# Research on Transmission Capacity Characteristics of the Cluster Flight Spacecraft Network

Authors: Li Su; Jian Wu; Jinrong Mo
Journal: International Journal of Antennas and Propagation
 (2021)

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## Abstract

In order to analyze the transmission capacity performance of the cluster flight spacecraft network, there are two different types of outage performance theory which are derived in this paper. First of all, by applying the mean value theorem of integrals, the expression of the outage probability of decode-and-forward relaying is derived. Subsequently, according to the Macdonald random variable form, the expression of the outage probability of amplify-and-forward is derived. By simulating the transmission capacity of decode-and-forward, the transmission capacity characteristics of a single hop and dual hops are analyzed. The simulation results showed that transmission capacity performance changes with the change of the time slot in the orbital hyperperiod, and the transmission capacity of a dual-hop relay has better performance than a single-hop transmission in the cluster flight spacecraft network.


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## Body

## 1. Introduction

In recent years, fractionated spacecraft with the cluster flight model has become a hot topic in the field of distributed space network due to its advantages of flexibility, rapid response, low cost, strong scalability, and long lifetime. The previous studies have made contribution to earth observation and space exploration [1–3]. In those studies, fractionated spacecraft distributes the functionality of a traditional large monolithic spacecraft into a number of heterogeneous modules. Each module can be regarded as a node through wireless communication, and the nodes construct the cluster flight spacecraft network (CFSN) [4–6]. However, cluster flight spacecrafts require mutual cooperation between nodes to realize information exchange [7, 8], navigation communication, and power sharing. These spacecrafts constitute a virtual satellite platform with information exchange structure [9–11]. In addition, like other distributed space systems, the cluster flight spacecraft is a resource-sharing and energy-limited system [12]. Therefore, under the condition of limited transmit power, the study on the transmission capacity performance of the CFSN is an important issue.The outage rate between nodes is closely related to the transmission capacity and the connection status of the nodes. Using transmission capacity as a wireless network performance metric, it has the following advantages: (a) in some important scenarios or in other strictly bounded conditions, transmission capacity can be deduced accurately. (b) The performance that depends on the basic network parameters is explained. (c) Design insights can be obtained from these performance expressions. For the study of network transmission capacity, after the innovative definition of distributed network capacity by Gupta, it has been extensively studied by several scholars [13]. For example, transmission capacity is used to measure the parameters of distributed network capacity, which was redefined by Weber et al. [14], that is, the number of successful communications that the network can achieve per unit area when the outage probability of communication is limited. Random geometry theory is used to model the network and study the transmission capacity, and it is widely recognized in industrial production. However, there are some shortcomings. In the model of Weber et al. [14], an outage receiver can be randomly selected to transmit information within the maximum relay distance of any transmitter. The problem is that the choice is not directional, and the information cannot be guaranteed to reach the final receiver [15]. Therefore, a new model, which is trying to solve this problem, was defined by Andrews et al. [16]. The destination receiver is specified at the limited distance from the transmitter, and fixed relay receivers with equal intervals are used for auxiliary information transmission. Meanwhile, random access transmission capacity is defined to measure the size of network capacity [16]. According to the random access transmission capacity and under the condition of hop length between the source and the destination on the line with equidistance, a simple upper-bound closed solution is derived based on key network parameters when the number of retransmissions is unlimited [17]. Taking outage constraints into consideration, the transmission capacity of dual-hop relay in wireless ad hoc networks was studied by Lee et al. Numerical simulation results show that the transmission capacity of the dual-hop relay is better than the direct transmission considering both noise and interference [18].The relative static environment is used for the above network. However, in the actual network, the distribution of nodes is random. Due to the high-speed movement of nodes, the topology is constantly changing in the CFSN. Only the determined network topology can be used for the study of network transmission capacity performance, and the network transmission capacity can be determined [19]. The topology of the network will be determined within a time slot in an orbital hyperperiod, and dynamic network topology is changed periodically. The research on the transmission capacity characteristics of the CFSN is carried out, which is based on the conditions determined by the network topology periodically [19, 20].


