# Optimization of Transmitter-Receiver Pairing of Spaceborne Cluster Flight Netted Radar for Area Coverage and Target Detection

Authors: Tingting Yan; Shengbo Hu; Jianan Cai; Jinrong Mo; Mingfei Xia
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we investigate the optimization problem of the transmitter-receiver pairing of spaceborne cluster flight netted radar (SCFNR) for area coverage and target detection. First of all, we propose the novel concept of SCFNR integrated cluster flight spacecraft with netted radar, the mobility model for bistatic radar pair with twin-satellite mode, and formulate the radar-target distance distribution function and radar-target distance product distribution function with geometric probability method. Secondly, by dividing surveillance region into grids, we define the 0-1 grid coverage matrix for bistatic radar and the transmitter-receiver pairing matrix for SCFNR with using radar equation and the radar-target distance distribution function, and we describe the optimal problem of transmitter-receiver pairing of SCFNR for area coverage and target detection by definingK-grid coverage matrix. Thirdly, we propose a new algorithm integrated particle swarm optimization with Hungarian algorithm (PSO-HA) to address the optimal problem, which is actually one-to-one pairing problem. Finally, we validate the effectiveness and reasonability of the proposed algorithm through numerical analysis.


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## Body

## 1. Introduction

As a distributed space system, the spaceborne netted radar is composed of several spatially separated, mutually independent, and cooperative radars in space. Compared with the traditional radar, spaceborne netted radar has advantages of high flexibility, reliability, and antistealth ability [1–3]. In addition, it also has the advantage of being all-weather, wide coverage, and satisfying specific coverage requirements due to its location in outer space [4, 5]. On the other hand, the cluster flight spacecraft has been one of the hot issues regarding the distributed space network, because of its advantages of flexibility, rapid response, low cost, strong scalability, and long lifetime [6–8]. Unlike traditional satellite formation flying applications, cluster flight spacecraft requires nodes to maintain bounded relative distances between tens or hundreds of kilometers and to keep loose geometry for the entire mission lifetime, so that orbit controlling and relative position sensing for the spacecraft can be performed well [6–8]. Some researches are mentioned cluster flight spacecraft. The paper [6] presents cluster-keeping algorithms aimed at minimizing fuel consumption. The paper [9] provides a cooperative control framework aimed at synchronizing the mean-orbital element convergence among cluster-flying satellites. The paper [10] studies the relationship between first docking time and spatial initial distribution and the relationship between first separating time and spatial initial distribution for cluster flight spacecraft. The influence of node transmit power on the QoS performance of cluster flight spacecraft network is analyzed in [11]. In order to improve the performance of cluster flight spacecraft network, the nodal distance distributions are studied in [12]. Hence, we propose the novel concept of SCFNR integrated cluster flight spacecraft with the spaceborne netted radar, and the optimization problem of SCFNR on coverage is addressed.Coverage is one of the important issues about radar for target detection, localization, and tracking. According to radar equations, the coverage problem of the spaceborne netted radar is related to many factors such as orbit, antenna gain, transmitted power, and radar cross section. To meet the need of improving the target detection and position, it expects that more radars of SCFNR can cover the surveillance region on the earth, which is completely different from line-of-sight (LOS) coverage of satellite constellation [13, 14]. When the antenna gain, the transmitted power, and radar cross section are constant, the coverage of SCFNR completely depends on the product of transmitter-target and target-receiver distance and the spacecraft orbit. Our previous research shows that the geometry configuration of SCFNR is characterized by high spatiotemporal dynamic and random, which complicates coverage problem of SCFNR. So, the problem about the coverage about SCFNR is more challenging.The netted radar is a case of multistatic radar [15–19], where transmitters can collaborate with several receivers at different locations. According to the pairing method of transmitter-receiver, the netted radar is mainly classified into three categories: a group of bistatic radars, a single transmitter with several receivers, and a single receiver with several transmitters. To improve the performance of SCFNR, it expects that more radars of SCFNR can cover the surveillance region, and this can be described by the maximum intersection coverage. Actually, the maximum intersection coverage is the classical maximum k-subset intersection (MSI) in graph theory, and it is also a combinatorial optimization problem [20]. To the best of our knowledge, there is not seen much on solving MSI problems. In [21], the authors introduce a GRASP heuristic and propose an integer programming formulation MSI problem. However, to solve the MSI problem about SCFNR is more difficult due to the geometry configuration with spatiotemporal dynamic and random.To the best of our knowledge, this makes the first paper to investigate the SCFNR coverage problem. The main contributions of our work are summarized as follows:(1)
We propose the novel concept of SCFNR integrated cluster flight spacecraft with netted radar, and the mobility model for bistatic radar pair is established by twin-satellite mode. The distribution function of the product of transmitter-target and receiver-target distance is derived using the method of geometric probability.(2)
According to radar equation, we propose the concept of 0-1 grid coverage matrix for bistatic radar by dividing the surveillance region into grids, and the definition of the transmitter-receiver pairing matrix for SCFNR is given using bistatic radar pairs. These provide an important theoretical basis for optimizing the transmitter-receiver pairing of SCFNR for area coverage and target detection.(3)
We describe the optimal problem of transmitter-receiver pairing of SCFNR for area coverage and target detection by definingK-grid coverage matrix. Also, we propose a new algorithm integrated PSO-HA to address the optimal problem. We validate the effectiveness of the proposed algorithms through numerical calculation.The rest of the paper is organized as follows: Section2 reviews the related work. Section 3 proposes the novel concept of SCFNR, establishes mobility model for bistatic radar pair, and derives the distribution function of the product of transmitter-target and receiver-target distance. Section 4 defines the coverage matrix of bistatic radar and pairing matrix of SCFNR based on Section 3 and describes the optimal problem of transmitter-receiver pairing of SCFNR for area coverage and target detection. Section 5 presents the PSO-HA algorithm. Section 6 verifies the effectiveness of the proposed algorithm, and coverage and detection results using numerical calculation are given. Finally, we conclude the paper in Section 7.


## 2. Related Works

In recent years, with continuing advances in communication technology and micro-electromechanical systems (MEMS) technology, multistatic radar sensing technology has received considerable attention, especially bistatic radar sensing coverage. For instance, in [22], the authors considered the problem of deploying a network of bistatic radars in a region to maximize the worst-case intrusion detectability. They studied the coverage problem of a bistatic radar sensor network and the optimal placement of bistatic radars on a line segment to minimize its vulnerability. In [23], Wang et al. studied the belt barrier coverage with the minimum total placement cost in bistatic radar sensor networks. They proposed a line-based equipartition placement strategy such that all radars placed on a deployment line can form a barrier with some breadth and one or more such placement lines can form a belt barrier with the required breadth. In [24], the authors studied area coverage in bistatic radar sensor networks. They investigated the geometrical relationship between the c-coverage area of a bistatic radar and the distance between its component transmitter and receiver. Then, they reduced the problem dimension by transforming the area coverage problem to point coverage problem by employing the intersection point concept. In [25], the authors studied the worst-case coverage under deterministic deployment, aiming to find optimal deployment locations of radar transmitters and receivers such that the worst-case intrusion detectability was maximized. Then, by developing a novel 2-site Voronoi diagram with graph search techniques, they designed an algorithm to find approximate worst-case intrusion detectability. In [23], the authors studied the belt barrier coverage in bistatic radar sensor networks, which was dependent on the distance between a pair of radar transmitter and receiver. In [26], Wang et al. studied barrier coverage in bistatic radar sensor networks. They formulated the barrier coverage problem as minimum weight barrier coverage problem. By constructing a directed coverage graph, minimum weight barrier coverage problem was transformed into finding k node-disjoint shortest paths. Next, they proposed an energy-efficient algorithm to solve the problem within polynomial.In addition, intelligent coverage becomes a practical research topic in dynamic sensors network. The paper [27] provided wireless signal coverage schemes for point-to-point and point-to-region and determined the required horizontal rotation angle and pitch rotation angle of the directional antenna intelligent coverage. In [28], the authors established the mobile sensor non-cooperative game model. Then, a local information-based topology control (LITC) algorithm based on this model was proposed, in which sensors move to enhance coverage by exchanging information with neighbors. Also, the application of PSO is wide in complex system. In [29], a systematic data-driven adaptive neuro-fuzzy inference system (ANFIS) modelling methodology was proposed, and a high-performance PSO-LSE method was developed to improve the structure and to identify the consequent parameters of ANFIS model. In [30], the authors proposed an algorithm combined with belief-desire-intention agent with a quantum-behaved particle swami optimization (QPSO) algorithm to optimize a marine generator excitation controller, and the QPSO algorithm was highly robust because its performance was insensitive to the accuracy of system parameters. For intelligent coverage in complex dynamic environment, in [31], a novel trajectory scheduling method based on coverage rate for multiple mobile sinks was presented, especially for large-scale wireless sensor networks, and an improved PSO combined with mutation operator was introduced to search the parking positions with optimal coverage rate. Predictably, considering complex dynamic sensor network, PSO in intelligent coverage is helpful.For a long time, some works on spaceborne radar coverage are mainly focused on optimizing the orbital design according to the observation and detection requirements. For example, in [32], the authors presented a feasibility analysis of a spaceborne bistatic radar mission for soil moisture retrieval, and they studied the assessment of the spatial coverage from orbital design. In [33], the authors adopted bistatic geometry from space platforms, and they implemented bistatic synthetic aperture radar observation. In [34], based on the analysis of radar cross-section (RCS) characteristic of geostationary orbital targets, the orbital altitude and revisiting period of space-based radar was designed in detail, and they discussed the relationship between image’s resolutions of spaceborne inverse synthetic aperture radar and system parameters. In [35], the authors established a spaceborne-airborne bistatic radar model, and then they analyzed moving target detecting performance of the space-time adaptive processing technology.For cluster flight spacecraft, related researches focus more on orbital control and node connection. The paper [6] presented a methodological development of cluster flight algorithms for disaggregated satellite systems in low Earth orbits. To obtain distance-bounded relative motion, a new constraint on the initial conditions of the modules was developed. In [9], the authors developed the implementable cluster flight-control methods with realistic orbital and actuator modelling. They offered two distributed orbit control laws with fixed-magnitude thrust for satellite cluster flight based on mean-orbital elements. Recently, the team of this paper has done some works on the cluster flight spacecraft network. For example, in [10], the authors proposed the constraint condition of orbital elements for noise-limited fractionated spacecraft network percolating and path formation time. The numerical results showed that the network topology for fractionated spacecraft is time varying and dynamic. The paper [11] investigated the transmit power allocation problem to minimize the average packet error rate at the access point in the cluster flight spacecraft network. Due to the complexity of the calculation, the probability density function of the distance between nodes was fitted using eighth-order polynomial.


