# Netted Radar Tracking with Multiple Simultaneous Transmissions against Combined PDS Interception

Authors: Zijian Wu; Fei Wang; Jianjiang Zhou
Journal: Journal of Sensors
 (2020)

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## Abstract

One of the advantages of a netted airborne radar system (NARS) is escaping interception of the passive detection system (PDS) while tracking a target. A significant tactic to realize tracking without PDS interception is to study the low probability of interception (LPI) time of NARS. Firstly, this paper analyses the power, frequency, and platform interception probabilities of a combined PDS consisting of a radar-warning receiver (RWR) and an electronic support measurement (ESM). Secondly, this paper takes interactive multiple models (IMM) to describe the target tracking process and introduces a binary hypothesis test for chi square as well as noncentralized chi square distributions as a detection criterion of NARS during target tracking after the design of adaptive dwell time and the maximum illumination interval algorithm. Finally, based on experiential moving platform interception probabilities of a RWR and an ESM, a simplified math model is presented to estimate LPI time of NARS when the parameters are partially known. Simulations illustrate that the simultaneous management of radiation power and time is crucial for NARS against combined PDS interception.


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## Body

## 1. Introduction

Low probability of interception (LPI) technologies are necessary for an advanced aircraft to protect itself from a passive detection system (PDS) [1–3]. In order to improve the surviving ability of aircrafts, the airborne radar has to take advantage of LPI tactics to conceal its positions and action purposes.As for the radiation power control of a phased-only array radar, Zhang and Zhu proposed a multilevel power control strategy which showed better tracking performance and radiation power saving [4]. Kamble et al. analysed the relationship between emission power and target identification rate with a phase-coded radar waveform based on high resolution range profile (HRRP) [5]. Shi et al. designed a scheme to allocate power based on LPI radar in a complex electromagnetic background [6]. Wang et al. studied the joint allocation of beam pointing and dwell time of phased array radar in multitarget tracking [7]. Shi et al. proposed an optimized method of parameters including radar’s illumination interval, dwell time, and radiated power in target tracking, which showed the achievement of minimum interception probability [8]. And the illumination interval represents the time gap between two consecutive illuminations of the radar system transmitters.The ESM with a frequency search mode has a relatively longer intercept range than the detection range of radar so that the probability of interception in the power domain can be considered independent of range. We then assume that the ESM’s power interception probability equals to one when the radar is transmitting signals. However, the bandwidth of the ESM is narrow compared with the search range; thus, the frequency interception probability is very low. On the other hand, the RWR has a wider bandwidth compared with the ESM, and the frequency interception probability can be considered 100% when the radar is transmitting signals. However, its power intercept ability grew in inverse proportion to the distance between radar and target. Then, the netted airborne radar system (NARS) is attracting more and more researchers’ attention; each radar NARS can not only detect targets independently but also be managed by some fusion rules cooperatively. NARS always include different types of radar nodes, which can work under different mechanisms, frequency bands, working modes, and polarization modes. After fusing data from all radar nodes, NARS can improve the detection and LPI performances at the same time. The general rules to improve LPI performance of NARS are to allocate its power, time, and antenna resources reasonably. Godrich et al. proposed a power resource allocation scheme to meet the localization accuracy by minimizing the total transmitted energy of NARS [9]. Chavali and Nehorai studied a competitive resource allocation tactic of NARS using game theory [10]. Gao et al. proposed three allocation schemes about resources of NARS to improve its target detection performance by combining antenna and dwell time resources [11]. Barbary and Zong deduced a maximum nonparametric detection performance of NARS, which can be used to control radiation power when targets’ distances are known [12]. Shi et al. and Liu et al. proposed an effective way to minimize the intercept factor while maintaining tracking performance, and results proved it worked well [13, 14]. And the intercept factor can evaluate the LPI performance of the radar network system. Wang et al. proposed a track mission shifting tactic of a distributed airborne radar system or distributed airborne formation radar system based on LPI time [15]. And the LPI time refers to the time during which a certain airborne radar system is tracking the target without being intercepted by a passive detection system.The resource management models and algorithms of NARS are similar to the MIMO radar system. As for the MIMO radar system, Song et al. integrated the propagation losses into the MIMO radar signal model and investigate the power allocation problems [16]. Liao et al. proposed an interception probability model of the MIMO radar system by analysing radar signals in time, frequency, space, and power domains [17]. Yang et al. optimized LPI performance of the MIMO radar system by allocating its antenna, power, and dwell time resources in models of searching and detecting MIMO radar system [18]. Yu et al. improved the performance gain of CS-MIMO radars by allocating optimal power among the transmit antennas [19]. Garcia et al. proposed an algorithm to allocate power and bandwidth of the MIMO radar system reasonable with CRLB, which can get the best localization performance of multiple targets [20]. Ma et al. considered a joint scheme of antenna subset selection and optimal power allocation, and results showed better localization performance compared with other strategies and were close to the optimal solution [21]. Considering prior knowledge, Yan et al. studied the allocation of radar resources for different purposes when the MIMO radar system tracks multiple targets [22, 23]. A maximum signal-to-interference-plus-noise ratio (SINR) based on frequency diverse property of the MIMO radar system was deduced by Xiang and Chen, which was designed to control the power of the MIMO radar system under a determined SINR [24]. Tang et al. designed a new MIMO radar waveform which was used to improve the Rician targets’ detection performance under a limited power [25].However, most papers are in lack of radiation time analysis of NARS because of the common assumption that the passive detection system is unknown, which leads to the absence of LPI time analysis. Since modern intelligence system and early warning system have made it possible for NARS to obtain some properties of a PDS, this paper attempts to study LPI time of NARS when those properties are partially known. Some work has been done in [15], which only took the ESM into account. For the reason that modern aircraft are always equipped with both a RWR system and an ESM system, this paper takes a combined PDS made of the two systems into account. Since the probability of intercept of the ESM is independent of range and the RWR has an intercept ability that grew in inverse proportion to the distance between radar and target, the paper took different characteristics of the RWR and the ESM into account and changed the optimum LPI detection strategy of the network against interception. The simulations assume that NARS composed of four radars formed a square formation moving towards the target. Firstly, through deducing the short-time interception probability of the ESM based on the adaptive time resource constraint by NARS, this paper defines a platform interception probability of a combined PDS as the function of the radiation time by NARS, the power interception probability of a RWR, and the frequency interception probability of an ESM. Secondly, considering the definition of platform interception probability, this paper establishes a simplified math model to study the LPI time of NARS under some assumptions of experiential moving platform interception probabilities of a combined PDS.The main contributions of this paper are as follows: (1) This paper establishes a math model to analyse the LPI time of NARS confronting a combined PDS made of a RWR and an ESM. (2) Corresponding to the maximum illumination interval tactic adopted by NARS, this paper proposes a short-time average interception probability to replace the common interception probability of an ESM.