## 2. Transmission Capacity Analysis

### 2.1. Earth-Centered Inertial Coordinate System

Earth-centered inertial (ECI) coordinate system is defined in the following standard manner: the fundamental plane is the equatorial plane, thex-axis points towards the vernal equinox, the z-axis points towards the geographic north pole, and y=z×x, as shown in Figure 1. The vector α of classical orbital elements in the ECI coordinate, which describes natural orbits of the cluster flight spacecraft, is defined as(1)α≜a,e,β,ω,f,ΩT,where a is the semimajor axis, e is the eccentricity, β is the inclination, ω is the argument of perigee, f is the true anomaly, and Ω is the right ascension of the ascending node (RAAN).Figure 1
The schematic of the orbital elements in the ECI coordinate.If the vectorr=x,y,zT denotes the position of any satellite in the CFSN in the ECI coordinate, v=dr/dt is the velocity, and η=x2+y2 is the equatorial projection of the position vector, as well as the maximal and minimal equatorial projections of the position vector of the satellite at time t, given by ηmax=maxtηt and ηmin=mintηt, where ηt=x2t+y2t.


### 2.2. Nodal Mobility Model

The mobility modelMt for the CFSN can be defined as follows [19].Definition 1.
In ECI coordinates, if the position sets ofN nodes in the CFSN are R0=r10,r20,…,rN0 at initial time T0, the position set is Rk=r1k,r2k,…,rNk, and the positions are uniformly distributed within sphere Bri0,ai=1,2,…,N at time Tk, where ri0 and M−m/4 are the center and radius of the sphere, respectively, in Figure 2. Moreover, positions among all nodes are mutually independent and independent of all previous locations.
For the CFSN, the topological graph is denoted byGV,E, where V=v1,v2,…,vN denotes the node set.Figure 2
Distribution area of node positions of the CFSN.


### 2.3. The Definition of Transmission Capacity

Without loss of generality, transmission capacity reflects the number of node connections, which is associated with the spatial intensity, spectral efficiency, and outage probability. For the CFSN, as the number of nodes is limited, it cannot be characterized by spatial intensity. Therefore, the following definition is given.Definition 2.
For a given CFSN consisting ofN nodes, if the outage of the node vi is εi, the transmission rate is Rbi, and the access channel bandwidth is Bi, the transmission capacity of the node vi is(2)Ci=RiBbi1−εibpsHz,i=1,…,N.
Therefore, the transmission capacity of the network isC=∑i=1NCi.
Note: the transmission rate and access channel bandwidth in Definition2 should satisfy the constraints of Shannon theory.


## 2.1. Earth-Centered Inertial Coordinate System

Earth-centered inertial (ECI) coordinate system is defined in the following standard manner: the fundamental plane is the equatorial plane, thex-axis points towards the vernal equinox, the z-axis points towards the geographic north pole, and y=z×x, as shown in Figure 1. The vector α of classical orbital elements in the ECI coordinate, which describes natural orbits of the cluster flight spacecraft, is defined as(1)α≜a,e,β,ω,f,ΩT,where a is the semimajor axis, e is the eccentricity, β is the inclination, ω is the argument of perigee, f is the true anomaly, and Ω is the right ascension of the ascending node (RAAN).Figure 1
The schematic of the orbital elements in the ECI coordinate.If the vectorr=x,y,zT denotes the position of any satellite in the CFSN in the ECI coordinate, v=dr/dt is the velocity, and η=x2+y2 is the equatorial projection of the position vector, as well as the maximal and minimal equatorial projections of the position vector of the satellite at time t, given by ηmax=maxtηt and ηmin=mintηt, where ηt=x2t+y2t.


## 2.2. Nodal Mobility Model

The mobility modelMt for the CFSN can be defined as follows [19].Definition 1.
In ECI coordinates, if the position sets ofN nodes in the CFSN are R0=r10,r20,…,rN0 at initial time T0, the position set is Rk=r1k,r2k,…,rNk, and the positions are uniformly distributed within sphere Bri0,ai=1,2,…,N at time Tk, where ri0 and M−m/4 are the center and radius of the sphere, respectively, in Figure 2. Moreover, positions among all nodes are mutually independent and independent of all previous locations.
For the CFSN, the topological graph is denoted byGV,E, where V=v1,v2,…,vN denotes the node set.Figure 2
Distribution area of node positions of the CFSN.