## 3. The Concept of SCFNR

As mentioned above, the spaceborne netted radar is composed of several spatially separated, mutual independent, and cooperative radars in space. The spaceborne netted radar has the advantages of high flexibility, reliability, and antistealth ability. In addition, it also has the advantage of being all-weather, wide coverage, and satisfying specific coverage. On the other hand, the cluster flight spacecraft has many advantages such as flexibility, rapid response, low cost, strong scalability, and long lifetime. What’s more, cluster flight spacecraft can perform orbit controlling and relative position sensing easily. Hence, we propose the novel concept of SCFNR integrated advantages of both cluster flight spacecraft and the spaceborne netted radar.Generally, netted radar has the following three cases: (1) a group of bistatic radars, where the output of the bistatic radars are processed centrally to obtain a decision regarding the presence of a target and to estimate parameters. In this case, it is assumed that the transmitters do not interfere with each other, which is typically achieved either by using separate frequency bands or orthogonal transmitted waveforms. At the same time, each receiver is assumed to be able to receive the signals from each transmitter; (2) a single transmitter with several receivers, typically in the case of a high-value unit equipped with the transmitter, for instance an airborne warning and control system, and receivers cooperating to achieve the detection; (3) a single receiver with several transmitters, where a single receiver receives waveforms from several transmitters in different frequency bands to information fusion.In this paper, we adopt SCFNR with bistatic radar pairs. It is assumed that one-to-one pairing method is taken by SCFNR in any slot of the orbital hyperperiod. So, we assume that each pair of transmitter and receiver can potentially form a bistatic radar. We further assume that orthogonal transmissions are used for interference avoidance. In view of this, we assume that one transmitter can only be connected to one receiver, and the corresponding bistatic radar is formed in any slot of the orbital hyperperiod. Therefore, given a SCFNR consisting ofN radars, if N is even, then the pairing of bistatic radars is N/2 pairs, and if N is odd, then the pairing of bistatic radars is (N − 1)/2 pairs and a monostatic radar. Since the monostatic radar can be considered as a bistatic radar with a baseline length 0, it can also be considered that (N + 1)/2 pairs of bistatic radars is formed.Based on the above, this paper focuses on the optimization problem of the transmitter-receiver pairing of SCFNR for area of interest coverage and target detection, that is, how to pair transmitter-receiver properly to satisfy the requirements of area coverage and detection in any slot of the orbital hyperperiod. First, the mobility model for SCFNR is presented and analyzed. Of course, the mobility model for bistatic radar pair can be given, and also the distribution function of the product of transmitter-target and receiver-target distance needs to be derived.


### 3.1. The Mobility Model for Bistatic Radar Pair

To accomplish the cluster flight model within bounded distance, the twin-satellite model is adopted to study the mobility model for bistatic radar pair. As shown in Figure1, the transmitter or receiver position is uniformly distributed on sphere within (M − m)/4. M is the upper bound of transmitter-receiver distance in SCFNR, and m is the lower bound.Figure 1
The mobility model for bistatic radar pair.Based on orbit dynamics theory, the orbital hyperperiod can be divided intoT0,T1,T2,…,TT, times for fractionated spacecraft [7, 36]. So, there are T time slots in an orbital period. The orbital hyperperiod is ℋ=TT−T0, time slot σk=Tk−1,Tkk=1,2,…,T [7]. So, the mobility model of SCFNR can be defined as follows.Definition 1.
In earth-centered inertial (ECI) coordinates, if the position set ofN transmitters and receivers in CFSNR is S0=S10,S20,…,SN0 at initial time T0, the position set is Sk=S1k,S2k,…,SNk, and the positions are uniformly distributed within sphere BSi0,Ri=1,2,…,N at time k, where Si0 and R=M−m/4 are the center and radius of the sphere, respectively. Moreover, positions among all transmitters and receivers are mutually independent and independent of all previous locations.


### 3.2. The Probability Distribution Function of the Distance Product

We consider a SCFNR scenario as shown in Figure2. Let T be the transmitter set and R be the receiver set. Transmitters and receivers are located at different locations. We use TR to denote all transmitter-receiver pairs. If transmitter Ti∈T and receiver Rj∈R choose the same channel, then the bistatic radar TiRj∈TR is formed by Ti and Rj, and different channels can be considered as orthorhombic channels to avoid interference. Without ambiguity, in any time slot of orbital hyperperiod for SCFNR, the position of transmitter and receiver is denoted by SiT and SjR, respectively, where i≠j.Figure 2
The SCFNR scenario.Thus, in SCFNR scenario, let P be a target position in the surveillance region. According to [37], for a bistatic radar TiRj∈TR, the signal-to-noise ratio (SNR) of P can be given as(1)SNR=KBSiTP2PSjR2,where SiTP and PSjR denote transmitter-target and target-receiver distances, respectively. KB is a constant related to the physical-layer parameters of the bistatic radar, such as transmit power, antenna gains of transmitter and receiver, and radar cross-section. However, we are not interested in the abovementioned physical-layer parameters, but transmitter-target and target-receiver distances. For convenience, we assume that the constant is identical for any bistatic radar, i.e., homogeneous bistatic radar also.As seen from equation (1), the SCFNR performance is determined by SiTPPSjR, i.e., the product of transmitter-target and target-receiver distances. According to Definition 1, the product is random. Therefore, we need to analyze its distribution.For convenience, the 2D scenario about transmitterTi and target P in SCFNR is described in Figure 3. Ti is assumed to be uniformly located in a circle of the two-dimensional plane, and the P is assumed to be fixed. In Figure 3, let SiTP=di, hi be the distance between P and initial orbital position of Ti, where di∈hi−R,hi+Rhi>R. Actually, if P is a target position of earth surface, then hi can be considered as the orbit height of Ti at initial time.Figure 3
The transmitter-target distance distribution in SCFNR.Therefore, the transmitter-target distancedi has the distance function given by probability distribution, that is,(2)FDidi≜PDi≤di,hi−R≤di≤hi+R,hi>R.Here,FDidi is calculated with geometric probability method [38, 39].Now, we extend the 2D scenario in Figure3 into 3D scenario. Let Ω be the sphere O and C0 be the intersection volume between sphere O and the sphere of radius di centered at P. Equation (2) can be rewritten as(3)FDidi=μC0μΩ,where μΩ=4πR3/3 is the measure of Ω.In order to calculateFDidi, μC0 can be divided into two cases: (1) di∈hi−R,hi; (2) di∈di∈hi,hi+R. Thus, Theorem 1 about distribution function of di can be proved.Theorem 1.
In SCFNR, if the initial position of transmitterSiT0 and mobility model M (t) are given, then the distribution function of random variable di is(4)FDidi=0,di<hi−R,2di3+ha3−3hadi2+2R3+hb3−3hbR24R3,hi−R≤di<hi,2di3+ha3−3hadi2+2R3−hc3+3hcR24R3,hi≤di≤hi+R,1,hi+R<di,where di is the radar-target distance, ha=hi2−R2+di2/2hi, hb=hi2+R2−di2/2hi, and hc=−hi2−R2+di2/2hi.Proof of Theorem 1.
The proof of Theorem 1 is given in Appendix A.
Similarly, ifPSjR=dj is the distance between target P and receiver Rj, then SiTPPSjR=didj=dij is the product of transmitter-target and target-receiver distances. Since di and dj are independent, Theorem 2 about distribution function of dij can be proved.Theorem 2.
In SCFNR, if the initial positions of transmitterSiT0, receiver SjR0, and mobility model M (t) are given, then distribution function of random variable dij is(5)FDijdij=0,dij<h1−Rh2−R,FDij1dij,h1−Rh2−R≤dij<h1+Rh2−R,FDij2dij,h1+Rh2−R≤dij<h1−Rh2+R,FDij3dij,h1−Rh2+R≤dij≤h1+Rh2+R,1,otherwise,where h1=maxhi,hj, h2=minhi,hj, and hj is the orbit height of radar receiver Rj at initial time,(6)FDij1dij=C11dij22+C21dij33+C31dij44+C41dij242lndijh1−Rh2−R−1+C51dij393lndijh1−Rh2−R−1C61dij4164lndijh1−Rh2−R−1+C01,FDij2dij=C12dij22+C22dij33+C32dij44+C02,FDij3dij=C13dij22+C23dij33+C33dij44+C43dij242lndijh1+Rh2+R−1+C53dij393lndijh1+Rh2+R−1+3lndijh1+Rh2+R+C63d34164lndijh1+Rh2+R−1+C03,C11=9h1−Rh2−R3h1h2+2h1R+2h2R+R216h1h2R6,C21=−9h1+h24h1h2R5,C31=116h1h2R6−27h1+9R2h1−2R+−27h2+9R2h2−2R,C41=9h12−R2h22−R216h1h2R6,C51=94R6,C61=916h1h2R6,C01=−h1−R3h2−R3h1+3Rh2+3R256h1h2R6,C12=916h1h2R6h12−R2h22−R2lnh1+Rh1−R−2Rh1h22−R2,C22=94R6lnh1+Rh2−R−184h1R5,C32=116h1h2R69R−27h12h1−2R+9R+27h22h2+2R+9lnh1+Rh1−R,C02=FDij1h1+Rh2−R−C12h1+R2h2−R22−C22h1+R3h2−R33−C32h1+R4h2−R44,C13=−9h1+Rh2+R3h1h2−2h1R−2h2R+R216h1h2R6,C23=−9h1+h24h1h2R5,C33=116h1h2R627h1+9R2h1+2R+27h2+9R2h2+2R,C43=−9h12−R2h22−R216h1h2R6,C53=−94R6,C63=−916h1h2R6,C03=FDij2h1−Rh2+R−C13h1−R2h2+R22−C23h1−R3h2+R33−C33h1−R4h2+R44−C43dij242lnh1−Rh1+R−1+C53dij393lnh1−Rh1+R−1+C63dij3163lnh1−Rh1+R−1.Proof of Theorem 2.
the proof of Theorem 2 is given in Appendix A.


