPI) is one of the important features of modern radars. LPI optimization strategy is proposed in radar network architectures in [2, 3], where transmit power is minimized among netted phased array radars. The paper [4] studies the problem of scheduling the searching, verification, and tracking tasks of the military surveillance radar. The continuous double auction parameter selection algorithm is presented in [5] to solve the problem of allocating resource and selecting operational parameters for phased array radar system. Networked phased array radars that are connected by a communication channel are studied in paper [6], which proposes two types of distributed management techniques for the coordinated radar resource management. Based on sparsity-aware matrix decomposition, an improved method to select informative radars for target tracking in radar networks in proposed in [7], which shows the efficiency and improvement of the tracking performance. The paper [8] presents an optimal solution to the power allocation problem in distributed active multiple-radar systems subject to different power constraints, using a linear fusion rule and a simple objective function. In paper [9], a novel general criterion with the consideration of correlation in the measurement errors from different radars is proposed, which is applicable for a more general sensor network with an excellent tracking performance.Sensor data fusion techniques are widely used in target tracking of radar network, which can be loosely defined as how to best extract useful information from multiple radar observations. Information filtering, which is essentially a Kalman filter expressed in terms of the inverse of the covariance matrix, has been widely used in multiple sensor estimation [10]. A novel approach to track-to-track fusion in a high-level sensor data fusion architecture for automotive surround environment perception using information matrix fusion (IMF) is presented in [11]. The paper [12] utilizes the iterative joint-integrated probabilistic data association technique for multisensor distributed fusion systems. The paper [13] develops the square-root extensions of unscented information filter and central difference information filter, which have better numerical properties than the original versions. For nonlinear system, the paper [14] presents a new state estimation algorithm called square-root cubature information filter, which is further extended for use in multisensory state estimation. To achieve further improvements in tracking accuracy, a self-adaptive algorithm for maneuver strategy transition probability matrix is proposed in [15], by utilizing the compression ratio of the maneuver strategy error. Many different architectures are presented in [16] for the target tracking in 3D Cartesian coordinates, with the measurements of all the sensors in polar coordinates.However, those papers focus on the data selection after radar detection and do not consider the difference and accuracy of the sensors. Less working radars means much better LPI performance. In order to save the radar resource with excellent LPI performance, a radar selection algorithm is proposed based on an improved information filtering. The remainder of this paper is organized as follows. Section2 presents the improved information filtering method and radar selection method. Simulations of the proposed algorithms and comparison results with other methods are provided in Section 3. The conclusions are presented in Section 4.


## 2. Radar Selection Based on Improved Information Filter

Interacting multiple model (IMM) method is used for tracking maneuvering target. Information filter (IF), which is the dual Kalman filter, has attracted much attention for tracking fusion using multiple sensors [17]. But it cannot present excellent tracking performance for maneuvering target. In order to accomplish tracking fusion of maneuvering targets in radar network, a novel radar selection method based on interacting multiple model information filtering (IMMIF) is proposed in this section.


### 2.1. Tracking Models and Measurement Noise in Radar Network

All the dynamic models areM=m1,m2,…,mr, mki is the ith model used at time k, the switch probability from model mkj to model mk+1i is Pmk+1j∣mki=πij, mk+1j, mki∈M, ∑i=1rπij=1,i=1,2,…,r. μjk is the probability of model j, μjk=Pmk=j∣Zk. Let Xk and Zk represent the state vector and the observation vector, respectively; the state equation and transfer equation at time k are
(1)Xk+1=ϕjk+1Xk+wk,Zk=HjXk+vk,where wk and vk are stationary white noise processes with covariance matrices Qk and Wk. ϕj is the transition matrix and Hj is the observation matrix. Every recurrence of the IMM algorithm contains interacting of input, model’s filtering, update of model probability, and interacting of output.The covariance matrixWk of measurement noise is controlled by the emitted power. As we know, radar equation at time k is as follows:
(2)Rk4=tBkPavkGTGRλ2σk4π3KTRSNRkL,where tBk is the single dwelling time of the beam from the normal direction at time k, Pavk is the average radiated power, GR is the receiver gain, σk is the radar cross section (RCS) of the target, K is Boltzmann constant, TR and L are, respectively, effective noise temperature and radar system loss, Rk is the detection range, GT is the transmit gain, SNRk represents the signal to noise ratio of the system at time k.The single pulse signal is radiated by the radar, and the covariance of the measurement noise can be denoted as:(3)Wk=c2Tp8SNRk003c2wc2Tp2SNRk,where Tp is the pulse width, c is the wave velocity, and wc is the carrier frequency. We can see that different SNRk can lead to different W.