## 2.3. The Definition of Transmission Capacity

Without loss of generality, transmission capacity reflects the number of node connections, which is associated with the spatial intensity, spectral efficiency, and outage probability. For the CFSN, as the number of nodes is limited, it cannot be characterized by spatial intensity. Therefore, the following definition is given.Definition 2.
For a given CFSN consisting ofN nodes, if the outage of the node vi is εi, the transmission rate is Rbi, and the access channel bandwidth is Bi, the transmission capacity of the node vi is(2)Ci=RiBbi1−εibpsHz,i=1,…,N.
Therefore, the transmission capacity of the network isC=∑i=1NCi.
Note: the transmission rate and access channel bandwidth in Definition2 should satisfy the constraints of Shannon theory.


## 3. Outage Performance Analysis

We consider a dual-hop system that consists of source, relay, and destination nodes in the CFSN. The dual hop is actually the process from the source node to the destination node through a relaying node. In general, relaying includes decode-and-forward and amplify-and-forward.


### 3.1. Decode-and-Forward Relaying

In a dual-hop network, relay nodes receive the signal transmitted from source nodes which can be considered as the first hop. For a CFSN consisting ofN nodes, the source node is vs, the relay node is vr, and other nodes are regarded as interferers. We consider interferers in a finite area with specific radius because nodes located far from a receiver cannot be an interferer. In the first hop, relay nodes receive the signal transmitted from source nodes, and the received signal yr can be expressed as(3)yr=hsrSs+∑k≠s,rNgkrSk+nr,where hsr is the channel gain between the source and the relay, Ss is the transmitted symbol of the source node with ESs2=Ps, and nr is an additive white Gaussian noise with an average power of σr2. In addition, gkr is the channel gain from the other node (interferer) to the relay, and average transmit power of an interferer is ESk2=Pk.Therefore, the received SINR at the relay can be written as(4)γr=Psdsr−2∑k≠s,rNPkdkr−2+σr2,where Ps is the transmit power of the source node, dsr is the distance between the source and the relay, and dkr is the distance between the interferer and the relay.For the second hop, that is, from the relay to the destination node, destination nodes receive the signal transmitted from relay nodes, and the received signal can be expressed as(5)yd=hrdTyr+∑l≠d,rNgldWl+nd,where Tyr is the re-encoded message, hrd is the channel gain between the relay and the destination, gld is the channel gain from interferer nodes to the destination, Wl is the interference suffered from the interferer vl, each average transmit power of an interferer vl is EWl2=Pl, and nd is an additive white Gaussian noise