Coverage in Cooperative LEO Satellite Networks

Abstract

Low-earth orbit (LEO) satellite networks ignite global wireless connectivity. However, signal outages and co-channel interference limit the coverage in traditional LEO satellite networks where a user is served by a single satellite. This paper explores the possibility of satellite cooperation in the downlink transmissions. Using tools from stochastic geometry, we model and analyze the downlink coverage of a typical user with satellite cooperation under Nakagami fading channels. Moreover, we derive the joint distance distribution of cooperative LEO satellites to the typical user. Our model incorporates fading channels, cooperation among several satellites, satellites’ density and altitude, and co-channel interference. Extensive Monte Carlo simulations are performed to validate analytical results. Simulation and numerical results suggest that coverage with LEO satellites cooperation considerably exceeds coverage without cooperation. Moreover, there are optimal satellite density and satellite altitude that maximize the coverage probability, which gives valuable network design insights.

Full text

Journal of Communications and Information Networks, Vol.8, No.4, Dec. 2023 Research paper
Coverage in Cooperative LEO Satellite Networks
Bodong Shang, Xiangyu Li, Caiguo Li, Zhuhang Li
Abstract—Low-earth orbit (LEO) satellite networks
ignite global wireless connectivity. However, signal out-
ages and co-channel interference limit the coverage in
traditional LEO satellite networks where a user is served
by a single satellite. This paper explores the possibility of
satellite cooperation in the downlink transmissions. Using
tools from stochastic geometry, we model and analyze
the downlink coverage of a typical user with satellite
cooperation under Nakagami fading channels. Moreover,
we derive the joint distance distribution of cooperative
LEO satellites to the typical user. Our model incorporates
fading channels, cooperation among several satellites,
satellites’ density and altitude, and co-channel interfer-
ence. Extensive Monte Carlo simulations are performed
to validate analytical results. Simulation and numerical
results suggest that coverage with LEO satellites cooper-
ation considerably exceeds coverage without cooperation.
Moreover, there are optimal satellite density and satellite
altitude that maximize the coverage probability, which
gives valuable network design insights.
Keywords—Low-earth orbit satellite, cooperative com-
munications, coverage probability, satellite-terrestrial net-
works, non-terrestrial networks
I. INTRODUCTION
One of the fundamental goals for sixth generation (6G)
networks is a radical increase in global coverage[1]. Low-
earth orbit (LEO) satellites, which establish a constellation,
are expected to be densely deployed. Such non-terrestrial net-
work (NTN) technology makes a paradigm shift in wireless
connectivity where users on the earth’s surface could directly
access the Internet through LEO satellites[2-4].
NTN has attracted a lot of attention in academia and in-
dustry. In the Release 14 stage, 3rd Generation Partner-
ship Project (3GPP) proposes satellite communication re-
quirements and application prospects as a fifth generation
Manuscript received Sep. 19, 2023; revised Oct. 18, 2023; accepted
Nov. 01, 2023. The associate editor coordinating the review of this paper
and approving it for publication was S. Zhou.
B. D. Shang, X. Y. Li, C. G. Li, Z. H. Li. Eastern Institute for Ad-
vanced Study, Eastern Institute of Technology, Ningbo 315200, China (e-
mail: bdshang@eitech.edu.cn; xyli@eitech.edu.cn; licaiguo2023@163.com;
jdhdyxj@163.com)).
(5G) access method. Technical discussions on NTN were
held in Release 15[5] and Release 16[6], and standard proto-
col modifications began in Release 17[7-8]. The development
of global LEO satellites continues to be hot, and many par-
ties have participated in the wave of satellite networking, in-
cluding SpaceX Starlink, Amazon Kuiper, OneWeb, Telesat,
and other companies that have successively planned satellite
launch plans for various applications in NTN.
Satellite systems have been studied in the past. In Ref. [9],
the authors investigated downlink coverage and rate in the
LEO satellite constellation based on the binomial point pro-
cess (BPP). However, the Rayleigh fading channels were as-
sumed in satellite-terrestrial links, which are not applicable
in practical systems but tractable in analysis. In Ref. [10],
a Poisson point process (PPP) model was introduced to an-
alyze the coverage of the LEO satellite networks. However,
the authors did not consider satellite cooperation by assuming
that users on the earth have access to their nearest satellite,
which facilitates the Laplace transform in coverage analysis.
In Ref. [11], cooperation between satellite and aerial relay-
ing links was considered, where a satellite and an unmanned
aerial vehicle (UAV) assist a group of other UAVs to forward
their data to a remote destination. However, Ref. [11] only
considered a single satellite, and the interference from other
satellites was not incorporated. Such a single satellite model
was also used in Ref. [12], where the authors derived outage
probability/coverage probability and symbol error rate over
satellite communication downlink channels when the users
are randomly located in single beam and multibeam areas. In
Ref. [13], the authors analyzed the satellite-to-airplane com-
munication in the Terahertz band by assuming that the air-
plane connects to its nearest satellite. It is worth noting that
the above works did not model and analyze cooperative LEO
satellites in downlink joint transmissions.
In the research conducted by the author in Ref. [14], a
comprehensive exploration was undertaken in the context of a
LEO downlink satellite-terrestrial millimeter wave (mmWave)
decode-and-forward relay network. The study’s conceptual
framework assumed a PPP distribution for the terrestrial re-
lay positions. The investigator utilized a sophisticated meta-
distribution methodology to meticulously extract and analyze
the coverage performance of the network. In the work pre-
sented in Ref. [15], the authors examined a complex network
configuration involving a single satellite, multiple terrestrial
users, and an aerial relay. The spatial arrangement of these ter-
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330 Journal of Communications and Information Networks
restrial users was represented using a homogeneous Poisson
point process (HSPPP). Utilizing the shadowed-Rician (SR)
fading model, the investigation systematically derived cover-
age and outage probabilities for both the satellite and terres-
trial components. Furthermore, the study extended its focus to
optimizing the end-to-end energy efficiency. Ref. [16] inves-
tigated downlink communication systems in LEO satellites,
wherein multiple satellites are uniformly distributed across a
sphere at a specific altitude, following a homogeneous BPP.
The authors analytically derived the precise outage probabil-
ity, and its approximate expression was derived through the
application of the Poisson limit theorem. The optimization
problem for system throughput was addressed using a pro-
posed iterative algorithm, resulting in near-optimal solutions.
Despite existing research effort[15] has addressed collabora-
tion between satellites and aerial platforms to optimize sig-
nal transmission efficiency, a notable gap persists in the litera-
ture concerning inter-satellite cooperation. Consequently, our
study endeavors to bridge this research gap by systematically
investigating and addressing this intricate facet.
Satellite cooperation is one of the ways to increase cov-
erage by enhancing the desired signal strength and reducing
co-channel interference. The densification of LEO satellites
facilitates satellite cooperation, especially for a user in remote
areas with many visible satellites. In situations marked by
extraordinary conditions, especially in remote regions char-
acterized by a sparse user population but abundant satellite
availability, the imperative to optimize the efficient utiliza-
tion of satellite communication resources becomes evident.
In pursuit of this objective, this paper introduces an innova-
tive model that harnesses multiple satellites for transmitting
identical data while employing a single-user signal reception
methodology, as shown in Fig. 1. Simultaneously, it is note-
worthy that typical users contend with interference from inter-
fering satellites. Consequently, we comprehensively analyze
the coverage probability within the model mentioned above.
The contributions of this paper are summarized as follows.
•Cooperative LEO satellites modeling: We introduce a
satellite cooperation system where several nearest satellites
jointly transmit data to the typical user. Moreover, we con-
sider the large-scale distance-dependent path-loss and small-
scale Nakagami-mfading in each serving and interfering link.
•Coverage probability analysis: We derive the coverage
probability under Nakagami-mfading channels by analyzing
the Laplace transform of interference and the desired signal