### 2.2. Information Filtering for Every Model

Utilizing all the states and model probabilities from last recurrence, the computation of input stateX^0jk−1 and covariance P0jk−1 of model j can be express as
(4)X^0jk−1=∑i=1rX^ik−1μijk−1,P0jk−1=∑i=1rPik−1+a.aT.μijk−1,a=X^ik−1−X^0jk−1,μijk−1=PMk−1=iMk=j,Zk−1=πijμik−1∑i=1rπijμik−1.Using information statey^0jk−1 and Fisher information Y0jk−1 replace state estimate X^0jk−1 and covariance P0jk−1, we can obtain information filtering result based on Kalman filter. The definition of information state y^0jk−1 and Fisher information Y0jk−1 is
(5)y^0jk−1=P0jk−1−1X^0jk−1,Y0jk−1=P0jk−1−1.Prediction and estimation ofy^ and Y and can be obtained by recursive iteration, combining with ik and Ik. The prediction of information state and Fisher information are given as
(6)y^jkk−1=Yjkk−1ϕjYjkk−1−1y^0jk−1,Yjkk−1=ϕjY0jk−1−1ϕjT+Qki−1.In the light of observation dataZmk from M different sensors in the sensor network, the estimation of information state and Fisher information for every model are as follows:
(7)y^jk=y^jkk−1+∑m=1Mijmk,Yjk=Yjkk−1+∑m=1MIjmk,where ijm and Ijm are the contributions that measurements Zk make to y^k and Y, respectively, which can be represented as follows.
(8)ijmk=HjmTWjmk‐1Zmk,(9)Ijmk=HjmTWjmk‐1Hjm.The model probabilityμjmk is recursively updated by the mth radar as
(10)μjmk=Λjk∑i=1rπijμik−1∑j=1r∑i=1rΛjkπijμik−1,where Λjk is the likelihood function of model j at time k.In the radar network, every radar will update a model probability, the final model probabilityμjk can be calculated as
(11)μjk=1M∑i=1Mμjmk.


### 2.3. Emitted Radar Selection Based on Information Computation

According to formula (2) and (3), the covariance Wkpre of the measurement noise can be predicted based on the prediction of target distance and target RCS. So Rk in (2) is replaced by Rkpre which is predicted by Rk−1 and vk−1. Rkpre is presented as
(12)Rkpre=Rk−1+vk−1T.Rk−1 and vk−1 are the target’s range and velocity which are estimated by the IMM tracking algorithm at time k−1 and T is the tracking interval. RCS σk is supposed to be the same as σk−1, which has been measured by the fusion center according to the echo.There areM radars in the network. When the observation noise is predicted by the mth radar, the information can be calculated using
(13)Ikm−pre=∑j=1rIjm−prekμjk,where Ijm−prek is the information gain which can be predicted by (9).An information vectorIk can be represented as
(14)Ikpre=Ik1−pre,Ik2−pre,…,IkM−pre.Ikmeanpreis the mean value of Ikm−pre, which can be represented as
(15)Ikmeanpre=1M∑m=1MIkm−pre.There areN radars whose information gain are larger than Ikmeanpre, which will be selected as the working radars at next time, N<M.