## 2. Platform Interception Probability Model of a Combined PDS

Based on previous work of other researchers, this paper assumed that the tracking ability of the network can well meet the task requirements. Relating works [26, 27] are as follows: Goodman used the likelihood ratio test (LRT) for multistatic airborne radar detection; the simulation results showed that diversity and geometry gain could be achieved through multistatic space-time adaptive processing (STAP). Rong et al. proposed a multiple access protocol for intraflight data link which meets the high-speed, secure, and reliable information sharing between the planes. This paper assumes that NARS is composed of Nt transmitters and Nr receivers, and each of which can independently transmit and process orthogonal waveforms at the same time. The main lobe of each radar antenna of NARS is tracking the target equipped with a combined PDS. Only a radiation signal of NARS satisfies the detected conditions of power, spatial, frequency, and time domains. The general detection probability Pi of a signal emitted by NARS could be expressed as
(1)Pi=Ps·Pf·Pd·Pt,where Ps, Pf, Pd, and Pt denote detection probabilities of spatial, frequency, power, and time domains.As for the RWR designed to detect radar signals instantly as soon as possible in a confrontation scene, bothPs and Pt in equation (1) should approximately equal to 1 because the platform of radar would not move quickly in the space domain and the radar signal would hold on for a while in the time domain. However, there have to be a trade-off between Pf and Pd because of the physical contradictory relationship between the bandwidth and sensitivity of a RWR or an ESM. To reduce that contradiction and improve the performance of the passive detection system, an effective method is to combine the high-sensitivity narrow-band ESM and the low-sensitivity wide-band RWR.


### 2.1. Power Interception Probability of a RWR

According to equation (1), both Pf and Pd are significant for the combined PDS; a low-sensitivity wide-band RWR is suitable to model the property of Pd of the combined PDS. Considering a classical detection equation of a linear detector, the interception probability of the RWR could be expressed as [28]
(2)Pd=2∫−lnPfa∞y·exp−y2+SNRiI02ySNRidy,where Pfa is the false alarm possibility, I0 denotes the modified first-order Bessel function, and SNRi represents the signal to noise (SNR) ratio. When SNRi is larger than 6 dB, a classical approximate expression [29] of equation (2), that is,
(3)Pd=0.5·erfc−lnpfa−SNRi+0.5,erfcz=1−2π∫0ze−v2dv.This paper proposes to usePd, Pfa, and SNRi of equation (3) to model the interception performance in the power domain of the combined PDS.