## 3.1. The Mobility Model for Bistatic Radar Pair

To accomplish the cluster flight model within bounded distance, the twin-satellite model is adopted to study the mobility model for bistatic radar pair. As shown in Figure1, the transmitter or receiver position is uniformly distributed on sphere within (M − m)/4. M is the upper bound of transmitter-receiver distance in SCFNR, and m is the lower bound.Figure 1
The mobility model for bistatic radar pair.Based on orbit dynamics theory, the orbital hyperperiod can be divided intoT0,T1,T2,…,TT, times for fractionated spacecraft [7, 36]. So, there are T time slots in an orbital period. The orbital hyperperiod is ℋ=TT−T0, time slot σk=Tk−1,Tkk=1,2,…,T [7]. So, the mobility model of SCFNR can be defined as follows.Definition 1.
In earth-centered inertial (ECI) coordinates, if the position set ofN transmitters and receivers in CFSNR is S0=S10,S20,…,SN0 at initial time T0, the position set is Sk=S1k,S2k,…,SNk, and the positions are uniformly distributed within sphere BSi0,Ri=1,2,…,N at time k, where Si0 and R=M−m/4 are the center and radius of the sphere, respectively. Moreover, positions among all transmitters and receivers are mutually independent and independent of all previous locations.


## 3.2. The Probability Distribution Function of the Distance Product

We consider a SCFNR scenario as shown in Figure2. Let T be the transmitter set and R be the receiver set. Transmitters and receivers are located at different locations. We use TR to denote all transmitter-receiver pairs. If transmitter Ti∈T and receiver Rj∈R choose the same channel, then the bistatic radar TiRj∈TR is formed by Ti and Rj, and different channels can be considered as orthorhombic channels to avoid interference. Without ambiguity, in any time slot of orbital hyperperiod for SCFNR, the position of transmitter and receiver is denoted by SiT and SjR, respectively, where i≠j.Figure 2
The SCFNR scenario.Thus, in SCFNR scenario, let P be a target position in the surveillance region. According to [37], for a bistatic radar TiRj∈TR, the signal-to-noise ratio (SNR) of P can be given as(1)SNR=KBSiTP2PSjR2,where SiTP and PSjR denote transmitter-target and target-receiver distances, respectively. KB is a constant related to the physical-layer parameters of the bistatic radar, such as transmit power, antenna gains of transmitter and receiver, and radar cross-section. However, we are not interested in the abovementioned physical-layer parameters, but transmitter-target and target-receiver distances. For convenience, we assume that the constant is identical for any bistatic radar, i.e., homogeneous bistatic radar also.As seen from equation (1), the SCFNR performance is determined by SiTPPSjR, i.e., the product of transmitter-target and target-receiver distances. According to Definition 1, the product is random. Therefore, we need to analyze its distribution.For convenience, the 2D scenario about transmitterTi and target P in SCFNR is described in Figure 3. Ti is assumed to be uniformly located in a circle of the two-dimensional plane, and the P is assumed to be fixed. In Figure 3, let SiTP=di, hi be the distance between P and initial orbital position of Ti, where di∈hi−R,hi+Rhi>R. Actually, if P is a target position of earth surface, then hi can be considered as the orbit height of Ti at initial time.Figure 3
The transmitter-target distance distribution in SCFNR.Therefore, the transmitter-target distancedi has the distance function given by probability distribution, that is,(2)FDidi≜PDi≤di,hi−R≤di≤hi+R,hi>R.Here,FDidi is calculated with geometric probability method [38, 39].Now, we extend the 2D scenario in Figure3 into 3D scenario. Let Ω be the sphere O and C0 be the intersection volume between sphere O and the sphere of radius di centered at P. Equation (2) can be rewritten as(3)FDidi=μC0μΩ,where μΩ=4πR3/3 is the measure of Ω.In order to calculateFDidi, μC0 can be divided into two cases: (1) di∈hi−R,hi; (2) di∈di∈hi,hi+R. Thus, Theorem 1 about distribution function of di can be proved.Theorem 1.
In SCFNR, if the initial position of transmitterSiT0 and mobility model M (t) are given, then the distribution function of random variable di is(4)FDidi=0,di<hi−R,2di3+ha3−3hadi2+2R3+hb3−3hbR24R3,hi−R≤di<hi,2di3+ha3−3hadi2+2R3−hc3+3hcR24R3,hi≤di≤hi+R,1,hi+R<di,where di is the radar-target distance, ha=hi2−R2+di2/2hi, hb=hi2+R2−di2/2hi, and hc=−hi2−R2+di2/2hi.Proof of Theorem 1.
The proof of Theorem 1 is given in Appendix A.
Similarly, ifPSjR=dj is the distance between target P and receiver Rj, then SiTPPSjR=didj=dij is the product of transmitter-target and target-receiver distances. Since di and dj are independent, Theorem 2 about distribution function of dij can be proved.Theorem 2.
In SCFNR, if the initial positions of transmitterSiT0, receiver SjR0, and mobility model M (t) are given, then distribution function of random variable dij is(5)FDijdij=0,dij<h1−Rh2−R,FDij1dij,h1−Rh2−R≤dij<h1+Rh2−R,FDij2dij,h1+Rh2−R≤dij<h1−Rh2+R,FDij3dij,h1−Rh2+R≤dij≤h1+Rh2+R,1,otherwise,where h1=maxhi,hj, h2=minhi,hj, and hj is the orbit height of radar receiver Rj at initial time,(6)FDij1dij=C11dij22+C21dij33+C31dij44+C41dij242lndijh1−Rh2−R−1+C51dij393lndijh1−Rh2−R−1C61dij4164lndijh1−Rh2−R−1+C01,FDij2dij=C12dij22+C22dij33+C32dij44+C02,FDij3dij=C13dij22+C23dij33+C33dij44+C43dij242lndijh1+Rh2+R−1+C53dij393lndijh1+Rh2+R−1+3lndijh1+Rh2+R+C63d34164lndijh1+Rh2+R−1+C03,C11=9h1−Rh2−R3h1h2+2h1R+2h2R+R216h1h2R6,C21=−9h1+h24h1h2R5,C31=116h1h2R6−27h1+9R2h1−2R+−27h2+9R2h2−2R,C41=9h12−R2h22−R216h1h2R6,C51=94R6,C61=916h1h2R6,C01=−h1−R3h2−R3h1+3Rh2+3R256h1h2R6,C12=916h1h2R6h12−R2h22−R2lnh1+Rh1−R−2Rh1h22−R2,C22=94R6lnh1+Rh2−R−184h1R5,C32=116h1h2R69R−27h12h1−2R+9R+27h22h2+2R+9lnh1+Rh1−R,C02=FDij1h1+Rh2−R−C12h1+R2h2−R22−C22h1+R3h2−R33−C32h1+R4h2−R44,C13=−9h1+Rh2+R3h1h2−2h1R−2h2R+R216h1h2R6,C23=−9h1+h24h1h2R5,C33=116h1h2R627h1+9R2h1+2R+27h2+9R2h2+2R,C43=−9h12−R2h22−R216h1h2R6,C53=−94R6,C63=−916h1h2R6,C03=FDij2h1−Rh2+R−C13h1−R2h2+R22−C23h1−R3h2+R33−C33h1−R4h2+R44−C43dij242lnh1−Rh1+R−1+C53dij393lnh1−Rh1+R−1+C63dij3163lnh1−Rh1+R−1.Proof of Theorem 2.
the proof of Theorem 2 is given in Appendix A.