### 2.4. Tracking Fusion Based on the Prediction for Error Covariance Matrix

The selected radars are used for target tracking. Then, the observation vectorZk will be obtained. Using information filter for every model, the final estimation X^nk of the nth radar can be represented as
(16)X^nk=∑j=1rX^jkμjk,where X^jk=Yjky^jk.Different radars have different predicted covariance matrix. The predicted covariance matrix is given as(17)Pn−prek=∑j=1rμjkPjn−prek,where μjk is model probability at time k, the predicted covariance matrix of every model can be represented as Pjn−prek=Yjk∣k−1+∑m=1MIjn‐prek−1.Thenn estimations X^nk and predicted covariance matrices Pn−prek and its trace TRn will be obtained, TRn = trace Pn−prek. Then, the reciprocal vector TRN=1/TR1,1/TR2,…,1/TRn is computed. The weight wn of the radar tracking data can be formulated as
(18)wn=1/TRisumTRN.The final fusion result can be obtained at last:(19)X^k=∑j=1rX^nkwn.


## 2.1. Tracking Models and Measurement Noise in Radar Network

All the dynamic models areM=m1,m2,…,mr, mki is the ith model used at time k, the switch probability from model mkj to model mk+1i is Pmk+1j∣mki=πij, mk+1j, mki∈M, ∑i=1rπij=1,i=1,2,…,r. μjk is the probability of model j, μjk=Pmk=j∣Zk. Let Xk and Zk represent the state vector and the observation vector, respectively; the state equation and transfer equation at time k are
(1)Xk+1=ϕjk+1Xk+wk,Zk=HjXk+vk,where wk and vk are stationary white noise processes with covariance matrices Qk and Wk. ϕj is the transition matrix and Hj is the observation matrix. Every recurrence of the IMM algorithm contains interacting of input, model’s filtering, update of model probability, and interacting of output.The covariance matrixWk of measurement noise is controlled by the emitted power. As we know, radar equation at time k is as follows:
(2)Rk4=tBkPavkGTGRλ2σk4π3KTRSNRkL,where tBk is the single dwelling time of the beam from the normal direction at time k, Pavk is the average radiated power, GR is the receiver gain, σk is the radar cross section (RCS) of the target, K is Boltzmann constant, TR and L are, respectively, effective noise temperature and radar system loss, Rk is the detection range, GT is the transmit gain, SNRk represents the signal to noise ratio of the system at time k.The single pulse signal is radiated by the radar, and the covariance of the measurement noise can be denoted as:(3)Wk=c2Tp8SNRk003c2wc2Tp2SNRk,where Tp is the pulse width, c is the wave velocity, and wc is the carrier frequency. We can see that different SNRk can lead to different W.


## 2.2. Information Filtering for Every Model

Utilizing all the states and model probabilities from last recurrence, the computation of input stateX^0jk−1 and covariance P0jk−1 of model j can be express as
(4)X^0jk−1=∑i=1rX^ik−1μijk−1,P0jk−1=∑i=1rPik−1+a.aT.μijk−1,a=X^ik−1−X^0jk−1,μijk−1=PMk−1=iMk=j,Zk−1=πijμik−1∑i=1rπijμik−1.Using information statey^0jk−1 and Fisher information Y0jk−1 replace state estimate X^0jk−1 and covariance P0jk−1, we can obtain information filtering result based on Kalman filter. The definition of information state y^0jk−1 and Fisher information Y0jk−1 is
(5)y^0jk−1=P0jk−1−1X^0jk−1,Y0jk−1=P0jk−1−1.Prediction and estimation ofy^ and Y and can be obtained by recursive iteration, combining with ik and Ik. The prediction of information state and Fisher information are given as
(6)y^jkk−1=Yjkk−1ϕjYjkk−1−1y^0jk−1,Yjkk−1=ϕjY0jk−1−1ϕjT+Qki−1.In the light of observation dataZmk from M different sensors in the sensor network, the estimation of information state and Fisher information for every model are as follows:
(7)y^jk=y^jkk−1+∑m=1Mijmk,Yjk=Yjkk−1+∑m=1MIjmk,where ijm and Ijm are the contributions that measurements Zk make to y^k and Y, respectively, which can be represented as follows.
(8)ijmk=HjmTWjmk‐1Zmk,(9)Ijmk=HjmTWjmk‐1Hjm.The model probabilityμjmk is recursively updated by the mth radar as
(10)μjmk=Λjk∑i=1rπijμik−1∑j=1r∑i=1rΛjkπijμik−1,where Λjk is the likelihood function of model j at time k.In the radar network, every radar will update a model probability, the final model probabilityμjk can be calculated as
(11)μjk=1M∑i=1Mμjmk.