with an average power of σd2.Similar to equation (4), the received SINR at the destination can be written as(6)γd=Prdrd−2∑l≠d,rNPldld−2+σd2,where Pr is the transmit power of the relay node, drd is the distance between the relay and the destination, and dld is the distance between the interferer and the destination.As can be seen from [14], the distance between nodes in equations (4) and (6) is Gaussian distribution, so the distribution of γr and γd is still very complex.Since the distance between nodes is bounded in the CFSN, from the perspective of improving the robustness, we consider the case with the maximum interference, that is,dkr−2=dld−2=m−2. Therefore, equations (4) and (6) can be further changed into(7)γrm=Psdsr−2∑k≠s,rNPkm−2+σr2,γdm=Prdrd−2∑l≠d,rNPlm−2+σd2.According to the empirical statistical method described in the literature [21], the empirical distribution function of the distance between nodes is solved [22]. If the distance distribution between nodes is fdd, then(8)fdd=ae−d−b2/c2,d≥0.Therefore, by using the mean value theorem of integrals, the following theorem can be obtained.Theorem 1.
For the decode-and-forward dual-hop CFSN, when the interference is maximum, the cumulative distribution function and the probability density function of the received SINR at the relayX∼γrm are given by(9)FXx=1−aPsm−2∑k≠s,rNPk+σr2x−me−ξ1−b2/c2,(10)fXx=a2PSm−2∑k≠s,rNPk+σr2⋅1x5e−ξ1−b2/c2,where m≤ξ1≤Ps/m−2∑k≠s,rNPk+σr2⋅1/x≤M.Proof.
(11)FXx=PrSINR<x=PrPsd−2m−2∑k≠s,rNPk+σr2<x=PrPsd−2<m−2∑k≠s,rNPk+σr2x=PrPsd2<m−2∑k≠s,rNPk+σr2x=Prd2>Psm−2∑k≠s,rNPk+σr2x=1−Prd<Psm−2∑k≠s,rNPk+σr2x=1−∫−∞Ps/m−2∑k≠s,rNPk+σr2xfdddd=1−∫mPs/m−2∑k≠s,rNPk+σr2xfdddd.
According to the mean value theorem of integrals, we can obtain(12)FXx=1−aPsm−2∑k≠s,rNPk+σr2x−me−ξ1−b2/c2.
The derivation of the cumulative distribution functionFXx with respect to x is proved.
Similar to the derivation of Theorem1, the following theorem can be obtained.Theorem 2.
For the decode-and-forward dual-hop CFSN, when the interference is maximum, the cumulative distribution function and the probability density function of the received SINR at the destinationY∼γdm can be written as(13)FYy=1−aPrm−2∑l≠d,rNPl+σd2y−me−ξ2−b2/c2,fYy=a2Prm−2∑l≠d,rNPl+σd2⋅1y5e−ξ2−b2/c2,where m≤ξ2≤Pr/m−2∑l≠d,rNPl+σd2⋅1/y≤M.Therefore, according to Theorems1 and 2, when the interference is maximum, the end-to-end SINR of dual-hop decode-and-forward relaying in the CFSN can be written as(14)γsd=minγrm,γdm.Then, the outage probability can be represented as(15)Poutagedβ=FXβ+FYβ−FXβFYβ=2−aPsm−2∑k≠s,rNPk+σr2β−me−ξ1−b2/c2+Prm−2∑l≠d,rNPl+σd2β−me−ξ2−b2/c2−1−aPsm−2∑k≠s,rNPk+σr2β−me−ξ1−b2/c21−aPrm−2∑l≠d,rNPl+σd2β−me−ξ2−b2/c2=1−a2Psm−2∑k≠s,rNPk+σr2β−mPrm−2∑l≠d,rNPl+σd2β−me−ξ1−b2/c2e−ξ2−b2/c2,where β is the threshold between nodes communicated successfully.