## 4. The Area of Interest Coverage of SCFNR

According to equation (1), the larger the product of transmitter-target and target-receiver distance SiTPPSjR, the smaller the received SNR, and the probability of the P detected by transmitter-receiver is smaller too. Conversely, the smaller the SiTPPSjR, the larger the probability will be.So, we define point coverage of SCFNR for target detection.Definition 2.
Given a threshold valuec and a point target P, if there exists a bistatic radar TiRj∈TRi≠j and the product of transmitter-target and target-receiver distances in any time slot of orbital hyperperiod is satisfied(7)SiTPPSjR<c,then the bistatic radar TiRj∈TR can provide point coverage to point P.
For the sake of analysis, according to the idea of grid, the surveillance region is divided into grids with equal borders, the border length of the grid is able to be elected in accordance with the range resolution of radar. That is, in ECI coordinate, the surveillance region of interestA (see Figure 4) is encoded in accordance with horizontal encoding gx1≤gx≤Nx and vertical encoding gy1≤gy≤Ny; the grid coordinate gx,gy is denoted by Agxgy. Thus, the region of interest A can be determined uniquely by all grids and expressed as follows:(8)A=Agxgy|1≤gx≤Nx,1≤gy≤Ny.
So, for a bistatic radarTiRj∈TR, SNR of each grid can be given as follows:(9)SNRgxgy=KBSiTAgxgy2AgxgySjR2.
LetΓ be the SNR threshold, then c=KB/Γ, and a grid target is covered by a bistatic radar TiRj∈TR, if SNRgxgy≥Γ. Then, the definition of 0-1 grid coverage matrix on SCFNR can be described as follows:Figure 4
Sketch of coverage region in SCFNR scenario.Definition 3.
For a bistatic radarTiRj∈TR in SCFNR, given Agxgy∈A, the 0-1 grid coverage matrix is denoted by Uij=uij,gxgyNx×Ny, where(10)uij,gxgy=1,FDijSiTAgxgyAgxgySjR≤FDijc,0,others.
Ifuij,gxgy=1 in equation (10), it indicates that the grid Agxgy can be covered by the bistatic radar TiRj∈TR.
Additionally, to analyze the impact of transmitter-receiver pairs on coverage, 0-1 pairing matrix, which describes the transmitter-receiver pairs selected in SCFNR, can be defined as follows.Definition 4.
For SCFNR, suppose the cardinalities bothT and R are N. If Ti and Rj are selected as a bistatic radar, let mij=1; otherwise, mij=0. Then, 0-1 pairing matrix of transmitter-receivers is denoted by M=mijN×N.
Note that same grids may be covered by different bistatic radar pairs. Thus, based on Definitions2 and 4, the introduction of cumulative coverage times wgxgy describes the coverage level of SCFNR at grid Agxgy, that is,(11)wgxgy=∑i=1N∑j=1Nmijuij,gxgy.
As seen from equation (11), wgxgy∈0,1,…,N. On the basis of this, K-grid coverage matrix of SCNFR can be defined as follows.Definition 5.
For SCFNR, given a valueKK≤N, if the variable cgxgy is satisfied as(12)cgxgy=1,wgxgy≥K,0,otherwise,then the matrix C∈ℝNx×Ny; C=cgxgy is called K-grid coverage matrix.
In equation (12), the total number of elements with 1 in C represents the grid number satisfying K-grid coverage, and the total number of elements with 0 in C represents the grid number unsatisfying K-grid coverage, that is,(13)g0=NxNy−∑gx=1Nx∑gy=1Nycgxgy.
From the point of optimizing system, there isg0⟶0. If the values of g0 approach 0, then SCFNR can provide completely K-grid coverage to the region A; otherwise, SCFNR fails to provide K-grid coverage to the region A. Therefore, the normalized g0 is taken as g1 to measure coverage performance of SCFNR, that is,(14)g1=1−1NxNy∑gx=1Nx∑gy=1Nycgxgy.
Also, using the radar equation and conditional probability, letlij=SiTAgxgyAgxgySjR; the detection probability of bistatic radar TiRj∈TR to grid Agxgy is given by(15)pij,gxgy=Prdij≥lij|dij≤c.
Thus, the detection probability of SCFNR radar to gridAgxgy is as follows:(16)pgxgy=1−∏i=1N∏j=1N1−mijpij,gxgy.
For the sake of optimization analysis, the worst-case detection probability of all grids is taken as the second objective function to measure the region detection performance of SCFNR, that is,(17)pnet≜mingx,gypgxgy.
To sum up, in SCFNR coverage scenario,K-grid coverage and detection probability (e.g., g1 and pnet) are functions of pairing matrix M. Therefore, pairing transmitter-receivers with minimum g1 and maximum pnet can be optimized as follows:(18)ming1M=1−1NxNy∑gx=1Nx∑gy=1Nycgxgyming2M=1−pnets.t.C1:∑i=1Nmij=1,∀j∈1,…,N,C2:∑j=1Nmij=1,∀i∈1,…,N.
The constraintsC1 and C2 denote that each selected transmitter or receiver can only be associated to one receiver or transmitter. This is actually one-to-one pairing problem [40].


## 5. Algorithm Design

Obviously, the problem described in equation (18) is a multiobjective optimization problem. Due to the conflicting nature of the two objectives, no solution optimizing all objective functions simultaneously exists in general. Instead, balance among objective functions is taken into account, which is called trade-off analysis in multiobjective optimization, i.e., Pareto optimal solutions [41]. The basic idea is based on a distance measure to determine the solution near by the ideal solution. Here, the weighted Lp norm is taken as the distance measure, that is,(19)gM=∑ib=12ζibgib−gib∗p1/p,where gib∗ is the ideal value of ib-th goal, and g1∗=g2∗=0, and ζibis the weight factor of the ib-th goal.The problem described in equation (19) is a combinatorial optimization problem. For this problem, particle swarm optimization (PSO) has been proved as an effective tool [42–44]. PSO is based on the behavior of birds flocking [45]. Each particle represents a potential solution to optimization task and all particles fly in the search space to find the optimal solution. But, its solution does not satisfy the constraints C1 and C2. As mentioned before, the constraints C1 and C2 described in equation (18) are a one-to-one paring problem, which can be solved using Hungarian algorithm (HA) [40]. As a combinatorial optimization method, HA can finish the one-to-one paring task in polynomial time. Therefore, the PSO-HA integrated PSO with HA is proposed. The outline of PSO-HA is given as follows.Step 1.
Initialization.
Suppose swarm size isL, particle is 11≤l≤L, the maximum number of iterations is Tmax, iteration time is t1≤t≤Tmax, acceleration factors are c1 and c2, and the position and velocity of l-th particle are M¯l and vl, respectively. Let t = 1, and set the parameter values: L, Tmax, M¯lt, vlt, c1, and c2.
As a note,M¯l and vl are both N×N matrixes, in which each element of matrixes is generated randomly. Here, each element of M¯l and vl is set in the range [0, 1] and [−0.5, 0.5], respectively.Step 2.
Update positionM¯l by using PSO.
In each iteration,M¯l and vl are updated as follows:(20)vlt+1=ωt×vlt+c1r1tρlt−M¯lt+c1r1tρgt−M¯lt,M¯lt+1=M¯lt+vlt+1,where ρl is the current position of l-th particle, ρg is the best position of all particles it has visited so far, ωt is the inertia weight which decreases with iteration time as ωt=0.9−0.5×t/Tmax [46], and r1t and r2t are random independent variables in the range [0, 1].Step 3.
Generate one-to-one paring matrixMl by using HA.
The updating resultM¯l is taken as the cost matrix of l-th particle in HA, and then the optimization problem on one-to-one paring is formulated as follows:(21)min∑i=1N∑j=1Nm¯ijlmijls.t.C1:∑i=1Nmij=1,∀j∈1,…,N,C2:∑j=1Nmij=1,∀i∈1,…,N,where m¯ijl is the element of matrix M¯l, mijl is the element of matrix Ml, and mijl is either 0 or 1.
Note that the paring matrixMl, which satisfies constraints C1 and C2, is obtained by using HA.Step 4.
Calculate the fitness functiongMl of l-th particle and determine the best solution Mg (i.e., optimal paring matrix).
Taking the objective function, i.e., equation (19), as the fitness function of l-th particle (denoted by gMl), the current paring matrix Ml of l-th particle and optimal paring matrix Mg of all particles are updated as follows:(22)Mlt+1=Mlt+1,ifgMlt+1≤gMlt,Mlt,others,Mgt+1=Mlt+1whenmingMlt+11≤l≤L.Step 5.
Ift≤Tmax, then increment t and go to Step 2; otherwise, end.