### 2.2. Frequency Interception Probability of an ESM

Equation (3) shows interception performance in the power domain by a RWR of the combined PDS. Then, this paper proposes to describe the frequency interception property Pf by an ESM. Frequency search is the most popular search mode of the ESM, and the average interception time on the frequency domain is widely used to describe an ESM performance. The relationship between it and interception is that the shorter the average interception time is, the higher the interception probability of an ESM will be.Due to the high sensitivity of ESM in the frequency search mode, this paper suggests thatPs=1, Pd=1, and Pt=1. Therefore, the interception probability of a high-sensitivity ESM is defined as
(4)pfi=ΔfsBs,where Δfs is ESM’s stepping frequency and Bs is an ESM’s frequency search range and is greater than the radar’s frequency hopping range. Supposing that there are Nt radars in NARS transmitting orthogonal waveforms with Nt′ (Nt′≤Nt) carrier frequencies at the same time, then the intercepted probability of NARS by an ESM could be
(5)pf=1−1−pfiNt′.IfNt′pfi<0.3, equation (5) could be approximated as
(6)pf≈Nt′pfi.Then, the ESM search period is(7)Tf=τrempf=τremNt′pfi,where Tf is the frequency search period and τrem is dwell time of an ESM. When the illumination period is a constant, the interception definition of the ESM can be given by De Martino [30]. However, NARS often need to track the target with an adaptive illumination interval; thus, the short-time average interception time of the ESM is
(8)T¯c=TfT¯rsτe′−τrem,where T¯rs and τe′=τe/Ntη are the short-time average illumination interval and dwell time of NRAS on the target, respectively. Let 0.5≤η<1 represent noncoherent integration efficiency, τe denotes the sum of dwell time of NARS; thus, the NARS’s short-time average intercepted probability by an ESM is
(9)Pfx=1−exp−xT¯c,x>0,0,x<0,where x denotes the total scan time of ESM.IfT¯rs is a constant, equation (8) can be rewritten as
(10)T¯c∝τremNt′pfiτeNtη−τrem.IfNt=Nt′=1, then
(11)T¯c∝τrempfiτe−τrem.Sinceτrem are smaller than τe and τe/Ntη, if Ntη≥Nt′, then
(12)Nt′pfiτeNtη−τrem≤pfiτe−τrem,which means that the LPI performance of NARS is proportional to the number of aircrafts in NARS. Equations (10) and (11) show that minimizing the number of carrier frequencies and minimizing radiation time of NARS are good ways to escape from a high-sensitivity ESM with a frequency search mode.


### 2.3. Platform Interception Probability of a Combined PDS

Although LPI performance is important to protect NARS from combined PDS, NARS must use a reasonable SNR to detect targets. Thus,SNRnet of NARS is defined as the ideal coherent integration for all radar nodes, that is,
(13)SNRnet=∑i=1Nt∑j=1NrSNRij,where SNRij=PtiGtiGrjGrpjσijλi2τei/4π3kToBriFriRti2Rrj2LijTri, Pti, Gti, τei, Tri, λi, Bri, and Fri are emission power, antenna gain, dwell time, pulse repetition time, wavelength, bandwidth of match filter, and noise coefficient of the ith radar; Grj and Grpj represent antenna gain and processor gain of the jth radar receiver; σij is radar cross-section (RCS) of a target between the ith radar and the jth radar receiver; k is Boltzmann constant; To is system noise temperature; Lij is system loss between the ith radar and the jth radar; Rti is distance from the ith radar to the target; and Rrj is distance between the target and the jth radar. If the coherent integration process of NARS is not perfect, that loss could be reflected in Lij.


## 2.1. Power Interception Probability of a RWR

According to equation (1), both Pf and Pd are significant for the combined PDS; a low-sensitivity wide-band RWR is suitable to model the property of Pd of the combined PDS. Considering a classical detection equation of a linear detector, the interception probability of the RWR could be expressed as [28]
(2)Pd=2∫−lnPfa∞y·exp−y2+SNRiI02ySNRidy,where Pfa is the false alarm possibility, I0 denotes the modified first-order Bessel function, and SNRi represents the signal to noise (SNR) ratio. When SNRi is larger than 6 dB, a classical approximate expression [29] of equation (2), that is,
(3)Pd=0.5·erfc−lnpfa−SNRi+0.5,erfcz=1−2π∫0ze−v2dv.This paper proposes to usePd, Pfa, and SNRi of equation (3) to model the interception performance in the power domain of the combined PDS.