### 3.2. Amplify-and-Forward Relaying

In the first hop, the received signal at the amplify-and-forward relaying is identical to (4). If the amplification factor of the relay node is assumed to be Grd, expressed as(16)Grd=Pryr2=PrPsdsr−2+∑k≠s,rNPkdkr−2+σr2,then the received signal of amplify-and-forward relaying at the destination can be written as(17)yda=hrdGrdhsrSs+∑k≠s,rgkrSk+nr+∑l≠d,rgldWl+nd=hrdhsrGrdSs+hrdGrd∑k≠s,rgkrSk+hrdGrdnr+∑l≠d,rgldWl+nd.End-to-end SINR at the destination of amplify-and-forward relaying can be given by(18)γda=Psdsr−2drd−2Grd2drd−2Grd2∑k≠s,rPkdkr−2+∑l≠d,rPldld−2+drd−2Grd2σr2+σd2.As for the type of amplify-and-forward relaying, we also consider the case with the maximum interference, that is,dkr−2=dld−2=m−2. Then, end-to-end SINR of amplify-and-forward relaying can be written as(19)γdma=Psdsr−2drd−2Grd2drd−2Grd2∑k≠s,rPkm−2+∑l≠d,rPlm−2+drd−2Grd2σr2+σd2.We rearrange the end-to-end SINR (19) to make the Macdonald random variable form as [23](20)γdma=Psdsr−2drd−2⋅Pr/Psdsr−2+∑k≠s,rNPkdkr−2+σr2m−4⋅Pr/Psdsr−2+∑k≠s,rNPkdkr−2+σr2⋅∑k≠s,rPk+∑l≠d,rPlm−2+m−2Pr/Psdsr−2+∑k≠s,rNPkdkr−2+σr2⋅σr2+σd2=Psdsr−2Prdrd−2∑k≠s,rNPkm−2+Psdsr−2∑l≠d,rNPlm−2+∑k≠s,rNPkm−2∑l≠d,rNPlm−2+σr2∑l≠d,rNPlm−2⋯⋅Prdrd−2+σr2Prdrd−2+σd2Psdsr−2+σd2∑k≠s,rNPkdkr−2+σr2σd2=Psdsr−2Psdsr−2Prdrd−2+σd2+∑l≠d,rNPlm−2Psdsr−2+σr2+Psdsr−2+σr2Prdrd−2+σd2⋯⋅Prdrd−2−Psdsr−2Prdrd−2+∑k≠s,rNPkm−2∑l≠d,rNPlm−2=Psdsr−2⋅Prdrd−2∑l≠d,rNPlm−2Psdsr−2+∑k≠s,rNPkm−2+σr2+Psdsr−2+∑k≠s,rNPkm−2+σr2Prdrd−2+σd2−Psdsr−2Prdrd−2=Psdsr−2⋅Prdrd−2Psdsr−2+∑k≠s,rNPkm−2+σr2Prdrd−2+∑l≠d,rNPlm−2+σd2−Psdsr−2Prdrd−2=1Psdsr−2+∑k≠s,rNPkm−2+σr2/Psdsr−2Prdrd−2+∑l≠d,rNPlm−2+σd2/Prdrd−2−1=1∑k≠s,rNPkm−2+σr2/Psdsr−2+∑l≠d,rNPlm−2+σd2/Prdrd−2+∑k≠s,rNPkm−2+σr2∑l≠d,rNPlm−2+σd2/Psdsr−2⋅Prdrd−2=Psdsr−2/∑k≠s,rNPkm−2+σr2⋅Prdrd−2/∑l≠d,rNPlm−2+σd2Psdsr−2/∑k≠s,rNPkm−2+σr2+Prdrd−2/∑l≠d,rNPlm−2+σd2+1.According to Section3.1, X∼Psdsr−2/∑k≠s,rNPkm−2+σr2 and Y∼Prdrd−2/∑l≠d,rNPlm−2+σd2. Hence, end-to-end SINR of amplify-and-forward relaying can be written as(21)γdma=XYX+Y+1.The random variablesX and Y were defined in decode-and-forward relaying, and the cumulative distribution function and the probability density function of them are the same as those in Theorems 1 and 2.Theorem 3.
(outage probability of amplify-and-forward relaying). For the amplify-and-forward dual-hop CFSN, when the interference is maximum, from the cumulative distribution function and the probability density function ofX and Y, we can compute the outage probability of amplify-and-forward relaying as(22)Poutrageaβ=1+ame−ξ2−b2/c2+a2mABD8⋅β4−a2ABCD2⋅∫β∞y−ββy5y+1dy,where A=Pr/m−2∑l≠d,rNPl+σd2, B=e−ξ2−b2/c2, C=PS/m−2∑k≠s,rNPk+σr2, and D=e−ξ1−b2/c2.Proof.
According to the cumulative distribution function and the probability density function ofX and Y, we can compute the outage probability of amplify-and-forward relaying as(23)Poutrageaβ=PrXYX+Y+1<β=FYβ+∫β∞FXβy+1y−βfYydy=1−aBA1β−m+∫β∞1−aD⋅Cy−ββy+1−m⋅fYydy=1+amB−∫β∞aD⋅Cy−ββy+1−m⋅fYydy=1+amB−∫β∞a2ABD2⋅1y5⋅Cy−ββy+1−mdy=1+ame−ξ2−b2/c2+∫β∞a2mABD2⋅1y5dy−∫β∞a2ABCD2⋅y−ββy5y+1dy=1+ame−ξ2−b2/c2+a2mABD8⋅β4−a2ABCD2⋅∫z∞y−ββy5y+1dy,where A=Pr/m−2∑l≠d,rNPl+σd2, B=e−ξ2−b2/c2, C=PS/m−2∑k≠s,rNPk+σr2, and D=e−ξ1−b2/c2. Hence, we can obtain the outage probability of amplify-and-forward relaying in Theorem 3.