## 6. Simulation Analysis

In order to simulate and analyze multiobjective pairing optimization, i.e., coverage and detection performances of SCFNR, in time slot of the orbital hyperperiod, we establish the SCFNR scenario by STK (Satellite Tool Kit) first. Then, we use PSO-HA to find optimal pairing matrix in Windows 10 and MATLAB R2017b environment. At the same time, area of interest coverage and detection probability are analyzed numerically.


### 6.1. Parameters Setting

#### 6.1.1. Orbital Elements in SCFNR

Suppose SCFNR is composed of 4 pairs of homogeneous bistatic radars. LetT=T1,T2,T3,T4, R=R1,R2,R3,R4,m=30km, and M=850km. According to the orbit design of cluster flight spacecraft proposed in [10], all near circular orbital elements of SCFNR are listed in Table 1.Table 1
All near circular orbital elements of SCFNR.
ParameterSemimajor axis (km)Eccentricity (deg)Inclination (deg)Argument of perigee (deg)True anomaly (deg)Right ascension of ascending node (deg)T17378.140.02350.000000.000000.00000T27378.140.02350.001631.139473.38820T37378.140.02350.000680.476301.82400T47378.140.0235359.997−2.16501.70983R17378.140.02350.001062.506022.50602R27378.140.02350.000220.153831.08022R37378.140.0235359.999−0.797634.02245R47378.140.02350.003442.39289−1.18115According to Table1, all orbital periods can be calculated and are approximated as 6310 seconds using STK, so we believe the orbital hyperperiods of the SCFNR are also 6310 s. In addition, as shown in Figure 5, we can also calculate all relative distances between transmitters and receivers in 172 days by STK. It is observed that the relative distance between any transmitter-receiver always remains below 850 km and above 30 km.Figure 5
The relative distance between any transmitter-receiver within 172 days.


#### 6.1.2. The Target Grid and Other Parameters

Suppose that the longitude and latitude of surveillance regions are in the range [0, 0.07865345] (rad) and [0, 0.07865345] (rad), respectively. That is, surveillance region is set as square with the size of500×500km on the earth surface. The region with longitude and latitude can be divided into Nx×Ny grids. Let Nx=100 and Ny=100. So, according to coordinate transforming relations between spherical coordinates and rectangular coordinates, each grid can be computed in the ECI coordinate.For radar equation and PSO-HA, the parameters are listed in Table2. In this case, the distribution function of dij for SCFNR in equation (5) can be calculated, as presented in equation (23). At the same time, as shown in Figure 6, we give the curve of distribution function of dij for SCFNR. (23)FDijdij=0,dij<7.04878×105,F1dij,7.04878×105≤dij<1.04910×106,F2dij,1.04910×106≤dij≤1.56142×1061,otherwise,where(24)F1dij≈8.30169×10−9×dij2−7.33464×10−15×dij3−6.49148×10−21×dij4+3.82234×10−9×dij2×logdij7.04878×105+1.01050×10−14×dij3×logdij7.04878×105+1.73646×10−21×dij4×logdij7.04878×105+46.52592,F2dij≈−1.14225×10−8×dij2−5.97942×10−16×dij3+5.07373×10−21×dij4−3.82234×10−9×dij2×logdij1.56143×106−1.01050×10−14×dij3×logdij1.56143×106−1.73646×10−21×dij4×logdij1.56143×106−32.88394.Table 2
Parameter setting.
ParameterValueKB45M4Γ12.5 dBh11044.5702 kmh21044.5702 kmc1.14932×106FDijc0.62269L10Tmax100P2c11.49445c21.49445R205 kmTable 3
gM for 24 cases of pairing schemes.
Paring schemeζ1=0.7ζ1=0.5ζ1=0.3ζ2=0.3ζ2=0.5ζ2=0.7T1R1, T2R2, T3R3, T4R40.155400.255000.35597T1R1, T2R2, T3R4, T4R30.164690.240260.32681T1R1, T2R3, T3R4, T4R20.133680.194920.26509T1R1, T2R3, T3R2, T4R40.150690.242290.33688T1R1, T2R4, T3R2, T4R30.155010.221660.29994T1R1, T2R4, T3R3, T4R20.127470.193740.26614T1R2, T2R1, T3R3, T4R40.098530.157540.21879T1R2, T2R1, T3R4, T4R30.113940.135840.17271T1R2, T2R4, T3R3, T4R10.111830.183080.25544T1R2, T2R4, T3R1, T4R30.117650.144360.18576T1R2, T2R3, T3R1, T4R40.098440.150160.20646T1R2, T2R3, T3R4, T4R10.118290.189770.26374T1R3, T2R1, T3R2, T4R40.108550.174870.24324T1R3, T2R1, T3R4, T4R20.104590.150330.20370T1R3, T2R2, T3R4, T4R10.140800.232390.32475T1R3, T2R2, T3R1, T4R40.126400.205940.28708T1R3, T2R4, T3R1, T4R20.136270.210100.28956T1R3, T2R4, T3R2, T4R10.129350.213100.29770T1R4, T2R3, T3R2, T4R10.148100.241480.33667T1R4, T2R3, T3R1, T4R20.113590.155520.20788T1R4, T2R1, T3R2, T4R30.127590.166160.21857T1R4, T2R1, T3R3, T4R20.098470.138390.18638T1R4, T2R2, T3R1, T4R30.139890.191680.25627T1R4, T2R2, T3R3, T4R10.153990.254570.35585Figure 6
The distance distribution function (equation (23)) associated with dij in SCFNR.Figure 7
The distributions of coverage level and detection probability withζ1=0.7, ζ2=0.3, and Γ=12.5dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)Figure 8
The distributions of coverage level and detection probability withζ1=0.5, ζ2=0.5, and Γ=12.5dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)


### 6.2. Numerical Result and Analysis

#### 6.2.1. Transmitter-Receiver Pairing Scheme

Case1:gM with different weight values in the slot 1 for the 1st orbital hyperperiod.Considering three conditions, i.e.,ζ1>ζ2, ζ1=ζ2, and ζ1<ζ2, we calculate the optimal pairing matrix in the same slot of its orbital hyperperiod under the same simulation environment as described in Section 6.1.Letζ1=0.7 and ζ2=0.3; the optimal pairing matrix is given as follows:(25)Mg=0100001010000001.Equation (25) indicates that optimal pairing scheme of bistatic radars is T1R2, T2R3, T3R1, T4R4, and gM=0.098349 with g1 = 0.06270 and g2 = 0.29371.Letζ1=0.5 and ζ2=0.5; the optimal pairing matrix is given as follows:(26)Mg=0100100000010010.Equation (26) indicates that optimal pairing scheme of bistatic radars is T1R2, T2R1, T3R4, T4R3, and gM=0.135841 with g1 = 0.1259 and g2 = 0.24075.Letζ1=0.3 and ζ2=0.7; the optimal pairing matrix is given as follows:(27)Mg=0100100000010010.Equation (27) indicates that the optimal pairing scheme of bistatic radars is T1R2, T2R1, T3R4, T4R3, and gM=0.172705 with g1 = 0.1259 and g2 = 0.24075.For the sake of comparison, Table3 lists all pairing schemes for 4 pairs of homogeneous bistatic radars, i.e., 24 cases of pairing schemes and corresponding gM using the same parameters.As shown in Table3, we mark the gM in bold font. It can be seen that, under conditions of same weight values, the corresponding paring schemes are consistent with equations (25) and (26), respectively. Therefore, PSO-HA is effective and reasonable.Case 2:gM with different weight values in the slots 1 and 2 of different orbital hyperperiods.Whenζ1=0.7 and ζ2=0.3, the optimal pairing scheme and corresponding gM can be listed in Table 4, and when ζ1=0.5 and ζ2=0.5, the optimal pairing scheme and corresponding gM can be listed in Table 5.Table 4
Optimal pairing scheme and correspondinggM with ζ1=0.7 and ζ2=0.3.
Orbital hyperperiodTime slotOptional pairing schemegMThe 2nd orbital hyperperiod1T1R2, T2R1, T3R3, T4R40.0635862T1R1, T2R2, T3R3, T4R40.139745The 3th orbital hyperperiod1T1R2, T2R1, T3R3, T4R40.0478432T1R3, T2R4, T3R1, T4R30.123073The 12th orbital hyperperiod1T1R2, T2R3, T3R4, T4R10.0097152T1R1, T2R4, T3R3, T4R20.198599The 16th orbital hyperperiod1T1R2, T2R3, T3R4, T4R10.0688672T1R1, T2R3, T3R4, T4R20.295543Table 5
Optimal pairing scheme and correspondinggM with ζ1=0.5 and ζ2=0.5.
Orbital hyperperiodTime slotOptional pairing schemegMThe 2nd orbital hyperperiod1T1R3, T2R1, T3R4, T4R20.0985362T1R2, T2R1, T3R3, T4R40.072687The 3th orbital hyperperiod1T1R3, T2R4, T3R2, T4R10.1561372T1R2, T2R4, T3R1, T4R30.156137The 12th orbital hyperperiod1T1R2, T2R3, T3R4, T4R10.0153812T1R1, T2R4, T3R3, T4R20.251995The 16th orbital hyperperiod1T1R4, T2R3, T3R1, T4R20.0905442T1R1, T2R3, T3R4, T4R20.380717As Tables4 and 5 show, the optimal pairing schemes are various in different slots of different orbital hyperperiods, andζ2=0.3 is different as well. From this result, we conclude that geometric topology of SCFNR with high dynamic and random leads to optimal pairing scheme with dynamic and random.