distribution. An approximated but tractable expression of cov-
erage probability is derived. Furthermore, the joint distance
distribution of the serving satellite and the expectation of in-
terference power are given. In addition, the coverage of two
and three cooperative satellites are given, respectively.
•Network design insights: Simulation results demonstrate
that coverage probability with satellite cooperation is signif-
LEO satellite
Interfering link
Desired link
User
Fig. 1 An illustration of cooperative LEO satellite networks
icantly improved compared to that without satellite coopera-
tion. Specifically, coverage can be enhanced by more than 100
percent when only two satellites work cooperatively. More-
over, there is an optimal combination of satellite altitude and
the number of satellites in cooperative LEO satellite networks
that maximize the coverage probability.
The paper is organized as follows. Section II presents the
system model including network model, channel model, and
signal model. Section III develops intermediate results that
will be used in coverage derivations. Section IV derives cov-
erage probability for a typical user in cooperative LEO satel-
lite networks. Simulation and numerical results are discussed
in section V. Section VI concludes the paper.
II. SY ST EM MO DE L
In this section, we introduce system model in cooperative
LEO satellite networks. In this system, we consider the LEO
satellites are coordinated via a central base station which is
deployed either on the ground or on one of LEO/MEO satel-
lites with more advanced computing capabilities and sufficient
power. The locations of the central base station can be known.
When the typical user confirms several nearest LEO satellites
for service, it will initiate a connection request to the cen-
tral base station. The central base station will then coordinate
these serving LEO satellites to simultaneously transmit sig-
nals to the typical user. Similar cooperative modes of LEO
satellites can be referred to Refs. [17-18].
A. Network Model
In our analysis, we begin with the foundational assumption
that both serving and interfering satellites are positioned on
the surface of a sphere, characterized by a radius denoted as
RS, shown in Fig. 2. The spatial distribution of serving satel-
lites is rigorously modeled using a homogeneous spatial Pois-
son point process (SPPP), wherein the density parameter gov-
erns the distribution. Consequently, the set ΦS={X1,··· ,X2}
represents the discrete locations of serving satellites. Here,
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Coverage in Cooperative LEO Satellite Networks 331
Earth
Satellite sphere
LEO satellite
RS
RE
Desired link Interfering link
N
r1
rN
Fig. 2 Geometry of cooperative LEO satellites
each for i∈Nis considered as an independent and uniformly
distributed point on the surface of the sphere. The variable sig-
nifies the number of serving satellites allocated to the typical
user k, following a Poisson distribution. As a result, |ΦS|=N.
Similarly, the quantity |ΦI|denotes the number of interfering
satellites that affect the typical user k. This parameter also
adheres to a Poisson distribution. Each satellite in the system
operates with a transmit power represented as P
T, while the
satellite transmit antenna gain is denoted by GT. Additionally,
it is essential to consider the geographical placement of a typ-
ical user on the earth’s surface, which is characterized by a
radius RE. Our model assumes that wireless transmissions to
the user exclusively originate from satellites positioned above
its local horizon.
We consider a typical user positioned at coordinates
(0,0,RE). As illustrated in Fig. 2, we define a typical spher-
ical cap Athat lies within the field of view at the location of
the typical user. For clarity, this spherical cap signifies the sec-
tion of the surface of the sphere with radius RSthat intersects
with a tangent plane to the earth. The center of this tangent
plane is located at coordinates (0,0,RE). The surface area of
this typical spherical cap can be written as
|A|=2π(RS−RE)RS.(1)
Moreover, we define a spherical cap Ariwith a distance ri
from the typical receiver’s location. This cap encompasses all
points located within a distance less than rifrom the typical
receiver’s position, the surface area of Arican be written as
|Ari|=2πRS−RE−(RS2−RE2)−r2
i
2RERS.(2)
We note that only the satellites on the typical spherical cap A
are visible to the typical user.
B. Channel Model
The wireless channel propagation is characterized by a
composite model encompassing path-loss attenuation and
small-scale fading. We employ the free-space path-loss model
to account for large-scale fading effects, defined as
β(ri) = β0r−α
i.(3)
In this equation, β0represents the path loss at a reference dis-
tance, risignifies the distance between the ith nearest satellite
and the typical user, and αdenotes the path loss exponent. We
adopt the Nakagami-mdistribution to introduce the element of
randomness associated with small-scale channel fading. Let
hidenote the fading coefficient between ith nearest satellite
and the typical user. We model hias hi∼Nakagami(m,Ω),
where |hi|2∼Gamma(k,θ), with k=m,θ=Ω/m. Under
the assumption of E[|hi|2] = 1, the probability density func-
tion (PDF) of hitakes the following form
f(hi) = 2mmh2m−1
i
Γ(m)Ωe−mh2
i
Ω,(4)
where Γ(m)represents the gamma function defined as Γ(m) =
R∞
0tm−1e−tdt. The Nakagami-mdistribution offers versatility
in modeling a wide range of small-scale fading phenomena.
Notably, it converges to the Rayleigh distribution for m=1
and to the Rician-Kdistributions for m= (K+1)2/(2K+1).
For Shadowed-Rician (SR) fading in satellite-terrestrial com-
munications, the SR fading can be approximated by a Nak-
agami distribution with appropriate shape and scale parame-
ters. The general expression for channel coefficients linking
satellites and the typical user can be succinctly written as
g(Xi) = qβ0r−α
ihi.(5)
C. Signal Model and SINR
We consider that a typical user is being served by Nnearest
satellites while simultaneously experiencing interference from
ΦIsatellites. Accordingly, the received signal at the typical
user located at (0,0,RE)is expressed as follows.
y=
N
∑
i=1qP
TGTβ0r−α
ihix+I+n0.(6)
Let xrepresent the transmitted signal originating from the
serving satellites to the typical user. The simultaneous trans-
mission of identical signals to the typical user by multiple col-
laborative satellites designates transmitting satellites as ‘serv-
ing satellites’. However, owing to the LEO satellite mega con-
stellation and LEO satellites near random distribution given a
time slot, within the observable region of the typical user, nu-
merous satellites concomitantly use the same channel for sig-
nal transmission to other users, termed ‘interfering satellites’,
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332 Journal of Communications and Information Networks
these entities contribute to the aggregated interference signal
power received by the typical user, so it is imperative to under-
take an analysis of co-channel interference for the typical user.
Let Idenote the interfering signal emanating from interfering
satellites to the typical user. The power of the interfering sig-
nal at the typical user is designated as
P
I=
|ΦI|
∑
i=1
P
TGTβ0r−α
I,i|hI,i|2,(7)
where rI,isignifies the distance between the ith interfering
satellite and the typical user, and hI,irepresents the small-scale
fading coefficient between the ith interfering satellite and the
typical user.
The signal-to-interference-plus-noise ratio (SINR) at the
typical user equipped with a single antenna is
SINR =∑N
i=1qP
TGTβ0r−α
ihi
2
∑|ΦI|
i=1P
TGTβ0r−α
I,i|hI,i|2+σ2
n0
,(8)
where σ2
n0denotes the variance of the noise. It is worth not-
ing that the ground user’s location and satellite ephemeris are
known, and thus the timing advance can be compensated in
cooperative transmission from multiple satellites. In addition,
the robust downlink synchronization performance can be pro-
vided by the synchronization signal block (SSB) design in
NTN[5-6].
D. Coverage Probability
The performance metric, in this paper, known as coverage
probability, is formally defined as follows
Pcov =P{SINR ⩾γth},(9)
where γth represents the minimum SINR required for success-
ful data transmission. In simpler terms, when the SINR of
the typical user, considering both its Nserving satellites and
|ΦI|interfering satellites, exceeds the threshold value γth, it
is considered to be within the coverage area of the satellite
communication network.
III. INT ER ME DI ATE RES ULTS
In this section, we derive some intermediate technical re-
sults that will be used in the calculation of coverage probabil-
ity in the sequel.
Lemma 1 The joint distance distribution of Nnearest serv-
ing LEO satellites is shown as follows.
fR(r) =
N
∏
i=1
2πλS
RS
RE
eπλS
RS
RE(R2
S−R2
E)eλS|Ari−1|
e2πλSRS(RS−RE)−eλS|Ari−1|rie−πλS
RS
REr2
i,(10)
where r= [r1,r2,··· ,rN], eλS|Ari−1|=e2πλSRS(RS−RE)×
eπλS
RS
RE[(R2
S−R2
E)−r2
i−1].
Proof We first derive the distribution of the two nearest
satellites, then derive the distribution of the Nclosest satel-
lites. The distribution of the closest satellites is given by
fR1(r1) = 