## 3.1. Decode-and-Forward Relaying

In a dual-hop network, relay nodes receive the signal transmitted from source nodes which can be considered as the first hop. For a CFSN consisting ofN nodes, the source node is vs, the relay node is vr, and other nodes are regarded as interferers. We consider interferers in a finite area with specific radius because nodes located far from a receiver cannot be an interferer. In the first hop, relay nodes receive the signal transmitted from source nodes, and the received signal yr can be expressed as(3)yr=hsrSs+∑k≠s,rNgkrSk+nr,where hsr is the channel gain between the source and the relay, Ss is the transmitted symbol of the source node with ESs2=Ps, and nr is an additive white Gaussian noise with an average power of σr2. In addition, gkr is the channel gain from the other node (interferer) to the relay, and average transmit power of an interferer is ESk2=Pk.Therefore, the received SINR at the relay can be written as(4)γr=Psdsr−2∑k≠s,rNPkdkr−2+σr2,where Ps is the transmit power of the source node, dsr is the distance between the source and the relay, and dkr is the distance between the interferer and the relay.For the second hop, that is, from the relay to the destination node, destination nodes receive the signal transmitted from relay nodes, and the received signal can be expressed as(5)yd=hrdTyr+∑l≠d,rNgldWl+nd,where Tyr is the re-encoded message, hrd is the channel gain between the relay and the destination, gld is the channel gain from interferer nodes to the destination, Wl is the interference suffered from the interferer vl, each average transmit power of an interferer vl is EWl2=Pl, and nd is an additive white Gaussian noise with an average power of σd2.Similar to equation (4), the received SINR at the destination can be written as(6)γd=Prdrd−2∑l≠d,rNPldld−2+σd2,where Pr is the transmit power of the relay node, drd is the distance between the relay and the destination, and dld is the distance between the interferer and the destination.As can be seen from [14], the distance between nodes in equations (4) and (6) is Gaussian distribution, so the distribution of γr and γd is still very complex.Since the distance between nodes is bounded in the CFSN, from the perspective of improving the robustness, we consider the case with the maximum interference, that is,dkr−2=dld−2=m−2. Therefore, equations (4) and (6) can be further changed into(7)γrm=Psdsr−2∑k≠s,rNPkm−2+σr2,γdm=Prdrd−2∑l≠d,rNPlm−2+σd2.According to the empirical statistical method described in the literature [21], the empirical distribution function of the distance between nodes is solved [22]. If the distance distribution between nodes is fdd, then(8)fdd=ae−d−b2/c2,d≥0.Therefore, by using the mean value theorem of integrals, the following theorem can be obtained.Theorem 1.
For the decode-and-forward dual-hop CFSN, when the interference is maximum, the cumulative distribution function and the probability density function of the received SINR at the relayX∼γrm are given by(9)FXx=1−aPsm−2∑k≠s,rNPk+σr2x−me−ξ1−b2/c2,(10)fXx=a2PSm−2∑k≠s,rNPk+σr2⋅1x5e−ξ1−b2/c2,where m≤ξ1≤Ps/m−2∑k≠s,rNPk+σr2⋅1/x≤M.Proof.
(11)FXx=PrSINR<x=PrPsd−2m−2∑k≠s,rNPk+σr2<x=PrPsd−2<m−2∑k≠s,rNPk+σr2x=PrPsd2<m−2∑k≠s,rNPk+σr2x=Prd2>Psm−2∑k≠s,rNPk+σr2x=1−Prd<Psm−2∑k≠s,rNPk+σr2x=1−∫−∞Ps/m−2∑k≠s,rNPk+σr2xfdddd=1−∫mPs/m−2∑k≠s,rNPk+σr2xfdddd.
According to the mean value theorem of integrals, we can obtain(12)FXx=1−aPsm−2∑k≠s,rNPk+σr2x−me−ξ1−b2/c2.
The derivation of the cumulative distribution functionFXx with respect to x is proved.
Similar to the derivation of Theorem1, the following theorem can be obtained.Theorem 2.
For the decode-and-forward dual-hop CFSN, when the interference is maximum, the cumulative distribution function and the probability density function of the received SINR at the destinationY∼γdm can be written as(13)FYy=1−aPrm−2∑l≠d,rNPl+σd2y−me−ξ2−b2/c2,fYy=a2Prm−2∑l≠d,rNPl+σd2⋅1y5e−ξ2−b2/c2,where m≤ξ2≤Pr/m−2∑l≠d,rNPl+σd2⋅1/y≤M.Therefore, according to Theorems1 and 2, when the interference is maximum, the end-to-end SINR of dual-hop decode-and-forward relaying in the CFSN can be written as(14)γsd=minγrm,γdm.Then, the outage probability can be represented as(15)Poutagedβ=FXβ+FYβ−FXβFYβ=2−aPsm−2∑k≠s,rNPk+σr2β−me−ξ1−b2/c2+Prm−2∑l≠d,rNPl+σd2β−me−ξ2−b2/c2−1−aPsm−2∑k≠s,rNPk+σr2β−me−ξ1−b2/c21−aPrm−2∑l≠d,rNPl+σd2β−me−ξ2−b2/c2=1−a2Psm−2∑k≠s,rNPk+σr2β−mPrm−2∑l≠d,rNPl+σd2β−me−ξ1−b2/c2e−ξ2−b2/c2,where β is the threshold between nodes communicated successfully.


