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IMPROVED GRAPH-BASED USER SCHEDULING FOR SUM-RATE MAXIMIZATION IN
LEO-NTN SYSTEMS
Bilal Ahmad ⋆, Daniel Gaetano Riviello⋆, Alessandro Guidotti †, Alessandro Vanelli-Coralli ⋆.
Department of Electrical, Electronic, and Information Engineering (DEI), University of Bologna ⋆
National Inter-University Consortium for Telecommunications (CNIT), Bologna, Italy †
{bilal.ahmad6, daniel.riviello, a.guidotti, alessandro.vanelli}@unibo.it
ABSTRACT
In this paper, we study the problem of user scheduling for
Low Earth Orbit (LEO) Multi-User (MU) Multiple-Input-
Multiple-Output (MIMO) Non-Terrestrial Network (NTN)
systems with the objective of maximizing the sum-rate ca-
pacity while minimizing the total number of clusters. We
propose an iterative graph-based maximum clique scheduling
approach with constant graph density. Users are grouped to-
gether based on the channel coefficient of correlation (CoC)
as dissimilarity metric and served by the satellite via Space
Division Multiple Access (SDMA) by means of Minimum
Mean Square Error (MMSE) digital beamforming on a clus-
ter basis. Clusters are then served in different time slots via
Time Division Multiple Access (TDMA). The results, pre-
sented in terms of per-cluster sum-rate capacity and per-user
throughput, show that the presented approach can signifi-
cantly improve the system performance.
Index Terms—LEO, MU-MIMO, User Scheduling,
Beamforming, MMSE
1. INTRODUCTION
With the approval of 3GPP Rel-17 and the inclusion of Non-
Terrestrial Networks (NTN) in 5G-Advanced and 6G systems,
the satellite industry is undergoing a rapid growth, paving the
way for global access to 6G services. Specifically, NTNs in
Low Earth Orbit (LEO) have revealed their potential as they
provide high data rates with lower latency and better link bud-
get [1]. To further increase the data rate, aggressive (full)
frequency reuse schemes provide a more effective use of the
bandwidth, thus improving the spectral efficiency, at the cost
of an increased interference that must be managed at the re-
ceiver via Multi-User Detection (MUD) and/or transmitter via
precoding/beamforming, [2].
Although precoding techniques can improve the perfor-
mance, there must be a trade-off between computational com-
Part of this work has been funded by the 6G-NTN project, which re-
ceived funding from the Smart Networks and Services Joint Undertaking
(SNS JU) under the European Union’s Horizon Europe research and inno-
vation programme under Grant Agreement No 101096479.
plexity and performance reduction. For example, Dirty Paper
Coding (DPC) achieves the channel capacity, but due to its
high computational complexity, it is still not widely used in
existing practical systems [3]. Thanks to lower computational
costs and close to the ideal performance, linear precoding
and detection schemes, such as Zero Forcing (ZF), MMSE
or Signal-to-Leakage-plus-Noise Ratio (SLNR) beamform-
ing [4, 5], are more desirable in NTN MU-MIMO systems
as compared to non-linear precoding [6]. A thorough survey
on MIMO techniques applied to SatCom is provided in [7],
where both fixed and mobile satellite systems are evaluated
and the main channel impairments are addressed.
Conceptually, adding more antennas at the satellite in-
creases the system capacity, thus enhancing energy efficiency
and significantly boosting the system throughput [8]. Design-
ing algorithms that can address the issue of scarce system re-
sources in the MU-MIMO LEO system is therefore a funda-
mental, and challenging, technology for NTN. Since the num-
ber of User Terminals (UTs) on the ground is much larger
than the on-board satellite antennas, efficient user schedul-
ing is required. Scheduling can be accomplished by grouping
users into different clusters: i) UTs within the same cluster
are multiplexed and served together via Space Division Mul-
tiple Access (SDMA), i.e., digital beamforming or Multi-User
MIMO (MU-MIMO) techniques; and ii) the different clusters
are then served on different time slots via Time Division Mul-
tiple Access (TDMA) [9, 10, 11]. The design of an optimal
user grouping strategy is known to be an NP-complete prob-
lem which can be solved only through exhaustive search [12].