ν(λS,RS)re−λSπRS
REr2,Rmin ⩽r1⩽Rmax,
0,otherwise,
(11)
where
ν(λS,RS) = 2πλS
RS
RE
exp(λSπRS
RE(R2
S−R2
E))
exp(2λSπRS(RS−RE)) −1,
Rmin =RS−RE,Rmax =qR2
S−R2
E.
The joint distance distribution of two nearest satellites to the
typical user is expressed as
fR(r1,r2) = fR2|R1(r2|r1)fR1(r1),(12)
where fR2|R1(r2|r1)is the conditional distance distribution
of the second nearest satellite. The complementary cumula-
tive distribution function (CCDF) of the conditional r2condi-
tioned on r1is given by
P{R>r2|Φ(A/Ar1)>0}=
P{Φ(Ar2/Ar1) = 0|Φ(A/Ar1)>0}=
P{Φ(Ar2/Ar1) = 0,Φ(A/Ar1)>0}
{Φ(A/Ar1)>0}=
P{Φ(Ar2/Ar1) = 0}{Φ(A/Ar2)>0}
{Φ(A/Ar1)>0}=
P{Φ(Ar2/Ar1) = 0}(1− {Φ(A/Ar2) = 0})
{Φ(A/Ar1) = 0}=
exp(−λS|Ar2/Ar1|)−(1−exp(−λS|A/Ar2|))
1−exp(−λS|A/Ar1|)=
exp(−λS|Ar2/Ar1|)−exp(−λS|A/Ar1|)
1−exp(−λS|A/Ar1|),(13)
where A/Brepresents the area that Aexcludes B. The cumu-
lative distribution function (CDF) of r2conditioned on r1is
given by
FR2|Φ(A/Ar1)>0(r2|r1) =
1−P{R>r2|Φ(A/Ar1)>0}=
1−exp(−λS|Ar2/Ar1|)
1−exp(−λS|A/Ar1|)=
1−exp(−λS|Ar2|)exp(λS|Ar1|)
1−exp(−λS|A|)exp(λS|Ar1|)=
e4πλSRS(RS−RE)−e−πλS
RS
RE[r2
2+r2
1−2(R2
S−R2
E)]
e4πλSRS(RS−RE)−e−πλS
RS
RE[r2
1−(R2
S−R2
E)] ,(14)
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Coverage in Cooperative LEO Satellite Networks 333
and we have
|Ari|=2π(RS−RE−hri)RS,(15)
where hri=(R2
S−R2
E)−r2
i
2RE.
Then, the probability density function (PDF) of r2condi-
tioned on r1is obtained by calculating the derivation of its
CDF, as follows.
fR2|Φ(A/Ar1)>0(r2|r1) =
∂FR|Φ(A/Ar1)>0(r2|r1)
∂r=
∂
∂r2
1−e−2πλSRS(RS−RE)e−πλS
RS
RE[r2
2−(R2
S−R2
E)]eλS|Ar1|
1−e−2πλSRS(RS−RE)eλS|Ar1|=
2πλS
RS
RE
eπλS
RS
RE(R2
S−R2
E)eλS|Ar1|
e2πλSRS(RS−RE)−eλS|Ar1|r2e−πλS
RS
REr2
2.(16)
Finally, the joint distribution of r1and r2is obtained as fol-
lows.
fR(r1,r2) =
2πλS
RS
RE
eπλS
RS
RE(R2
S−R2
E)eλS|Ar1|
e2πλSRS(RS−RE)−eλS|Ar1|r2e−πλS
RS
REr2
2·
2πλS
RS
RE
eπλS
RS
RE(R2
S−R2
E)
e2πλSRS(RS−RE)−1r1e−λSπRS
REr2
1.(17)
Based on the above result, we can generalize to any number N
of cooperating LEO satellites. The joint distance distribution
of Nnearest LEO satellites is
fR(r) =
N
∏
i=1
2πλS
RS
RE
eπλS
RS
RE(R2
S−R2
E)eλS|Ari−1|
e2πλSRS(RS−RE)−eλS|Ari−1|rie−πλS
RS
REr2
i,
(18)
which completes the proof.
When the LEO satellites are densely deployed, the typical
user has a high probability to observe its serving satellites.
Therefore, we have a simplified expression of the joint dis-
tance distribution of Nnearest LEO satellites, as shown in
Lemma 2.
Lemma 2 In dense LEO satellite networks, the joint dis-
tance distribution of Nnearest serving LEO satellites is ap-
proximated by
fR(r)≈2πλS
RS
REN
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)·
e−πλS
RS
REr2
N
N
∏
i=1
ri.(19)
Proof When the LEO satellites become dense, the denomi-
nator in Lemma 1 approaches to one, and thus we have
fR(r)≈2πλS
RS
RE
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)N
·
N
∏
i=1
eλS|Ari−1|rie−πλS
RS
REr2
i.(20)
Lemma 2 is obtained from (20) with some mathematical ma-
nipulations.
IV. COVERAGE PRO BABILITY
Now we are in the position of deriving coverage probability
of cooperative LEO satellite networks.
A. Main Results
The coverage probability of a typical user is derived in the
following.
Pcov (γth;λS,RS,PT,GT,m,N) =
P{SINR ⩾γth}=
P