## 2.2. Frequency Interception Probability of an ESM

Equation (3) shows interception performance in the power domain by a RWR of the combined PDS. Then, this paper proposes to describe the frequency interception property Pf by an ESM. Frequency search is the most popular search mode of the ESM, and the average interception time on the frequency domain is widely used to describe an ESM performance. The relationship between it and interception is that the shorter the average interception time is, the higher the interception probability of an ESM will be.Due to the high sensitivity of ESM in the frequency search mode, this paper suggests thatPs=1, Pd=1, and Pt=1. Therefore, the interception probability of a high-sensitivity ESM is defined as
(4)pfi=ΔfsBs,where Δfs is ESM’s stepping frequency and Bs is an ESM’s frequency search range and is greater than the radar’s frequency hopping range. Supposing that there are Nt radars in NARS transmitting orthogonal waveforms with Nt′ (Nt′≤Nt) carrier frequencies at the same time, then the intercepted probability of NARS by an ESM could be
(5)pf=1−1−pfiNt′.IfNt′pfi<0.3, equation (5) could be approximated as
(6)pf≈Nt′pfi.Then, the ESM search period is(7)Tf=τrempf=τremNt′pfi,where Tf is the frequency search period and τrem is dwell time of an ESM. When the illumination period is a constant, the interception definition of the ESM can be given by De Martino [30]. However, NARS often need to track the target with an adaptive illumination interval; thus, the short-time average interception time of the ESM is
(8)T¯c=TfT¯rsτe′−τrem,where T¯rs and τe′=τe/Ntη are the short-time average illumination interval and dwell time of NRAS on the target, respectively. Let 0.5≤η<1 represent noncoherent integration efficiency, τe denotes the sum of dwell time of NARS; thus, the NARS’s short-time average intercepted probability by an ESM is
(9)Pfx=1−exp−xT¯c,x>0,0,x<0,where x denotes the total scan time of ESM.IfT¯rs is a constant, equation (8) can be rewritten as
(10)T¯c∝τremNt′pfiτeNtη−τrem.IfNt=Nt′=1, then
(11)T¯c∝τrempfiτe−τrem.Sinceτrem are smaller than τe and τe/Ntη, if Ntη≥Nt′, then
(12)Nt′pfiτeNtη−τrem≤pfiτe−τrem,which means that the LPI performance of NARS is proportional to the number of aircrafts in NARS. Equations (10) and (11) show that minimizing the number of carrier frequencies and minimizing radiation time of NARS are good ways to escape from a high-sensitivity ESM with a frequency search mode.


## 2.3. Platform Interception Probability of a Combined PDS

Although LPI performance is important to protect NARS from combined PDS, NARS must use a reasonable SNR to detect targets. Thus,SNRnet of NARS is defined as the ideal coherent integration for all radar nodes, that is,
(13)SNRnet=∑i=1Nt∑j=1NrSNRij,where SNRij=PtiGtiGrjGrpjσijλi2τei/4π3kToBriFriRti2Rrj2LijTri, Pti, Gti, τei, Tri, λi, Bri, and Fri are emission power, antenna gain, dwell time, pulse repetition time, wavelength, bandwidth of match filter, and noise coefficient of the ith radar; Grj and Grpj represent antenna gain and processor gain of the jth radar receiver; σij is radar cross-section (RCS) of a target between the ith radar and the jth radar receiver; k is Boltzmann constant; To is system noise temperature; Lij is system loss between the ith radar and the jth radar; Rti is distance from the ith radar to the target; and Rrj is distance between the target and the jth radar. If the coherent integration process of NARS is not perfect, that loss could be reflected in Lij.