#### 6.2.2. Coverage Level and Detection Probability

Case 1: whenΓ=12.5dB,Using the optimal pairing matrices given by equations (25)–(27), we calculate the corresponding distributions of coverage level and detection probability as shown in Figures 7 and 8. The coverage level and detection probability corresponding to equations (26) and (27) are the same due to the same optimal pairing matrices in equations (26) and (27). So, we only need to give the two distributions with ζ1=0.7, ζ2=0.3 and ζ1=0.5, ζ2=0.5, respectively.In Figures7 and 8, it is observed that either the distribution of coverage level or the distribution of detection probability is roughly the same. Besides, the higher the coverage level, the higher the detection probability.Case 2: whenΓ=12.0dB,In this case, keeping other parameters unchanged, the distributions of coverage level and detection probability are shown in Figures9–11.Figure 9
The distributions of coverage level and detection probability withζ1=0.7, ζ2=0.3, and Γ=12.0dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)Figure 10
The distributions of coverage level and detection probability withζ1=0.5, ζ2=0.5, and Γ=12.0dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)Figure 11
The distributions of coverage level and detection probability withζ1=0.3, ζ2=0.7, and Γ=12.0dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)In Figures9–11, it is observed that the three distributions of coverage level and detection probability are roughly the same with different weight values. However, there is considerable difference between the distributions with Γ=12.5dB and Γ=12.0dB, and we find that the SNR threshold has a great influence on coverage level and detection probability in SCFNR. We also observe that the smaller the threshold Γ, the larger the coverage level and detection probability. These show that the proposed PSO-HA, coverage, and detection probability model are reasonable and effective, especially for coverage and detection performance measured by distance function.In addition, using PSO-HA to solve optimal pairing matrix, considering three weight values, we give the relationship between iteration andgM in slot 1 for the 1st orbital hyperperiod. As shown in Figure 12, the iterative process has good convergence.Figure 12
The relationship between iteration and objective function with different weight values.
(a)(b)(c)


## 6.1. Parameters Setting

### 6.1.1. Orbital Elements in SCFNR

Suppose SCFNR is composed of 4 pairs of homogeneous bistatic radars. LetT=T1,T2,T3,T4, R=R1,R2,R3,R4,m=30km, and M=850km. According to the orbit design of cluster flight spacecraft proposed in [10], all near circular orbital elements of SCFNR are listed in Table 1.Table 1
All near circular orbital elements of SCFNR.
ParameterSemimajor axis (km)Eccentricity (deg)Inclination (deg)Argument of perigee (deg)True anomaly (deg)Right ascension of ascending node (deg)T17378.140.02350.000000.000000.00000T27378.140.02350.001631.139473.38820T37378.140.02350.000680.476301.82400T47378.140.0235359.997−2.16501.70983R17378.140.02350.001062.506022.50602R27378.140.02350.000220.153831.08022R37378.140.0235359.999−0.797634.02245R47378.140.02350.003442.39289−1.18115According to Table1, all orbital periods can be calculated and are approximated as 6310 seconds using STK, so we believe the orbital hyperperiods of the SCFNR are also 6310 s. In addition, as shown in Figure 5, we can also calculate all relative distances between transmitters and receivers in 172 days by STK. It is observed that the relative distance between any transmitter-receiver always remains below 850 km and above 30 km.Figure 5
The relative distance between any transmitter-receiver within 172 days.


### 6.1.2. The Target Grid and Other Parameters

Suppose that the longitude and latitude of surveillance regions are in the range [0, 0.07865345] (rad) and [0, 0.07865345] (rad), respectively. That is, surveillance region is set as square with the size of500×500km on the earth surface. The region with longitude and latitude can be divided into Nx×Ny grids. Let Nx=100 and Ny=100. So, according to coordinate transforming relations between spherical coordinates and rectangular coordinates, each grid can be computed in the ECI coordinate.For radar equation and PSO-HA, the parameters are listed in Table2. In this case, the distribution function of dij for SCFNR in equation (5) can be calculated, as presented in equation (23). At the same time, as shown in Figure 6, we give the curve of distribution function of dij for SCFNR. (23)FDijdij=0,dij<7.04878×105,F1dij,7.04878×105≤dij<1.04910×106,F2dij,1.04910×106≤dij≤1.56142×1061,otherwise,where(24)F1dij≈8.30169×10−9×dij2−7.33464×10−15×dij3−6.49148×10−21×dij4+3.82234×10−9×dij2×logdij7.04878×105+1.01050×10−14×dij3×logdij7.04878×105+1.73646×10−21×dij4×logdij7.04878×105+46.52592,F2dij≈−1.14225×10−8×dij2−5.97942×10−16×dij3+5.07373×10−21×dij4−3.82234×10−9×dij2×logdij1.56143×106−1.01050×10−14×dij3×logdij1.56143×106−1.73646×10−21×dij4×logdij1.56143×106−32.88394.Table 2
Parameter setting.
ParameterValueKB45M4Γ12.5 dBh11044.5702 kmh21044.5702 kmc1.14932×106FDijc0.62269L10Tmax100P2c11.49445c21.49445R205 kmTable 3
gM for 24 cases of pairing schemes.
Paring schemeζ1=0.7ζ1=0.5ζ1=0.3ζ2=0.3ζ2=0.5ζ2=0.7T1R1, T2R2, T3R3, T4R40.155400.255000.35597T1R1, T2R2, T3R4, T4R30.164690.240260.32681T1R1, T2R3, T3R4, T4R20.133680.194920.26509T1R1, T2R3, T3R2, T4R40.150690.242290.33688T1R1, T2R4, T3R2, T4R30.155010.221660.29994T1R1, T2R4, T3R3, T4R20.127470.193740.26614T1R2, T2R1, T3R3, T4R40.098530.157540.21879T1R2, T2R1, T3R4, T4R30.113940.135840.17271T1R2, T2R4, T3R3, T4R10.111830.183080.25544T1R2, T2R4, T3R1, T4R30.117650.144360.18576T1R2, T2R3, T3R1, T4R40.098440.150160.20646T1R2, T2R3, T3R4, T4R10.118290.189770.26374T1R3, T2R1, T3R2, T4R40.108550.174870.24324T1R3, T2R1, T3R4, T4R20.104590.150330.20370T1R3, T2R2, T3R4, T4R10.140800.232390.32475T1R3, T2R2, T3R1, T4R40.126400.205940.28708T1R3, T2R4, T3R1, T4R20.136270.210100.28956T1R3, T2R4, T3R2, T4R10.129350.213100.29770T1R4, T2R3, T3R2, T4R10.148100.241480.33667T1R4, T2R3, T3R1, T4R20.113590.155520.20788T1R4, T2R1, T3R2, T4R30.127590.166160.21857T1R4, T2R1, T3R3, T4R20.098470.138390.18638T1R4, T2R2, T3R1, T4R30.139890.191680.25627T1R4, T2R2, T3R3, T4R10.153990.254570.35585Figure 6
The distance distribution function (equation (23)) associated with dij in SCFNR.Figure 7
The distributions of coverage level and detection probability withζ1=0.7, ζ2=0.3, and Γ=12.5dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)Figure 8
The distributions of coverage level and detection probability withζ1=0.5, ζ2=0.5, and Γ=12.5dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)


## 6.1.1. Orbital Elements in SCFNR

Suppose SCFNR is composed of 4 pairs of homogeneous bistatic radars. LetT=T1,T2,T3,T4, R=R1,R2,R3,R4,m=30km, and M=850km. According to the orbit design of cluster flight spacecraft proposed in [10], all near circular orbital elements of SCFNR are listed in Table 1.Table 1
All near circular orbital elements of SCFNR.
ParameterSemimajor axis (km)Eccentricity (deg)Inclination (deg)Argument of perigee (deg)True anomaly (deg)Right ascension of ascending node (deg)T17378.140.02350.000000.000000.00000T27378.140.02350.001631.139473.38820T37378.140.02350.000680.476301.82400T47378.140.0235359.997−2.16501.70983R17378.140.02350.001062.506022.50602R27378.140.02350.000220.153831.08022R37378.140.0235359.999−0.797634.02245R47378.140.02350.003442.39289−1.18115According to Table1, all orbital periods can be calculated and are approximated as 6310 seconds using STK, so we believe the orbital hyperperiods of the SCFNR are also 6310 s. In addition, as shown in Figure 5, we can also calculate all relative distances between transmitters and receivers in 172 days by STK. It is observed that the relative distance between any transmitter-receiver always remains below 850 km and above 30 km.Figure 5
The relative distance between any transmitter-receiver within 172 days.