∑N
i=1qPTGTβ0r−α
ihi
2
∑|ΦI|
i=1PTGTβ0r−α
I,i|hI,i|2+σ2
n0
⩾γth



=
P(
N
∑
i=1qr−α
ihi
2
⩾γth
|ΦI|
∑
i=1
r−α
I,i|hI,i|2+γthσ2
n0
PTGTβ0).(21)
We derive the distribution of P
S=|∑N
i=1r−α/2
ihi|2which is
the left-hand side term of the last step of the above inequality.
Since hi∼Nakagami(m,Ω),we obtain an upper bound of P
S
shown as follows.
P
S=
N
∑
i=1
r−α
2
ihi
2
⩽
N
∑
i=1
r−α
i
N
∑
i=1
|hi|2=PUP
S,(22)
where ∑N
i=1|hi|2∼Γ(Nm,1/m),PUP
Sdenotes an upper bound
of P
S, and Γ(a,b)indicates the Gamma distribution with pa-
rameters aand b.
Theorem 1 The coverage probability for the typical user in
a downlink cooperative LEO satellite network where NLEO
satellites jointly transmit to it is given in (23) at the top of the
next page.
Proof The coverage probability can be reformulated as
P(PUP
S,Nak ⩾γth |ΦI|
∑
i=1
r−α
I,i|hI,i|2+σ2
n0
PTGTβ0!)=
P(N
∑
i=1
|hi|2⩾γth
∑N
i=1r−α
i |ΦI|
∑
i=1
r−α
I,i|hI,i|2+σ2
n0
PTGTβ0!).
(24)
Then, we derive the Laplace transform of the summation of
small-scale fading channel power gain, the Laplace transform
of interference and noise power.
By utilizing the Laplace transform, the coverage probabil-
ity is given in (25).
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334 Journal of Communications and Information Networks
Pcov ≈Z
rmin<r1<···<rN<rmax
fR(r)Z+∞
−∞













exp(−2πλSRS
REm−2
α2jπγth
∑N
i=1r−α
i
s2/αZ(2jπγths
m∑N
i=1r−α
i
)−2/αr2
max
(2jπγths
m∑N
i=1r−α
i
)−2/αr2
min
1−1
(1+u−2/α)mdu)
exp−2jπγth
∑N
i=1r−α
i
sσ2
n0
PTGTβ01
2jπs1
(1+−2jπs/m)Nm −1