A greedy user scheduling strategy for downlink MU-MIMO,
which takes into account heterogeneous users, has been pro-
posed in [10]. Multiple antenna downlink orthogonal cluster-
ing (MADOC), a well-established low-complexity algorithm
introduced in [11], builds on the previous work by taking user
fairness and group number minimization into account.
In this paper, we extend our previous work in [9] by
proposing a low-complexity graph-based iterative procedure
with constant graph density for user scheduling in MU-
MIMO LEO NTN systems and we aim at maximizing the
sum-rate capacity while preserving fairness among the users.
We model the clustering problem as an undirected and un-
weighted graph and we iteratively search for the maximum
clique, i.e., the largest fully connected subgraph to form clus-
ters of spatially separable users. We perform an heuristic
optimization, i.e., via exhaustive simulations, of the graph
density value, which maximizes the sum-rate capacity. We
show that the proposed approach can significantly improve
the performance w.r.t. our previous work [9] and MADOC.
2. SYSTEM MODEL
We focus on the downlink of a single LEO satellite operating
in S-band equipped with an antenna array. The array con-
sists of Nradiating elements which provide connectivity to
Kuniformly distributed on-ground single-antenna users un-
der the assumption that N≪K. Our additional assumptions
are based on the system architecture, which is thoroughly de-
scribed in [1, 9, 13]. In particular, we consider non-ideal
Channel State Information (CSI) with channel aging, i.e., we
take into account the latency ∆tbetween the channel esti-
mation time and the transmission time, which introduces a
misalignment between the channel on which the scheduling
and beamforming matrices are generated and the actual chan-
nel through which the transmission occurs [9, 4]. By default,
the antenna boresight direction is defined by the direction of
the Sub-Satellite Point (SSP) and the point Pis the posi-
tion of the UT on the ground. The UT direction is identified
by the (ϑ, φ)angles, where the boresight direction is (0,0).
The direction cosines for the considered UT are derived as
u=Py
∥P∥sin ϑsin φ, and v=Pz
∥P∥cos ϑ. The total array
response of the Uniform Planar Array (UPA) in the generic
i-th UT direction (ϑi, φi) can be expressed as the Kronecker
product of the array responses of two Uniform Linear Arrays
(ULAs) lying on the y- and z-axis [14]. We first define the
1×NHSteering Vector (SV) of the ULA along the y-axis
aH(θi, φi)and the 1×NVSV of the ULA along the z-axis
aV(θi)
aH(ϑi,φi)= h1, ejk0dHsin ϑisin φi, . . . , ejk0dH(NH−1) sin ϑisin φii
(1)
aV(ϑi)= h1, ejk0dVcos ϑi, . . . , ejk0dV(NV−1) cos ϑii.(2)
where k0= 2πλ is the wave number, NH,NVdenote the num-
ber of array elements on the horizontal (y-axis) and vertical
(z-axis) directions respectively with N=NH·NVand dH, dV
denote the distance between adjacent array elements on the y-
and z-axis respectively. We assume that the array is equipped
with directive antenna elements, whose radiation pattern is
denoted by gE(ϑi, φi). Finally, we can express the (1 ×N)
SV of the UPA at the satellite targeted for the i-th user as the
Kronecker product of the 2 SVs along each axis multiplied by
the element radiation pattern:
a(ϑi, φi) = gE(ϑi, φi)aH(ϑi, φi)⊗aV(ϑi)(3)
The CSI vector at feed level ˆ
hirepresents the channel be-
tween the Nradiating elements and the generic i-th on-
ground UT, with i= 1, .. . , K, and can be written as:
ˆ
hi=G(rx)
i
λ