## 6.1.2. The Target Grid and Other Parameters

Suppose that the longitude and latitude of surveillance regions are in the range [0, 0.07865345] (rad) and [0, 0.07865345] (rad), respectively. That is, surveillance region is set as square with the size of500×500km on the earth surface. The region with longitude and latitude can be divided into Nx×Ny grids. Let Nx=100 and Ny=100. So, according to coordinate transforming relations between spherical coordinates and rectangular coordinates, each grid can be computed in the ECI coordinate.For radar equation and PSO-HA, the parameters are listed in Table2. In this case, the distribution function of dij for SCFNR in equation (5) can be calculated, as presented in equation (23). At the same time, as shown in Figure 6, we give the curve of distribution function of dij for SCFNR. (23)FDijdij=0,dij<7.04878×105,F1dij,7.04878×105≤dij<1.04910×106,F2dij,1.04910×106≤dij≤1.56142×1061,otherwise,where(24)F1dij≈8.30169×10−9×dij2−7.33464×10−15×dij3−6.49148×10−21×dij4+3.82234×10−9×dij2×logdij7.04878×105+1.01050×10−14×dij3×logdij7.04878×105+1.73646×10−21×dij4×logdij7.04878×105+46.52592,F2dij≈−1.14225×10−8×dij2−5.97942×10−16×dij3+5.07373×10−21×dij4−3.82234×10−9×dij2×logdij1.56143×106−1.01050×10−14×dij3×logdij1.56143×106−1.73646×10−21×dij4×logdij1.56143×106−32.88394.Table 2
Parameter setting.
ParameterValueKB45M4Γ12.5 dBh11044.5702 kmh21044.5702 kmc1.14932×106FDijc0.62269L10Tmax100P2c11.49445c21.49445R205 kmTable 3
gM for 24 cases of pairing schemes.
Paring schemeζ1=0.7ζ1=0.5ζ1=0.3ζ2=0.3ζ2=0.5ζ2=0.7T1R1, T2R2, T3R3, T4R40.155400.255000.35597T1R1, T2R2, T3R4, T4R30.164690.240260.32681T1R1, T2R3, T3R4, T4R20.133680.194920.26509T1R1, T2R3, T3R2, T4R40.150690.242290.33688T1R1, T2R4, T3R2, T4R30.155010.221660.29994T1R1, T2R4, T3R3, T4R20.127470.193740.26614T1R2, T2R1, T3R3, T4R40.098530.157540.21879T1R2, T2R1, T3R4, T4R30.113940.135840.17271T1R2, T2R4, T3R3, T4R10.111830.183080.25544T1R2, T2R4, T3R1, T4R30.117650.144360.18576T1R2, T2R3, T3R1, T4R40.098440.150160.20646T1R2, T2R3, T3R4, T4R10.118290.189770.26374T1R3, T2R1, T3R2, T4R40.108550.174870.24324T1R3, T2R1, T3R4, T4R20.104590.150330.20370T1R3, T2R2, T3R4, T4R10.140800.232390.32475T1R3, T2R2, T3R1, T4R40.126400.205940.28708T1R3, T2R4, T3R1, T4R20.136270.210100.28956T1R3, T2R4, T3R2, T4R10.129350.213100.29770T1R4, T2R3, T3R2, T4R10.148100.241480.33667T1R4, T2R3, T3R1, T4R20.113590.155520.20788T1R4, T2R1, T3R2, T4R30.127590.166160.21857T1R4, T2R1, T3R3, T4R20.098470.138390.18638T1R4, T2R2, T3R1, T4R30.139890.191680.25627T1R4, T2R2, T3R3, T4R10.153990.254570.35585Figure 6
The distance distribution function (equation (23)) associated with dij in SCFNR.Figure 7
The distributions of coverage level and detection probability withζ1=0.7, ζ2=0.3, and Γ=12.5dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)Figure 8
The distributions of coverage level and detection probability withζ1=0.5, ζ2=0.5, and Γ=12.5dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)


## 6.2. Numerical Result and Analysis

### 6.2.1. Transmitter-Receiver Pairing Scheme

Case1:gM with different weight values in the slot 1 for the 1st orbital hyperperiod.Considering three conditions, i.e.,ζ1>ζ2, ζ1=ζ2, and ζ1<ζ2, we calculate the optimal pairing matrix in the same slot of its orbital hyperperiod under the same simulation environment as described in Section 6.1.Letζ1=0.7 and ζ2=0.3; the optimal pairing matrix is given as follows:(25)Mg=0100001010000001.Equation (25) indicates that optimal pairing scheme of bistatic radars is T1R2, T2R3, T3R1, T4R4, and gM=0.098349 with g1 = 0.06270 and g2 = 0.29371.Letζ1=0.5 and ζ2=0.5; the optimal pairing matrix is given as follows:(26)Mg=0100100000010010.Equation (26) indicates that optimal pairing scheme of bistatic radars is T1R2, T2R1, T3R4, T4R3, and gM=0.135841 with g1 = 0.1259 and g2 = 0.24075.Letζ1=0.3 and ζ2=0.7; the optimal pairing matrix is given as follows:(27)Mg=0100100000010010.Equation (27) indicates that the optimal pairing scheme of bistatic radars is T1R2, T2R1, T3R4, T4R3, and gM=0.172705 with g1 = 0.1259 and g2 = 0.24075.For the sake of comparison, Table3 lists all pairing schemes for 4 pairs of homogeneous bistatic radars, i.e., 24 cases of pairing schemes and corresponding gM using the same parameters.As shown in Table3, we mark the gM in bold font. It can be seen that, under conditions of same weight values, the corresponding paring schemes are consistent with equations (25) and (26), respectively. Therefore, PSO-HA is effective and reasonable.Case 2:gM with different weight values in the slots 1 and 2 of different orbital hyperperiods.Whenζ1=0.7 and ζ2=0.3, the optimal pairing scheme and corresponding gM can be listed in Table 4, and when ζ1=0.5 and ζ2=0.5, the optimal pairing scheme and corresponding gM can be listed in Table 5.Table 4
Optimal pairing scheme and correspondinggM with ζ1=0.7 and ζ2=0.3.
Orbital hyperperiodTime slotOptional pairing schemegMThe 2nd orbital hyperperiod1T1R2, T2R1, T3R3, T4R40.0635862T1R1, T2R2, T3R3, T4R40.139745The 3th orbital hyperperiod1T1R2, T2R1, T3R3, T4R40.0478432T1R3, T2R4, T3R1, T4R30.123073The 12th orbital hyperperiod1T1R2, T2R3, T3R4, T4R10.0097152T1R1, T2R4, T3R3, T4R20.198599The 16th orbital hyperperiod1T1R2, T2R3, T3R4, T4R10.0688672T1R1, T2R3, T3R4, T4R20.295543Table 5
Optimal pairing scheme and correspondinggM with ζ1=0.5 and ζ2=0.5.
Orbital hyperperiodTime slotOptional pairing schemegMThe 2nd orbital hyperperiod1T1R3, T2R1, T3R4, T4R20.0985362T1R2, T2R1, T3R3, T4R40.072687The 3th orbital hyperperiod1T1R3, T2R4, T3R2, T4R10.1561372T1R2, T2R4, T3R1, T4R30.156137The 12th orbital hyperperiod1T1R2, T2R3, T3R4, T4R10.0153812T1R1, T2R4, T3R3, T4R20.251995The 16th orbital hyperperiod1T1R4, T2R3, T3R1, T4R20.0905442T1R1, T2R3, T3R4, T4R20.380717As Tables4 and 5 show, the optimal pairing schemes are various in different slots of different orbital hyperperiods, andζ2=0.3 is different as well. From this result, we conclude that geometric topology of SCFNR with high dynamic and random leads to optimal pairing scheme with dynamic and random.