dsdr
(23)
PPUP
S,Nak ⩾γth (INak +N)=
Z
rmin<r1<···<rN<rmax
fR(r)Z+∞
−∞
LINak 2jπγth
∑N
i=1r−α
i
sLN2jπγth
∑N
i=1r−α
i
sL∑N
i=1|hi|2(−2jπs)−1
2jπsdsdr=
Z
rmin<r1<···<rN<rmax
fR(r)Z+∞
−∞
LINak 2jπγth
∑N
i=1r−α
i
sexp −2jπγth
∑N
i=1r−α
i
sσ2
n0
PTGTβ0!1
2jπs 1
(1+−2jπs/m)Nm −1!dsdr,
(25)
where
INak =∑|ΦI|
i=1r−α
I,i|hI,i|2,N=σ2
n0
PTGTβ0
,LN(s) = (e−sσ2
n0
PTGTβ0)=e−sσ2
n0
PTGTβ0,L∑N
i=1|hi|2(s) = 1
(1+s/m)Nm .
In the following, we derive the Laplace transform of the
interference power in (26).
LINak (s) =
EΦI,h{exp(−sINak)}=
EΦI(|ΦI|
∏
i=1
hnexp−sr−α
I,i|hI,i|2o)=
EΦI(|ΦI|
∏
i=1
1
(1+sr−α
I,i/m)m)=
exp−λSZr∈AI
1−1
(1+sr−α/m)mdr=
exp−2πλS
RS
REZrmax
rmin 1−1
(1+sr−α/m)mrdr=
exp
−2πλS
RS
RE
m−2
αs2
αZ(s
m)−2/αr2
max
(s
m)−2/αr2
min
1−1
(1+u−2/α)mdu
,
(26)
where rmin =RS−RE,rmax =qR2
S−R2
E.
By substituting (26) to (25), we obtain the desired results,
which completes the proof.
In Theorem 1, we give an analytical expression of cover-
age probability for the typical user based on Laplace trans-
form. However, Theorem 1 involves multiple integrals which
reduces the analytical tractability. In the following Theorem,
we give an approximated coverage probability based on the
fading distribution which is simple and concise.
Theorem 2 An approximated coverage probability for the
typical user in a downlink cooperative LEO satellite network
where NLEO satellites jointly transmit to it is given by
Pcov ≈1−(2πλSRS
RE)N
Γ(Nm)·
expπλS
RS
RER2
S−R2
E−2πλSRS(RS−RE)·
Z
rmin<r1<···<rN<rmax
γ

Nm,γth m(e
INak +σ2
n0
PTGTβ0)
∑N
i=1r−α
i

∏N
i=1ri
eπλS
RS
REr2
N
dr,
(27)
where e
INak =Ω2πλSRS
RERrmax
rmin Rrmax
rNr1−αfRN(rN)drdrNis the
average interference power received at the typical user,
fRN(rN)is the marginal PDF of rNwhich is given by
fRN(rN) = Z
rmin<r1<···<rN−1<rmax
fR(r)dr=
Z
rmin<r1<···<rN−1<rmax
N
∏
i=1
2πλS
RS
RE
eπλS
RS
RE(R2
S−R2
E)eλS|Ari−1|
e2πλSRS(RS−RE)−eλS|Ari−1|rie−πλS
RS
REr2
idr.(28)
Proof The coverage probability for a typical user is derived
in (29). In the first step of (29), we use the upper bound of
the desired signal power under Nakagami-mfading channel,
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Coverage in Cooperative LEO Satellite Networks 335
Pcov (γth;λS,RS,PT,GT,m,N)
P⩽(N
∑
i=1
r−α
i
N
∑
i=1
|hi|2⩾γth
|ΦI|
∑
i=1
r−α
I,i|hI,i|2+γthσ2
n0
PTGTβ0)=
P(N
∑
i=1
|hi|2⩾γth
∑N
i=1r−α
i |ΦI|
∑
i=1
r−α
I,i|hI,i|2+σ2
n0
PTGTβ0!)(a)
=
1−EN,I(1
Γ(Nm)γ N m,γthm1
∑N
i=1r−α
i |ΦI|
∑
i=1
r−α
I,i|hI,i|2+σ2
n0
PTGTβ0!!)(b)
=
1−1
Γ(Nm)Z
rmin<r1<···<rN<rmax
EI(γ Nm,γth m1
∑N
i=1r−α
i |ΦI|
∑
i=1
r−α
I,i|hI,i|2+σ2
n0
PTGTβ0!!)fR(r)dr
(c)
≈
1−1
Γ(Nm)Z
rmin<r1<···<rN<rmax
γ Nm,γth m1
∑N
i=1r−α
i e
INak +σ2
n0
PTGTβ0!!fR(r)dr(29)
where, in (a), we use the incomplete Gamma function to rep-
resent the distribution of the sum of small-scale channel power
gain, in (b), we use the joint distance distribution function
which is obtained in Lemma 1 and Lemma 2 to average the
distance between serving satellites and the typical user, in (c),
we calculate the average interference directly to achieve an
approximated coverage probability.
Note that the average interference power at the typical user
is obtained by using Campell theorem and averaging on rN,
which is shown as follows.
e
INak =P(|ΦI|
∑
i=1
r−α
I,i|hI,i|2)=Pn|hI,i|2o(|ΦI|
∑
i=1
r−α
I,i)=
ΩPrNλS
∂|Ar|
∂rZrmax
rN
r−αdr=
Ω2πλS
RS
REZrmax
rmin Zrmax
rN
r1−αfRN(rN)drdrN,(30)
where fRN(rN)is given in which is obtained from the joint
distance distribution.
After some mathematical manipulations, we obtain the de-
sired results in (27), which completes the proof.
B. Particular Cases
In this subsection, we derive some valid but closed-form
technical results in particular cases. These results help iden-
tify the fundamental coverage improvement when LEO satel-
lites collaborate in transmissions compared to the traditional
LEO satellite network without cooperation. We provide the
closed-form results with two and three satellites involving co-
operation in the following, respectively.
Corollary 1 When N=2, the coverage probability for the
typical user in a downlink cooperative LEO satellite network
where two LEO satellites jointly transmit to it is expressed as
follows.
Pcov (γth;λS,RS,PT,GT,m,N=2)≈
1−(2πλSRS
RE)2
Γ(2m)·
expπλS
RS
RER2
S−R2
E−2πλSRS(RS−RE)·
Zrmax
rmin Zr2
rmin
r1r2e−πλS
RS
REr2
2·
γ