## 3. The Analysis of LPI Time for NARS

### 3.1. The Design of Adaptive Dwell Time and Illumination Interval

In NARS, we use the binary hypothesis test to analyseH0 and H1, which are centralized chi square distribution and noncentralized chi square distribution, respectively,
(14)H0:r∼χ2Nt2,H1:r∼χ′2Nt2.After match filtering, the time whenjth radar gets the max output is t0. In equation (14), for H0, Er=Nt2 and varr=2Nt2 are the mean and variance of r. For H1, the mean and variance are Er=Nt2+λ and varr=2Nt2+4λ, λ=∑j=1Nt∑k=1Ntμjk2t0. When t=t0, the mean is μjkt0. For H0, μjkt0=0. And λ is determined by the minimum acceptable SNRnet.In this paper, adaptive dwell time and illumination interval design are mutually independent, so the NARS system can control the dwell time to meet the requirement of the tracking task. After estimating the information of RCS and the target distance, NARS first adjust the dwell time to satisfy requiredSNRnet for H1 in equation (14). Since the estimation values are not exactly accurate, the NARS will continue to illuminate the target for Δτe until the hypothesis test H1 is available if the current dwell time cannot meet the requirement.The following is a classic adaptive illumination interval design process [31]. The state and measure equations of the jth move model of the target are defined as
(15)X^jt=FjΔTXjt−ΔT+VjΔTt,(16)Zt=HjΔTXjt+WjΔTt,where X^jt and Xjt are predicted state vector and state vector of the jth model; Zt is measure vector at time t; and FjΔT, HjΔT, VjΔTt, and WjΔTt are the transfer matrix, measure matrix, system noise, and measure noise of the jth model, respectively, when the illumination interval of the radar transmitter is ΔT, which is the time gap between two consecutive illuminations of the NARS transmitters. ΔT=mΔTmin, m∈1,2,⋯,M, where ΔTmin and MΔTmin are the minimum and the maximum intervals available. Covariance matrices of VjΔTt and WjΔTt are QjΔTt and RjΔTt, respectively.Adaptive illumination interval design based on the interactive multimodel Kalman filter (IMMKF) process is as follows after combining with equations (15) and (16).Step 1.
SupposingΔT is the illumination interval from the previous time to current time and the posterior state vector X^jt−ΔT of the jth model of last time is
(17)X^jt−ΔT=∑i=1rX^it−ΔT·μi∣jt−ΔT,where μi∣jt is the model converting probability of the jth model transferred from the ith model.Step 2.
According to the following formula to compute predicted covariance error matrixPjΔTt−, residual matrix SjΔTt, and Kalman gain KjΔTt of the jth model,
(18)PjΔTt−=FjΔTPjt−ΔTFjΔTT+GjΔTQjΔTtGjΔTT,SjΔTt=HjΔTPjΔTtHjΔTT+RjΔTt,KjΔTt=PjΔTt−HjΔTTSjΔTt−1,where Pjt−ΔT denotes the covariance error matrix of last time and GjΔT denotes the weighing matrix of system noise of the jth model.Step 3.
Estimating state vectorX^jt and covariance error matrix Pjt of the jth model by using Step 2’s result:
(19)X^jt=FjΔTX^jt−ΔT+KjΔTtZt−HjΔTFjΔTX^jt−ΔT,Pjt=I−KjΔTtHjΔTPjΔTt−,where I denotes the unit matrix.Step 4.
Updating each moving model’s probability:(20)μjt=Λjt∑i=1rπijμit−1∑j=1r∑i=1rΛjtπijμit−1,where Λjt is likelihood function of the jth model at time t; πij is transform probability from the ith model to the jth model, ∑i=1Iπij=1,i=1,2,⋯,I; and μit is probability of the jth model at time t.Step 5.
Combining state vectors and covariance error matrices of all models toX^t and Pt:
(21)X^t=∑j=1rX^jtμjt,Pt=∑j=1rμjtPjt+bjbjT,where bj=X^jt−X^t.In Steps1–5, a process cycle of IMMKF is finished, and predicted upper limitation Ppredt determines illumination interval ΔT. Then, repeat Steps 1–5 to continue the next adaptive illumination interval.


### 3.2. LPI Time for NARS

According to the property of a RWR, the location error of the signal source is relatively large using the amplitude location algorithm so that a relatively long time about 20 to 100 seconds is needed to initialize tracking of the platform of the signal source. On the contrary, with the phase location algorithm, location error of the signal source by an ESM is relatively small so that it might be able to track the platform of the signal source after detecting the signal source for three times.Once the platform is tracked by a RWR or by an ESM, the combined PDS is able to identify the threat level of the signal source and then take actions. Since an ESM is sensitive to LPI time, the math model for LPI time estimation from equation (9) can be defined as
(22)maxSNRi,xT¯cns.t.0<Pf≤Pfth0<Pd≤Pdth0<Pti≤PtmaxTri≤τei≤cτ02vSNRnetth≤SNRnet,where SNRi and x refer to equations (3) and (9); T¯cn denotes the average intercepted time for n times; Pdth and Pfth represent the predicted acceptable interception probability thresholds of a RWR and an ESM by NARS; Pti, Tri, and τei are emission power, the minimum dwell time, and dwell time of the ith radar node; Ptmax is the maximum emission power of each radar node; c is light speed; τ0 is pulse width and τ0≤Δτe; and SNRnetth is predicted the minimum detection SNR by NARS:
(23)Pd=maxPdi,Pf=maxPfi,fori=1,2,⋯,Nt,where Pdi and Pfi denote predicted interception probabilities of a RWR and an ESM of the ith radar node.SNRi of a RWR corresponding to Pdi can be defined as
(24)SNRi=PtiGtiGrλi2GIP4π2kToFIBILIiRi2,where Gti, LIi, and Ri are main beam gain, propagation loss, and distance from the ith radar node to the RWR and Gr, BI, FI, and GIP are antenna gain, bandwidth, noise coefficient, and processor gain of the RWR.Assume that the distance between any two radar nodes of NARS is far less than that from NARS to the target. The Schleher intercept factor of NARSαnet [32] is used to evaluate the LPI state of NARS, that is,
(25)αnet=RimaxRnetmax.The NARS will be in LPI state confronting a RWR in the power domain whenαnet is far less than 1:
(26)Rimax=maxPtiGtiλi2/LIiGrGIP4π2PI1/2,Rnetmax≜σnetGnet2Grpnet4π3LnetPRnet∑i=1NtPtiτeiTriλi21/4,where Rimax is the maximum intercepted distance of NARS by a RWR; Rnetmax is an approximate maximum detection distance by NARS; PI is the RWR’s sensitivity; σnet, Gnet, Grpnet, Lnet, and PRnet are approximate equivalent RCS, antenna gain, processor gain, system loss, and sensitivity of NARS, respectively.