## 2.3. Emitted Radar Selection Based on Information Computation

According to formula (2) and (3), the covariance Wkpre of the measurement noise can be predicted based on the prediction of target distance and target RCS. So Rk in (2) is replaced by Rkpre which is predicted by Rk−1 and vk−1. Rkpre is presented as
(12)Rkpre=Rk−1+vk−1T.Rk−1 and vk−1 are the target’s range and velocity which are estimated by the IMM tracking algorithm at time k−1 and T is the tracking interval. RCS σk is supposed to be the same as σk−1, which has been measured by the fusion center according to the echo.There areM radars in the network. When the observation noise is predicted by the mth radar, the information can be calculated using
(13)Ikm−pre=∑j=1rIjm−prekμjk,where Ijm−prek is the information gain which can be predicted by (9).An information vectorIk can be represented as
(14)Ikpre=Ik1−pre,Ik2−pre,…,IkM−pre.Ikmeanpreis the mean value of Ikm−pre, which can be represented as
(15)Ikmeanpre=1M∑m=1MIkm−pre.There areN radars whose information gain are larger than Ikmeanpre, which will be selected as the working radars at next time, N<M.


## 2.4. Tracking Fusion Based on the Prediction for Error Covariance Matrix

The selected radars are used for target tracking. Then, the observation vectorZk will be obtained. Using information filter for every model, the final estimation X^nk of the nth radar can be represented as
(16)X^nk=∑j=1rX^jkμjk,where X^jk=Yjky^jk.Different radars have different predicted covariance matrix. The predicted covariance matrix is given as(17)Pn−prek=∑j=1rμjkPjn−prek,where μjk is model probability at time k, the predicted covariance matrix of every model can be represented as Pjn−prek=Yjk∣k−1+∑m=1MIjn‐prek−1.Thenn estimations X^nk and predicted covariance matrices Pn−prek and its trace TRn will be obtained, TRn = trace Pn−prek. Then, the reciprocal vector TRN=1/TR1,1/TR2,…,1/TRn is computed. The weight wn of the radar tracking data can be formulated as
(18)wn=1/TRisumTRN.The final fusion result can be obtained at last:(19)X^k=∑j=1rX^nkwn.


## 3. Simulation Results

In this section, Monte Carlo simulations are performed to analyze the performance of the proposed resource scheduling method.


### 3.1. Trajectory Design

Figure1 shows the target trajectory with its measurement results in 100 s. RCS of every radar is produced randomly during target tracking. All the radar positions are shown in Figure 1, which can be used to evaluate the LPI performance of the fusion methods.Figure 1
Target trajectory.


### 3.2. Radar Selection Results

To verify the effectiveness of the proposed approach based on the detailed mathematical expressions of Section2, computer simulations using MATLAB software are conducted for the comparisons with other methods [14–16]. Measurement data fusion [16] is used in all the methods in the simulations. The radiation label of the five radars is shown in Figure 2. We can see that the radars work in turn. The number of working numbers at every tracking time is illustrated in Figure 3; we can see that the number of working numbers is less than 5 at most times. Compared with the traditional fusion methods [14–16], which use all the sensors or radars for the fusion, we can see that the proposed method reduces much more working radars. As a result, the proposed method has better low probability intercept ability than others with less radiation time. In the proposed strategy, the radars can be selected to work according to targets’ positions and radar information gain in order to meet the requirement of desired tracking performance.Figure 2
Radiation label of the radars.Figure 3
Working radar number.