2m,γthm(e
INak +σ2
n0
PTGTβ0)
r−α
1+r−α
2

dr1dr2.(31)
Proof Based on the intermediate results derived in sec-
tion III, we can obtain the joint distance distribution of co-
operative LEO satellites to the typical user. When N=2, the
joint distance distribution is given by
fR(r1,r2) =
2πλS
RS
RE2
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)·
r1r2e−πλS
RS
REr2
2.(32)
Thus, the PDF of r2is derived as follows.
fR2(r2) =Zr2
rmin
fR(r1,r2)dr1=
2πλS
RS
RE
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)2
·
r2e−πλS
RS
REr2
2Zr2
rmin
eλS|Ar1|r1e−πλS
RS
REr2
1dr1=
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336 Journal of Communications and Information Networks
2πλS
RS
RE2
r2e−πλS
RS
REr2
2e−2πλSRS(RS−RE)·
eπλS
RS
RE(R2
S−R2
E)r2
2−(RS−RE)2,(33)
where eλS|Ar1|=e2πλSRS(RS−RE)e−πλS
RS
RE(R2
S−R2
E)eπλS
RS
REr2
1.
By substituting the PDF of r2to the expression of aver-
age interference power, we obtain the closed-form interfer-
ence power for N=2, shown in (34) at the top of the next
page. It is worth noting that the average interference power
incorporating lower incomplete Gamma functions can be cal-
culated efficiently.
Substituting (33) and (34) into (29), we can express (31),
which completes the proof.
Corollary 2 When N=3, the coverage probability for the
typical user in a downlink cooperative LEO satellite network
where three LEO satellites jointly transmit data to the user is
expressed as
Pcov (γth;λS,RS,PT,GT,m,N=3)≈
1−(2πλSRS
RE)3
Γ(3m)·
expπλS
RS
RER2
S−R2
E−2πλSRS(RS−RE)·
Zrmax
rmin Zr3
rmin Zr2
rmin
r1r2r3e−πλS
RS
REr2
3·
γ

3m,γthm(e
INak +σ2
n0
PTGTβ0)
r−α
1+r−α
2+r−α
3

dr1dr2dr3.(35)
Proof Based on the joint distance distribution of cooper-
ative LEO satellites to the typical user which was derived
in section III, we obtain a concise form of this distribution.
Specifically, when N=3, the joint distance distribution in
satellite cooperation is given by
fR(r1,r2,r3) = 2πλS
RS
RE3
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)·
r1r2r3e−πλS
RS
REr2
3.(36)
Thus, the PDF of r3is derived as follows.
fR3(r3) = Zr3
rmin Zr2
rmin
fR(r1,r2,r3)dr1dr2=
2πλS
RS
RE3
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)·
r3e−πλS
RS
REr2
3Zr3
rmin Zr2
rmin
r1r2dr1dr2=
πλS
RS
RE3
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)·
User
data1
LEO satellite
LEO satellite networks
×106
8
6
4
2
0
−2
−4
−6
−8
Z (Up)
×106
5
0
−5 ×106
−6
−8
−4−202468
Y (North) X (North)
Fig. 3 An illustration of simulation scenario
r3e−πλS
RS
REr2
3r4
3−2r2
minr2
3+r4
min.(37)
The average interference power at the typical user is
e
INak =2(πλSRS
RE)4
α−2e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)·
Zrmax
rmin
r3e−πλS
RS
REr2
3·
r4
3−2r2
minr2
3+r4
minr2−α
3−r2−α
max dr3.(38)
Substituting (37) and (38) into (29), we can express (35),
which completes the proof.
We have completed the main results of this paper, i.e.,
the derivation of coverage probability for a typical user with
LEO satellites cooperation under Nakagami-mfading chan-
nels. By setting different mvalues such as m=1 and m=∞
in the expressions, we can obtain the coverage probabilities
for Rayleigh and no-fading cases, respectively.
V. SIMULATION AND NUMERICAL RESULTS
In this section, we numerically evaluate and validate the
coverage probability against important parameters for the typ-
ical user in cooperative LEO satellites, including altitude,
Nakagami fading parameter m, number of cooperative satel-
lites and total number of satellites. In addition, we also illu-
minate a trade-off between the desired signal power and the
interference signal power under the impacts of the number of
satellites and the altitude of satellites.
We do Monte Carlo simulations in MATLAB for the com-
parison of the simulation and analysis results. As is shown in
Fig. 3, we first depict a 3D diagram for LEO satellite net-
works. The users and LEO satellites are respectively dis-
tributed on the earth surface sphere and satellite orbit sphere
independently, according to homogeneous SPPP with differ-
ent densities.
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Coverage in Cooperative LEO Satellite Networks 337
e
INak =
2πλS
RS
RE
2πλS
RS
RE2
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)Zrmax
rmin
r2e−πλS
RS
REr2
2r2
2−(RS−RE)2Zrmax
r2
r1−αdrdr2=
4
α−2πλS
RS
RE3
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)Zrmax
rmin
r2e−πλS
RS
REr2
2r2
2−(RS−RE)2r2−α
2−r2−α
max dr2=
4
α−2πλS
RS
RE3
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)·
Zrmax
rmin
r2e−πλS
RS
REr2
2r4−α
2−r2−α
2(RS−RE)2−r2−α
max r2
2+r2−α
max (RS−RE)2dr2=
4
α−2πλS
RS
RE3
e−2πλSRS(RS−RE)eπλS
RS
RE(R2
S−R2
E)
