## 3.1. The Design of Adaptive Dwell Time and Illumination Interval

In NARS, we use the binary hypothesis test to analyseH0 and H1, which are centralized chi square distribution and noncentralized chi square distribution, respectively,
(14)H0:r∼χ2Nt2,H1:r∼χ′2Nt2.After match filtering, the time whenjth radar gets the max output is t0. In equation (14), for H0, Er=Nt2 and varr=2Nt2 are the mean and variance of r. For H1, the mean and variance are Er=Nt2+λ and varr=2Nt2+4λ, λ=∑j=1Nt∑k=1Ntμjk2t0. When t=t0, the mean is μjkt0. For H0, μjkt0=0. And λ is determined by the minimum acceptable SNRnet.In this paper, adaptive dwell time and illumination interval design are mutually independent, so the NARS system can control the dwell time to meet the requirement of the tracking task. After estimating the information of RCS and the target distance, NARS first adjust the dwell time to satisfy requiredSNRnet for H1 in equation (14). Since the estimation values are not exactly accurate, the NARS will continue to illuminate the target for Δτe until the hypothesis test H1 is available if the current dwell time cannot meet the requirement.The following is a classic adaptive illumination interval design process [31]. The state and measure equations of the jth move model of the target are defined as
(15)X^jt=FjΔTXjt−ΔT+VjΔTt,(16)Zt=HjΔTXjt+WjΔTt,where X^jt and Xjt are predicted state vector and state vector of the jth model; Zt is measure vector at time t; and FjΔT, HjΔT, VjΔTt, and WjΔTt are the transfer matrix, measure matrix, system noise, and measure noise of the jth model, respectively, when the illumination interval of the radar transmitter is ΔT, which is the time gap between two consecutive illuminations of the NARS transmitters. ΔT=mΔTmin, m∈1,2,⋯,M, where ΔTmin and MΔTmin are the minimum and the maximum intervals available. Covariance matrices of VjΔTt and WjΔTt are QjΔTt and RjΔTt, respectively.Adaptive illumination interval design based on the interactive multimodel Kalman filter (IMMKF) process is as follows after combining with equations (15) and (16).Step 1.
SupposingΔT is the illumination interval from the previous time to current time and the posterior state vector X^jt−ΔT of the jth model of last time is
(17)X^jt−ΔT=∑i=1rX^it−ΔT·μi∣jt−ΔT,where μi∣jt is the model converting probability of the jth model transferred from the ith model.Step 2.
According to the following formula to compute predicted covariance error matrixPjΔTt−, residual matrix SjΔTt, and Kalman gain KjΔTt of the jth model,
(18)PjΔTt−=FjΔTPjt−ΔTFjΔTT+GjΔTQjΔTtGjΔTT,SjΔTt=HjΔTPjΔTtHjΔTT+RjΔTt,KjΔTt=PjΔTt−HjΔTTSjΔTt−1,where Pjt−ΔT denotes the covariance error matrix of last time and GjΔT denotes the weighing matrix of system noise of the jth model.Step 3.
Estimating state vectorX^jt and covariance error matrix Pjt of the jth model by using Step 2’s result:
(19)X^jt=FjΔTX^jt−ΔT+KjΔTtZt−HjΔTFjΔTX^jt−ΔT,Pjt=I−KjΔTtHjΔTPjΔTt−,where I denotes the unit matrix.Step 4.
Updating each moving model’s probability:(20)μjt=Λjt∑i=1rπijμit−1∑j=1r∑i=1rΛjtπijμit−1,where Λjt is likelihood function of the jth model at time t; πij is transform probability from the ith model to the jth model, ∑i=1Iπij=1,i=1,2,⋯,I; and μit is probability of the jth model at time t.Step 5.
Combining state vectors and covariance error matrices of all models toX^t and Pt:
(21)X^t=∑j=1rX^jtμjt,Pt=∑j=1rμjtPjt+bjbjT,where bj=X^jt−X^t.In Steps1–5, a process cycle of IMMKF is finished, and predicted upper limitation Ppredt determines illumination interval ΔT. Then, repeat Steps 1–5 to continue the next adaptive illumination interval.