## 3.2. Amplify-and-Forward Relaying

In the first hop, the received signal at the amplify-and-forward relaying is identical to (4). If the amplification factor of the relay node is assumed to be Grd, expressed as(16)Grd=Pryr2=PrPsdsr−2+∑k≠s,rNPkdkr−2+σr2,then the received signal of amplify-and-forward relaying at the destination can be written as(17)yda=hrdGrdhsrSs+∑k≠s,rgkrSk+nr+∑l≠d,rgldWl+nd=hrdhsrGrdSs+hrdGrd∑k≠s,rgkrSk+hrdGrdnr+∑l≠d,rgldWl+nd.End-to-end SINR at the destination of amplify-and-forward relaying can be given by(18)γda=Psdsr−2drd−2Grd2drd−2Grd2∑k≠s,rPkdkr−2+∑l≠d,rPldld−2+drd−2Grd2σr2+σd2.As for the type of amplify-and-forward relaying, we also consider the case with the maximum interference, that is,dkr−2=dld−2=m−2. Then, end-to-end SINR of amplify-and-forward relaying can be written as(19)γdma=Psdsr−2drd−2Grd2drd−2Grd2∑k≠s,rPkm−2+∑l≠d,rPlm−2+drd−2Grd2σr2+σd2.We rearrange the end-to-end SINR (19) to make the Macdonald random variable form as [23](20)γdma=Psdsr−2drd−2⋅Pr/Psdsr−2+∑k≠s,rNPkdkr−2+σr2m−4⋅Pr/Psdsr−2+∑k≠s,rNPkdkr−2+σr2⋅∑k≠s,rPk+∑l≠d,rPlm−2+m−2Pr/Psdsr−2+∑k≠s,rNPkdkr−2+σr2⋅σr2+σd2=Psdsr−2Prdrd−2∑k≠s,rNPkm−2+Psdsr−2∑l≠d,rNPlm−2+∑k≠s,rNPkm−2∑l≠d,rNPlm−2+σr2∑l≠d,rNPlm−2⋯⋅Prdrd−2+σr2Prdrd−2+σd2Psdsr−2+σd2∑k≠s,rNPkdkr−2+σr2σd2=Psdsr−2Psdsr−2Prdrd−2+σd2+∑l≠d,rNPlm−2Psdsr−2+σr2+Psdsr−2+σr2Prdrd−2+σd2⋯⋅Prdrd−2−Psdsr−2Prdrd−2+∑k≠s,rNPkm−2∑l≠d,rNPlm−2=Psdsr−2⋅Prdrd−2∑l≠d,rNPlm−2Psdsr−2+∑k≠s,rNPkm−2+σr2+Psdsr−2+∑k≠s,rNPkm−2+σr2Prdrd−2+σd2−Psdsr−2Prdrd−2=Psdsr−2⋅Prdrd−2Psdsr−2+∑k≠s,rNPkm−2+σr2Prdrd−2+∑l≠d,rNPlm−2+σd2−Psdsr−2Prdrd−2=1Psdsr−2+∑k≠s,rNPkm−2+σr2/Psdsr−2Prdrd−2+∑l≠d,rNPlm−2+σd2/Prdrd−2−1=1∑k≠s,rNPkm−2+σr2/Psdsr−2+∑l≠d,rNPlm−2+σd2/Prdrd−2+∑k≠s,rNPkm−2+σr2∑l≠d,rNPlm−2+σd2/Psdsr−2⋅Prdrd−2=Psdsr−2/∑k≠s,rNPkm−2+σr2⋅Prdrd−2/∑l≠d,rNPlm−2+σd2Psdsr−2/∑k≠s,rNPkm−2+σr2+Prdrd−2/∑l≠d,rNPlm−2+σd2+1.According to Section3.1, X∼Psdsr−2/∑k≠s,rNPkm−2+σr2 and Y∼Prdrd−2/∑l≠d,rNPlm−2+σd2. Hence, end-to-end SINR of amplify-and-forward relaying can be written as(21)γdma=XYX+Y+1.The random variablesX and Y were defined in decode-and-forward relaying, and the cumulative distribution function and the probability density function of them are the same as those in Theorems 1 and 2.Theorem 3.
(outage probability of amplify-and-forward relaying). For the amplify-and-forward dual-hop CFSN, when the interference is maximum, from the cumulative distribution function and the probability density function ofX and Y, we can compute the outage probability of amplify-and-forward relaying as(22)Poutrageaβ=1+ame−ξ2−b2/c2+a2mABD8⋅β4−a2ABCD2⋅∫β∞y−ββy5y+1dy,where A=Pr/m−2∑l≠d,rNPl+σd2, B=e−ξ2−b2/c2, C=PS/m−2∑k≠s,rNPk+σr2, and D=e−ξ1−b2/c2.Proof.
According to the cumulative distribution function and the probability density function ofX and Y, we can compute the outage probability of amplify-and-forward relaying as(23)Poutrageaβ=PrXYX+Y+1<β=FYβ+∫β∞FXβy+1y−βfYydy=1−aBA1β−m+∫β∞1−aD⋅Cy−ββy+1−m⋅fYydy=1+amB−∫β∞aD⋅Cy−ββy+1−m⋅fYydy=1+amB−∫β∞a2ABD2⋅1y5⋅Cy−ββy+1−mdy=1+ame−ξ2−b2/c2+∫β∞a2mABD2⋅1y5dy−∫β∞a2ABCD2⋅y−ββy5y+1dy=1+ame−ξ2−b2/c2+a2mABD8⋅β4−a2ABCD2⋅∫z∞y−ββy5y+1dy,where A=Pr/m−2∑l≠d,rNPl+σd2, B=e−ξ2−b2/c2, C=PS/m−2∑k≠s,rNPk+σr2, and D=e−ξ1−b2/c2. Hence, we can obtain the outage probability of amplify-and-forward relaying in Theorem 3.