γ3−α
2,πλS
RS
REr2
max−γ3−α
2,πλS
RS
REr2
min
2πλS
RS
RE3−α
2−
(RS−RE)2
2πλS
RS
RE2−α
2γ2−α
2,πλSRS
REr2
max−γ2−α
2,πλSRS
REr2
min−
r2−α
max
2πλS
RS
RE2γ2,πλSRS
REr2
max−γ2,πλSRS
REr2
min+
r2−α
max (RS−RE)2
2πλS
RS
REγ1,πλSRS
REr2
max−γ1,πλSRS
REr2
min
















(34)
−25 −20 −15 −10 −5 0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Coverage probability
N increases
No cooperation N = 1, (simulation)
Cooperation N = 2, (analysis)
Cooperation N = 2, (simulation)
Cooperation N = 3, (analysis)
Cooperation N = 3, (simulation)
SINR threshold th (dB)
Fig. 4 Coverage probability versus the SINR threshold under different num-
bers of serving satellites, where m=2
In simulations, we set RE=6 372 km, the path loss expo-
nent α=2.1, the density of LEO satellites λS= (1e−12)/m2,
the Nakagami fading parameters Ω=1, m=2, transmit
power at the LEO satellite P
T=1 Watt, transmit antenna gain
GT=30 dBi, unless specified otherwise.
Taking into account the impact of the serving satellite num-
ber, Fig. 4 shows the coverage probability under different
numbers of serving satellites, compared with traditional no-
cooperation scenario. It is found that with the increase of
serving satellite number, the coverage probability of cooperate
−25 −20 −15 −10 −5 0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Coverage probability
No cooperation N = 1, m = 4,
(simulation)
Cooperation N = 2, m = 1, (analysis)
Cooperation N = 2, m = 1, (simulation)
Cooperation N = 2, m = 2, (analysis)
Cooperation N = 2, m = 2, (simulation)
Cooperation N = 2, m = 4, (analysis)
Cooperation N = 2, m = 4, (simulation)
SINR threshold th (dB)
Fig. 5 Coverage probability versus the SINR threshold under different Nak-
agami fading parameters m
LEO satellite networks becomes higher under the given SINR
threshold.
Fig. 5 compares the coverage probability under different
Nakagami fading parameter m. It is clear that the case of m=4
outperforms other cooperation counterparts when the SINR
threshold is lower than 1 dB; however, its coverage probability
drops fastest as SINR threshold increases, thus it is surpassed
by that of other cases. Note that no-operation case falls far
behind the cooperation cases.
The impact of satellite altitudes on the coverage probability
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338 Journal of Communications and Information Networks
−
35
−
30
−
25
−
20
−
15
−
10
−
5 0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Coverage probability
No cooperation h = 500 km (sim.)
Cooperation h = 500 km (ana.)
Cooperation h = 500 km (sim.)
No cooperation h = 1 000 km (sim.)
Cooperation h = 1 000 km (ana.)
Cooperation h = 1 000 km (sim.)
No cooperation h = 1 500 km (sim.)
Cooperation h = 1 500 km (ana.)
Cooperation h = 1 500 km (sim.)
SINR threshold th (dB)
Fig. 6 Coverage probability versus the SINR threshold under different alti-
tudes of satellites, where N=2, m=2
of cooperative LEO satellite networks is compared in Fig. 6. It
is revealed that for both cooperative and non-cooperative sce-
narios, the coverage probability increases with the altitude;
however, for the same altitude, the cooperative case enjoy
apparently higher coverage probability than it opposite non-
cooperative case. This is because with a lower altitude, the
increase of desired signal power dominates the increase of in-
terference power, resulting in the improvement of SINR and
coverage probability.
In the case of serving satellite quantity is N=2, Nak-
agami fading parameter is m=2, and the SINR threshold be-
ing −3 dB, we conducted a comparative analysis of coverage
probability concerning satellite altitude across varying satel-
lite quantities, both with and without satellite cooperation.
This analysis aimed to substantiate that the implementation
of LEO satellite cooperation significantly enhances downlink
coverage, surpassing the coverage achievable in the absence
of cooperation.
Fig. 7 provides a comparative assessment, elucidating the
coverage probability in situations involving satellite coop-
eration (indicated by colored lines) versus non-cooperation
(depicted by black lines) with respect to satellite quantities
of 500 (represented by lines with circles) and 1 000 (indi-
cated by lines with triangles). The visual evidence gleaned
from the figure unequivocally highlights that the collaborative
transmission of data among serving satellites leads to a sub-
stantial enhancement in coverage probability for ground user,
as opposed to scenarios where such cooperation is lacking.
This performance improvement is conspicuously discernible
within the graphical representation.
Furthermore, upon examining the scenario of non-
cooperation among satellites in Fig. 7, denoted by the black
line with circles as opposed to the black line with triangles,
100 200 300 400 500 600 700 800
Performance gain
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Coverage probability
Altitude of satellites (km)
No cooperation, No. of satellite = 500
Cooperation, No. of satellite = 500
No cooperation, No. of satellite = 1 000
Cooperation, No. of satellite = 1 000
Fig. 7 Coverage probability versus the altitude of satellites with different
SINR thresholds, where N=2, m=2, γth =−3 dB
we can deduce that a smaller satellite number exhibits a higher
coverage probability. This observation also holds true in the
context of cooperation among serving satellites. This is be-
cause the network with a smaller satellite number has a lower
co-channel interference power than the network with a high
satellite number, and thus it is beneficial to the coverage.
Moreover, in instances where satellites do not engage in
cooperative behavior, there is a noticeable decline in cover-
age probability with increasing satellite altitude. Upon closer
examination of the blue and red lines depicted in Fig. 7, it be-
comes evident that even when satellites cooperate, the cover-
age probability experiences a decline as satellite altitude rises.
However, it is noteworthy that slight increments are observed
when satellite altitude remains below 150 km. This is be-
cause that when the satellite altitude is very low, the typical
user could not always see the sufficient serving satellites for
cooperation. When the satellite altitude is very high, the de-
sired signal power decreases due to the increased distance-
dependent large-scale fading. Therefore, there is an optimal