## 3.2. LPI Time for NARS

According to the property of a RWR, the location error of the signal source is relatively large using the amplitude location algorithm so that a relatively long time about 20 to 100 seconds is needed to initialize tracking of the platform of the signal source. On the contrary, with the phase location algorithm, location error of the signal source by an ESM is relatively small so that it might be able to track the platform of the signal source after detecting the signal source for three times.Once the platform is tracked by a RWR or by an ESM, the combined PDS is able to identify the threat level of the signal source and then take actions. Since an ESM is sensitive to LPI time, the math model for LPI time estimation from equation (9) can be defined as
(22)maxSNRi,xT¯cns.t.0<Pf≤Pfth0<Pd≤Pdth0<Pti≤PtmaxTri≤τei≤cτ02vSNRnetth≤SNRnet,where SNRi and x refer to equations (3) and (9); T¯cn denotes the average intercepted time for n times; Pdth and Pfth represent the predicted acceptable interception probability thresholds of a RWR and an ESM by NARS; Pti, Tri, and τei are emission power, the minimum dwell time, and dwell time of the ith radar node; Ptmax is the maximum emission power of each radar node; c is light speed; τ0 is pulse width and τ0≤Δτe; and SNRnetth is predicted the minimum detection SNR by NARS:
(23)Pd=maxPdi,Pf=maxPfi,fori=1,2,⋯,Nt,where Pdi and Pfi denote predicted interception probabilities of a RWR and an ESM of the ith radar node.SNRi of a RWR corresponding to Pdi can be defined as
(24)SNRi=PtiGtiGrλi2GIP4π2kToFIBILIiRi2,where Gti, LIi, and Ri are main beam gain, propagation loss, and distance from the ith radar node to the RWR and Gr, BI, FI, and GIP are antenna gain, bandwidth, noise coefficient, and processor gain of the RWR.Assume that the distance between any two radar nodes of NARS is far less than that from NARS to the target. The Schleher intercept factor of NARSαnet [32] is used to evaluate the LPI state of NARS, that is,
(25)αnet=RimaxRnetmax.The NARS will be in LPI state confronting a RWR in the power domain whenαnet is far less than 1:
(26)Rimax=maxPtiGtiλi2/LIiGrGIP4π2PI1/2,Rnetmax≜σnetGnet2Grpnet4π3LnetPRnet∑i=1NtPtiτeiTriλi21/4,where Rimax is the maximum intercepted distance of NARS by a RWR; Rnetmax is an approximate maximum detection distance by NARS; PI is the RWR’s sensitivity; σnet, Gnet, Grpnet, Lnet, and PRnet are approximate equivalent RCS, antenna gain, processor gain, system loss, and sensitivity of NARS, respectively.