### 3.3. Comparison of Tracking Performance

The proposed adaptive radar selection method is labeled as “adaptive fusion.” Traditional fusion methods using all the radars and using selected radars based on Section2.3 are labeled as “all radar-based traditional fusion” and “selected radar-based traditional fusion”, respectively. The root-mean-square error (RMSE) of time k can be formulated as (20)
(20)RMSEk=1Mc∑m=1Mcxk−x^km2,where Mc is the number of the Monte Carlo simulation, xk is the true state of the system, and x^km is the estimated vector at the mth simulation, Mc=100.Figures4 and 5 show the range RMSE using the three methods. We can see that the proposed method “adaptive fusion” presents best tracking accuracy with other methods.Figure 4
Range RMSE of X direction.Figure 5
Range RMSE of Y direction.From the simulation results, we can see the proposed radar selection strategy is effective, which can achieve an optimal trade-off between the working radar numbers and tracking accuracy. The proposed algorithm selects the radars which can provide more accurate tracking data and then improves the information filtering by designing the weight parameters which can realize the tracking fusion perfectly. But “all radar-based traditional fusion” uses more radars for target tracking without tracking data selection, as some radars obtain the tracking data with much measurement noises or process noises, which will deteriorate tracking fusion performance. And “selected radar-based traditional fusion” does not consider the validity of the radar tracking data, and the fusion is realized using average weight of every radar. In addition, the common information filtering method does not consider the contribution difference of the radars in the network.


## 3.1. Trajectory Design

Figure1 shows the target trajectory with its measurement results in 100 s. RCS of every radar is produced randomly during target tracking. All the radar positions are shown in Figure 1, which can be used to evaluate the LPI performance of the fusion methods.Figure 1
Target trajectory.


## 3.2. Radar Selection Results

To verify the effectiveness of the proposed approach based on the detailed mathematical expressions of Section2, computer simulations using MATLAB software are conducted for the comparisons with other methods [14–16]. Measurement data fusion [16] is used in all the methods in the simulations. The radiation label of the five radars is shown in Figure 2. We can see that the radars work in turn. The number of working numbers at every tracking time is illustrated in Figure 3; we can see that the number of working numbers is less than 5 at most times. Compared with the traditional fusion methods [14–16], which use all the sensors or radars for the fusion, we can see that the proposed method reduces much more working radars. As a result, the proposed method has better low probability intercept ability than others with less radiation time. In the proposed strategy, the radars can be selected to work according to targets’ positions and radar information gain in order to meet the requirement of desired tracking performance.Figure 2
Radiation label of the radars.Figure 3
Working radar number.


## 3.3. Comparison of Tracking Performance

The proposed adaptive radar selection method is labeled as “adaptive fusion.” Traditional fusion methods using all the radars and using selected radars based on Section2.3 are labeled as “all radar-based traditional fusion” and “selected radar-based traditional fusion”, respectively. The root-mean-square error (RMSE) of time k can be formulated as (20)
(20)RMSEk=1Mc∑m=1Mcxk−x^km2,where Mc is the number of the Monte Carlo simulation, xk is the true state of the system, and x^km is the estimated vector at the mth simulation, Mc=100.Figures4 and 5 show the range RMSE using the three methods. We can see that the proposed method “adaptive fusion” presents best tracking accuracy with other methods.Figure 4
Range RMSE of X direction.Figure 5
Range RMSE of Y direction.From the simulation results, we can see the proposed radar selection strategy is effective, which can achieve an optimal trade-off between the working radar numbers and tracking accuracy. The proposed algorithm selects the radars which can provide more accurate tracking data and then improves the information filtering by designing the weight parameters which can realize the tracking fusion perfectly. But “all radar-based traditional fusion” uses more radars for target tracking without tracking data selection, as some radars obtain the tracking data with much measurement noises or process noises, which will deteriorate tracking fusion performance. And “selected radar-based traditional fusion” does not consider the validity of the radar tracking data, and the fusion is realized using average weight of every radar. In addition, the common information filtering method does not consider the contribution difference of the radars in the network.


## 4. Conclusions

In this paper, we have presented a new radar selection method for the radar network based on the improved IMM information filtering method. During the target tracking, the relation model is built between the radar resource and tracking performance. Then the radars are selected by the predicted information gain, and the weight parameters for fusion are designed in order to obtain the best tracking accuracy. The simulation results show that the proposed algorithm reduces much more radars with excellent tracking performance.



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*Source: 6104849-2018-12-17.xml*