satellite altitude given the number of satellite.
Next, we investigate the relationship between the number
of satellites and the coverage probability under fixed satel-
lite altitudes. When the number of satellites increase from
60 to 6 000 as is shown in Fig. 8, a decreasing trend is wit-
nessed for each of the four cases, with and without coopera-
tion. Similarly, a tiny increase can be seen in coverage with
a low altitude under cooperation mode. On the contrary, the
no-cooperation satellite system experiences a continuous and
sharp decrease in coverage from approximately 0.7 and 0.8 to
0 as the number of satellites come to nearly 4 000. There is
also an apparent performance gain between cooperation and
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Coverage in Cooperative LEO Satellite Networks 339
1 000 2 000 3 000 4 000 5 000 6 000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Coverage probability
Number of satellites
No cooperation, altitude of satellite = 300 km
Cooperation, altitude of satellite = 300 km
No cooperation, altitude of satellite = 600 km
Cooperation, altitude of satellite = 600 km
Fig. 8 Coverage probability versus the altitude of satellites with different
SINR thresholds, where N=2, m=2, γth =−3 dB
Number of satellites
6 000
5 000
4 000
3 000
2 000
1 000
0500 1 000 1 500
Altitude of satellites (km)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 9 Coverage probability versus the altitude of satellites and the number
of satellites, where, where N=2, m=2, γth =−5 dB
no-cooperation cases.
In Fig. 9 and Fig. 10, we examine the coverage probability
versus the number of satellites and the altitude of satellites in
2D and 3D views, respectively. We observe that there is an
optimal combination of the number of satellites and satellite
altitude, which gives system design insights. The reasons be-
hind this are as follows. Regarding satellite altitude, there is
a trade-off between the probability of seeing sufficient serv-
ing satellites in cooperation and the desired signal power at
the typical user. Regarding the number of satellites, there is
a trade-off between the distance-dependent path loss (or the
desired signal power) and the interference power from other
satellites. Moreover, we observe from Fig. 9 and Fig. 10 that
when deploying dense LEO satellites for satellite-terrestrial
communications, it is better to reduce the satellite altitude.
This is because for densification of satellites, reducing satel-
lite altitude deceases the number of interfering satellites for
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.2
0.4
0.6
0.8
1.0
Coverage probability
1 000
500
0
Altitude of satellites (km) Number of satellites
6 000
4 000
2 000
0
Fig. 10 Coverage probability versus the altitude of satellites and the number
of satellites in a 3D plot, where N=2, m=2, γth =−5 dB
the typical user and increases the desired signal power. This
guided us to seize the low-earth orbit.
VI. CONCLUSION
In this paper, we proposed and modelled a novel satellite
cooperation system which incorporates several nearest satel-
lites to provide services to the typical user. Considering the
satellite altitude, Nakagami fading channels, the number of
serving satellites and all visible satellites, and other related pa-
rameters, we derived approximated but tractable and closed-
form expressions for the coverage probability of the satellite
cooperation system. It was shown that in the satellite co-
operation system, the increase of cooperative satellite num-
ber significantly brought about larger coverage for the typical
user, significantly surpassing that in non-cooperative satellite
downlink transmission system. Specifically, there is a trade-
off between the desired signal power and the interference sig-
nal power brought by the total number and the altitude of
satellites.
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ABO UT T HE AU TH OR S
Bodong Shang [corresponding author] received the
B.Eng. degree in the School of Information Sci-
ence and Technology at Northwest University, Xi’an,
China, the M.S. degree in the School of Telecom-
munications Engineering at Xidian University, Xi’an,
China, and the Ph.D. degree from the Department of
Electrical and Computer Engineering at Virginia Tech,
Blacksburg, USA. He was a Postdoctoral Research
Associate at Carnegie Mellon University, Pittsburgh,
USA. He is an Assistant Professor at Eastern Institute for Advanced Study
(EIAS), Eastern Institute of Technology (EIT), Ningbo, China. He received
the Chinese government award for outstanding self-financed students abroad
and held a summer research internship at Nokia Bell Labs, USA. His current
research areas are wireless communications and networking, including 6G,
non-terrestrial networks, Internet of vehicles, integrated sensing and commu-
nications, edge computing, and reconfigurable intelligent surfaces.
Xiangyu Li received the B.Eng. degree in Elec-
tronic Information Engineering from Nantong Univer-
sity, Nantong, China, in June 2021 and the M.S. degree
in Electrical and Computer Engineering from Georgia
Institute of Technology, Atlanta, United States, in May
2023. He is currently working toward the Ph.D. degree
with Shanghai Jiao Tong University, Shanghai, China,
in the EIT-SJTU Joint PhD Program. His research in-
terests include space-air-ground integrated networks,
massive MIMO, reconfigurable intelligent surfaces, physical layer security,
and performance analysis of wireless systems.
Caiguo Li received the B.S. degree in the College of
Computer and Information Engineering from Henan
University, China, in 2020, and the M.S. degree in
Computer Science and Technology from Shenzhen
University, China, in 2023. She is a Research Assistant
at the Eastern Institute for Advanced Study (EIAS),
Ningbo, China. Her research interests include edge
computing, wireless communication, and information
security.
Zhuhang Li received the B.S. degree from the Uni-
versity of Electronic Science and Technology of China
(UESTC), Chengdu, China, in 2022, and the M.S.
degree from the Nanyang Technological University,
Singapore, in 2023. She is a Research Assistant
at the Eastern Institute for Advanced Study (EIAS),
Ningbo, China. Her research interests include space-
air-ground-sea integrated networks, and Internet-of-
things.
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