### 6.2.2. Coverage Level and Detection Probability

Case 1: whenΓ=12.5dB,Using the optimal pairing matrices given by equations (25)–(27), we calculate the corresponding distributions of coverage level and detection probability as shown in Figures 7 and 8. The coverage level and detection probability corresponding to equations (26) and (27) are the same due to the same optimal pairing matrices in equations (26) and (27). So, we only need to give the two distributions with ζ1=0.7, ζ2=0.3 and ζ1=0.5, ζ2=0.5, respectively.In Figures7 and 8, it is observed that either the distribution of coverage level or the distribution of detection probability is roughly the same. Besides, the higher the coverage level, the higher the detection probability.Case 2: whenΓ=12.0dB,In this case, keeping other parameters unchanged, the distributions of coverage level and detection probability are shown in Figures9–11.Figure 9
The distributions of coverage level and detection probability withζ1=0.7, ζ2=0.3, and Γ=12.0dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)Figure 10
The distributions of coverage level and detection probability withζ1=0.5, ζ2=0.5, and Γ=12.0dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)Figure 11
The distributions of coverage level and detection probability withζ1=0.3, ζ2=0.7, and Γ=12.0dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)In Figures9–11, it is observed that the three distributions of coverage level and detection probability are roughly the same with different weight values. However, there is considerable difference between the distributions with Γ=12.5dB and Γ=12.0dB, and we find that the SNR threshold has a great influence on coverage level and detection probability in SCFNR. We also observe that the smaller the threshold Γ, the larger the coverage level and detection probability. These show that the proposed PSO-HA, coverage, and detection probability model are reasonable and effective, especially for coverage and detection performance measured by distance function.In addition, using PSO-HA to solve optimal pairing matrix, considering three weight values, we give the relationship between iteration andgM in slot 1 for the 1st orbital hyperperiod. As shown in Figure 12, the iterative process has good convergence.Figure 12
The relationship between iteration and objective function with different weight values.
(a)(b)(c)


## 6.2.1. Transmitter-Receiver Pairing Scheme

Case1:gM with different weight values in the slot 1 for the 1st orbital hyperperiod.Considering three conditions, i.e.,ζ1>ζ2, ζ1=ζ2, and ζ1<ζ2, we calculate the optimal pairing matrix in the same slot of its orbital hyperperiod under the same simulation environment as described in Section 6.1.Letζ1=0.7 and ζ2=0.3; the optimal pairing matrix is given as follows:(25)Mg=0100001010000001.Equation (25) indicates that optimal pairing scheme of bistatic radars is T1R2, T2R3, T3R1, T4R4, and gM=0.098349 with g1 = 0.06270 and g2 = 0.29371.Letζ1=0.5 and ζ2=0.5; the optimal pairing matrix is given as follows:(26)Mg=0100100000010010.Equation (26) indicates that optimal pairing scheme of bistatic radars is T1R2, T2R1, T3R4, T4R3, and gM=0.135841 with g1 = 0.1259 and g2 = 0.24075.Letζ1=0.3 and ζ2=0.7; the optimal pairing matrix is given as follows:(27)Mg=0100100000010010.Equation (27) indicates that the optimal pairing scheme of bistatic radars is T1R2, T2R1, T3R4, T4R3, and gM=0.172705 with g1 = 0.1259 and g2 = 0.24075.For the sake of comparison, Table3 lists all pairing schemes for 4 pairs of homogeneous bistatic radars, i.e., 24 cases of pairing schemes and corresponding gM using the same parameters.As shown in Table3, we mark the gM in bold font. It can be seen that, under conditions of same weight values, the corresponding paring schemes are consistent with equations (25) and (26), respectively. Therefore, PSO-HA is effective and reasonable.Case 2:gM with different weight values in the slots 1 and 2 of different orbital hyperperiods.Whenζ1=0.7 and ζ2=0.3, the optimal pairing scheme and corresponding gM can be listed in Table 4, and when ζ1=0.5 and ζ2=0.5, the optimal pairing scheme and corresponding gM can be listed in Table 5.Table 4
Optimal pairing scheme and correspondinggM with ζ1=0.7 and ζ2=0.3.
Orbital hyperperiodTime slotOptional pairing schemegMThe 2nd orbital hyperperiod1T1R2, T2R1, T3R3, T4R40.0635862T1R1, T2R2, T3R3, T4R40.139745The 3th orbital hyperperiod1T1R2, T2R1, T3R3, T4R40.0478432T1R3, T2R4, T3R1, T4R30.123073The 12th orbital hyperperiod1T1R2, T2R3, T3R4, T4R10.0097152T1R1, T2R4, T3R3, T4R20.198599The 16th orbital hyperperiod1T1R2, T2R3, T3R4, T4R10.0688672T1R1, T2R3, T3R4, T4R20.295543Table 5
Optimal pairing scheme and correspondinggM with ζ1=0.5 and ζ2=0.5.
Orbital hyperperiodTime slotOptional pairing schemegMThe 2nd orbital hyperperiod1T1R3, T2R1, T3R4, T4R20.0985362T1R2, T2R1, T3R3, T4R40.072687The 3th orbital hyperperiod1T1R3, T2R4, T3R2, T4R10.1561372T1R2, T2R4, T3R1, T4R30.156137The 12th orbital hyperperiod1T1R2, T2R3, T3R4, T4R10.0153812T1R1, T2R4, T3R3, T4R20.251995The 16th orbital hyperperiod1T1R4, T2R3, T3R1, T4R20.0905442T1R1, T2R3, T3R4, T4R20.380717As Tables4 and 5 show, the optimal pairing schemes are various in different slots of different orbital hyperperiods, andζ2=0.3 is different as well. From this result, we conclude that geometric topology of SCFNR with high dynamic and random leads to optimal pairing scheme with dynamic and random.


## 6.2.2. Coverage Level and Detection Probability

Case 1: whenΓ=12.5dB,Using the optimal pairing matrices given by equations (25)–(27), we calculate the corresponding distributions of coverage level and detection probability as shown in Figures 7 and 8. The coverage level and detection probability corresponding to equations (26) and (27) are the same due to the same optimal pairing matrices in equations (26) and (27). So, we only need to give the two distributions with ζ1=0.7, ζ2=0.3 and ζ1=0.5, ζ2=0.5, respectively.In Figures7 and 8, it is observed that either the distribution of coverage level or the distribution of detection probability is roughly the same. Besides, the higher the coverage level, the higher the detection probability.Case 2: whenΓ=12.0dB,In this case, keeping other parameters unchanged, the distributions of coverage level and detection probability are shown in Figures9–11.Figure 9
The distributions of coverage level and detection probability withζ1=0.7, ζ2=0.3, and Γ=12.0dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)Figure 10
The distributions of coverage level and detection probability withζ1=0.5, ζ2=0.5, and Γ=12.0dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)Figure 11
The distributions of coverage level and detection probability withζ1=0.3, ζ2=0.7, and Γ=12.0dB in grids. (a)wgxgy. (b)pgxgy.
(a)(b)In Figures9–11, it is observed that the three distributions of coverage level and detection probability are roughly the same with different weight values. However, there is considerable difference between the distributions with Γ=12.5dB and Γ=12.0dB, and we find that the SNR threshold has a great influence on coverage level and detection probability in SCFNR. We also observe that the smaller the threshold Γ, the larger the coverage level and detection probability. These show that the proposed PSO-HA, coverage, and detection probability model are reasonable and effective, especially for coverage and detection performance measured by distance function.In addition, using PSO-HA to solve optimal pairing matrix, considering three weight values, we give the relationship between iteration andgM in slot 1 for the 1st orbital hyperperiod. As shown in Figure 12, the iterative process has good convergence.Figure 12
The relationship between iteration and objective function with different weight values.
(a)(b)(c)


## 7. Conclusions

In this paper, we study the optimization problem of the transmitter-receiver pairing of SCFNR for area coverage and target detection. Firstly, the novel concept of SCFNR integrated cluster flight spacecraft with netted radar is proposed. By establishing the mobility model for bistatic radar pair with twin-satellite mode, we have derived the radar-target distance distribution function and radar-target distance product distribution function with geometric probability method. Secondly, the radar-target distance distribution function and radar-target distance product distribution function with geometric probability method are proposed; we present the 0-1 grid coverage matrix for the bistatic radar and the transmitter-receiver paring matrix for SCFNR using the radar equation and the radar-target distance distribution function. Next, we describe the optimal problem of transmitter-receiver pairing of SCFNR for area coverage and target detection by definingK-grid coverage matrix. Finally, we propose new PSO-HA for the problem. We validate the effectiveness and reasonability of the proposed algorithm through numerical analysis. The numerical results can also be concluded as follows:(1)
Geometric topology of SCFNR with characteristic great dynamic and random leads to optimal paring scheme with characteristic time varying and random;(2)
When the paring matrix is optimal, the coverage level and detection probability with different weigh values had a slight change;(3)
SNR threshold had a great discernible impact on coverage and detection. In other words, the smaller the SNR threshold, the more achievable its coverage and detection will be.In the future, to further develop the theory and application of SCFNR, we will study problems on target detection, localization, and tracking.



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*Source: 8863000-2021-03-10.xml*