## 4. Theoretical Results of Transmission Capacity

In order to facilitate subsequent calculations, in this paper, the decode-and-forward outage expression is selected as the simulation model for the experiment. According to Definition2 and equation (10), when the interference is maximum, the transmission capacity of a single hop from node vi to the destination node is(24)Ciβ=aRbiBiPsim−2∑k≠siNPk+σri2β−me−ξi1−b2/c2.In the same way, the transmission capacity of dual hops after decoding, forwarding, and relaying is(25)Cirβ=aRbiBiPsm−2∑k≠s,rNPk+σr2β−mPrm−2∑l≠d,rNPl+σd2β−me−ξ1−b2/c2e−ξ2−b2/c2.


## 5. Numerical Simulation

Gaussian approximation and power constraints are based on orbital parameters and distance distribution between nodes in previous work [21, 24]. We assume that Bi=4kHz, Rbi=1kbps, Psi=Pk=Pr=Pl, and β=12.07. The noise power is 0.01 W. The relationship between transmission capacity vi and transmission power Psi of available nodes is shown in Figures 3 and 4. It can be seen from Figures 3 and 4 that as the transmission power of the node increases, the transmission capacity of the node is gradually increased. A comparison curve of the transmission capacity performance of a single node in the CFSN with a single hop and dual hops is given in Figure 4. It shows that, within a certain power range, the transmission capacity of the dual hops tends to be stable. And the transmission capacity of the dual-hop relay is better than the performance of a single-hop transmission in the CFSN.Figure 3
Relationship between transmission capacity and transmit power.Figure 4
Comparison of single-hop and dual-hop transmission capacity.Under the same parameters, the transmission capacity of different orbital hyperperiods of the CFSN and the distribution of the transmission power of different nodes are shown in Figures5(a)–5(c).Figure 5
Time evolution characteristic of transmission capacity.
(a)(b)(c)As can be seen from Figures5(a)–5(c), the transmission capacity curves of the first orbital hyperperiod and the second orbital hyperperiod system are very close. This means that, as time evolves, stronger periodicity of the transmission capacity performance is shown, and according to these three figures, it can be seen that the transmission power has a greater impact on it. In conclusion, it is shown that, under the condition of power constraints, the greater the node’s transmit power, the better its transmission performance.


## 6. Conclusion

In this paper, in order to analyze the transmission capacity performance of the CFSN, the theoretical derivation of two different types of outage probability is given. Using the mean value theorem of integrals, the outage probability expression of the decode-and-forward relaying is obtained, and then the transmission performance is simulated and analyzed. The numerical results show that the transmission capacity of the node gradually increases with the increase of the transmit power of the node. Within a certain power range, the transmission capacity of dual hops tends to a stable value. In the CFSN, the transmission capacity performance exhibits strong periodicity, and the transmission capacity performance of the dual-hop relay is better than single-hop transmission.



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*Source: 8821423-2021-02-09.xml*