## 4. Simulations

Assuming that NARS is tracking a target equipped with combined PDS whose properties is partially known from electronic intelligence and early warning system. LetSNRnetth=3dB, Pd=0.9, Pfa=1×10−6, and SNRi=15dB. Main beam width and propagation loss of each radar node are 1° and 0.0126dB/km. Other simulation parameters are listed in Tables 1 and 2.Table 1
Parameters of an airborne radar.
Radar parameters
Parameter values
Radar parameters
Parameter values
Gti
34 dB
Grj
34 dB
GRPj
0 dB
λ
0.03 m
Rg
0.01
BR
0.5 MHz
τ0
2μs
Ptmax
20 kW
Tri
1 ms
FR
2 dBTable 2
Parameters of a RWR and an ESM.
RWR parameters
Parameter values
ESM parameters
Parameter values
Gr
0 dB
T¯f
200 ms
GIP
0 dB
τsm
1 ms
BI
1 GHz
τrem
1 ms
FI
5 dB
Bs
10 GHz
PI
-60 dBmw
Δfs
50 MHzThe scene of target tracking by NARS is illustrated in Figure1, where target moves from (120 km, 120 km) to (90 km, 120 km); initial positions of four radars in NARS are (0 km, 0 km), (40 km, 0 km), (0 km, 30 km), and (40 km, 30 km), respectively. Four radars formed a square formation at the speed of 250 m/s moving towards the target along a line.Figure 1
Trajectories of NARS and target on a planar scene.From Table3, there are three models and ΔTmin=1s, ΔTmax=5s. Γ1=sin0.05T/0.05, Γ2=cos0.05T−1/0.05, Γ3=cos0.05T, and Γ4=sin0.05T. The covariance matrix of measure noise and system noise is RjTt=400,0;0,400 and QjTt=10,0;0,10, respectively. Measure matrix HjT=1,0,0,0;0,0,1,0. Weighing matrix of system noise GjT=T2/2,0;T,0;0,T2/2;0,T. The initial error covariance matrix is Pj0=diag460,100,460,100; the predefined upper limit of error covariance matrix is trPpredt=650. The initial possibility matrix of each move model is μj0=0.3,0.3,0.4. The model transform matrix is πij=0.9,0.05,0.05;0.1,0.8,0.1;0.05,0.15,0.8.Table 3
Maneuvering process of target.
Time
State equations
0~5
F1T=1,T,0,0;0,1,0,0;0,0,1,T;0,0,0,1
6~20
F2T=1,Γ1,0,Γ2;0,Γ3,0,Γ4;0,−Γ2,1,Γ1;0,−Γ1,0,Γ3
21~35
F1T=1,T,0,0;0,1,0,0;0,0,1,T;0,0,0,1
36~54
F3T=1,−Γ1,0,Γ2;0,Γ3,0,−Γ4;0,−Γ2,1,−Γ1;0,Γ1,0,Γ3
55~100
F1T=1,T,0,0;0,1,0,0;0,0,1,T;0,0,0,1
101~120
F2T=1,Γ1,0,Γ2;0,Γ3,0,Γ4;0,−Γ2,1,Γ1;0,−Γ1,0,Γ3To improve LPI time of NARS, we take adaptively the sample interval and interactive multimodel Kalman filter to track the target. Figure2 shows that the nearest distance between NARS and the target is 79 km. Figure 3 shows relationships between distance and sample intervals of four radars. Figure 3 shows that the maximum tracking error is less than 60 m; Figure 3(b) shows that the maximum sample interval is 5ΔTmin. Figures 4 and 5 show emission power and dwell time of each radar changed by target distance and target tracking time, respectively, during the target tracking process.Figure 2
Time-based target tracking process by NARS.Figure 3
Time-based target tracking error and illumination interval by NARS.
(a)
(b)Figure 4
Distance-based emission power and dwell time of NARS.
(a)
(b)Figure 5
Time-based emission power and dwell time of NARS.
(a)
(b)From Tables1–3, with Tri=1ms, τrem=1ms, Bs=10GHz, Δfs=50MHz, and empirical values of possible short time τ¯ei=150ms from Figures 4(b) and 5(b), and possible short time T¯rs=4.5+5+4/3=4.5s from Figure 3(b). The preestimated Pf is about Δfs/Bs=0.005, and the mean of intercepted time of NARS by an ESM is about T¯c=200/150·4.5=6s from equation (8).Figure6 shows the interception factor of NARS, while the receiver sensitivity of the airborne radar and the RWR is -140 dBmw and -60 dBmw, respectively, and SNRi is 15 dB. From Figure 6, the LPI performance of NARS confronting a RWR is still smaller although the value of the interception factor is greater than that generated from Liao et al. [17]. The reason is that this paper has taken an ESM into account in Figure 7, which shows that the LPI time of NARS is much better than that of Liao et al. [17].Figure 6
Time-based interception factor comparison confronting a RWR.Figure 7
The mean of intercepted time of NARS by an ESM.After 500 Monte Carlo experiments, Figures7 and 8 shows cumulative intercepted time of all four radars in NARS by an ESM. Figure 7 shows that the average interception time for three times is 17 s; compared with the single radar, the interception time has increased by 6 times. Figure 8 is standard deviation which shows that the standard deviation of interception time for three times is 11 s. Because the three interception is independent of each other, the short average interception time for three times in equation (22) is about T¯cn=3·T¯c=18s and the standard deviation is about 3T¯c=10.4s, which are close to 17 s and 11 s, respectively.Figure 8
The standard deviation of intercepted time of NARS by an ESM.With the change of the detection SNR threshold of the RWR. Table4 shows the changed interception factor from equation (25) and changed average time of being intercepted by three times. According to the data in Table 4, the ratio of the average time of being intercepted by three times to the interception factor is approximately linear to the detection SNR threshold of a RWR. However, if NARS could be located by a RWR once SNR is over a key threshold, the linear relationship in Table 4 would be not valid any more.Table 4
LPI time by a combined PDS with differentSNRi.
SNRi threshold
Interception factor
Three-time interception
10 dB
0.1234
3.6676
12 dB
0.1553
7.4291
14 dB
0.1955
11.719
16 dB
0.2462
22.24
18 dB
0.3099
34.1784
20 dB
0.3902
50.9894
22 dB
0.4912
72.7639
24 dB
0.6184
124.5220


## 5. Conclusion

In order to study LPI time of NARS under the circumstances of a PDS, we modelled a PDS as the combination of a RWR and an ESM. The model makes the tracking task between aircrafts and target more realistic. Generally, NARS is required to decrease emission power to elude a RWR while increasing emission power to elude an ESM. For that inherent contradiction, LPI time is important to guide NARS to control its radiation power and time simultaneously and to avoid the threat from a combined PDS. This paper proposes a simplified math model to estimate LPI time of NARS when the parameters of combined PDS are partially known. Simulation results illustrate that the simultaneous management of radiation power and time can help NARS against combined PDS interception.



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*Source: 5932539-2020-01-04.xml*