4πdirLi
κBTi
e−j2π
λdia(ϑi, φi)(4)
in which, diis the slant range between the generic i-th user
and the satellite, λis the wavelength, κBTidenotes the equiv-
alent thermal noise power, with κbeing the Boltzmann con-
stant, Bthe user bandwidth (assumed to be the same for all
users), and Tithe equivalent noise temperature of the i-th
UT. G(rx)
idenotes the receiving antenna gain for the i-th UT,
while Lidenotes all the additional losses per user, such as at-
mospheric, antenna, and cable losses. By collecting all of
the KCSI vectors, it is possible to build a K×Ncom-
plex channel matrix at system level ˆ
H=hˆ
h⊺
1,ˆ
h⊺
2,...,ˆ
h⊺
Ki⊺
,
where the generic k-th row contains the CSI vector of the
k-th user and the generic n-th column contains the channel
coefficients from the n-th on-board feed towards the Kon-
ground users. Given the set of all users to be scheduled,
denoted with U={U1, U2, . . . , UK}, the Radio Resource
Management (RRM) algorithm defines a possible users’ par-
titioning {C1,C2,...,CP}where Cp⊆ U is defined as cluster
and |Cp|=Kpis defined as the cardinality of the p-th clus-
ter with p= 1, . . . , P . In order to minimize the the total
number of clusters P, we further assume that each user can
be assigned only to one cluster (as in [11, 9]), therefore clus-
ters are disjoint sets of users, Ci∩ Cj=∅,∀i, j. In each
time slot, users belonging to cluster Cpare selected, lead-
ing to a Kp×Ncomplex scheduled channel matrix ˆ
Hp⊆
ˆ
H, which contains only the rows of the scheduled users in
the p-th cluster. The selected beamforming algorithm com-
putes for each cluster a N×Kpcomplex beamforming ma-
trix Wp= [w(p)
1,w(p)
2,...,w(p)
Kp], where w(p)
idenotes the
N×1beamformer designed for the i-th user in the p-th clus-
ter. The matrix Wpprojects the Kpdimensional column
vector sp= [s1, s2, . . . , sKp]⊺containing the unit-variance
users’ symbols onto the N-dimensional space defined by the
antenna feeds. Thus, in the feed space, the computation of the
beamforming matrix allows for the generation of a dedicated
beam towards each user direction. The signal received by the
i-th user in the p-th cluster can be expressed as follows [9]
y(p)
k=hkw(p)
ksk+
Kp
X
i=1
i=k
hkw(p)
isi+z(p)
k(5)
The Kp-dimensional vector of received symbols in the p-th
cluster is
yp=HpWpsp+zp(6)
It shall be noted that the estimated channel matrix ˆ
Hat time
t0is used to compute the scheduling and the beamforming
matrices Wpin the estimation phase, while the beamformed
symbols are sent to the users at a time instant t0+∆t, in which
the scheduled channel matrices and vectors are different and
denoted as Hpand hk, respectively. The SINR for user k
belonging to cluster pcan be computed as
SINR(p)
k=
hkw(p)
k
2
1 + PKp
i=1
i=k
hkw(p)
i
2(7)
We can define the per-cluster sum-rate capacity as:
Γp=B
Kp
X
k=1
log2(1 + SINR(p)
k)(8)
where Bdenotes the bandwidth. Given that each user can be
assigned to only one cluster and taking into account the duty
cycle associated with each cluster in TDMA, we can define
the throughput experienced by the k-th user as:
Rk=B
Plog21 + SINR(p)
k(9)
The beamforming matrix Wp, which is computed on a cluster
basis, is based on the linear Minimum Mean Square Error
(MMSE) equation:
Wp=ˆ
HH
p(ˆ
Hpˆ
HH
p+αIKp)−1(10)
where IKpindicates the Kp×Kpidentity matrix and α=N
Pt
is the regularisation factor with Ptthe total on-board power.
We consider the following two options for power normal-
ization: Sum Power Constraint (SPC) and Maximum Power
Constraint (MPC), which are defined as:
f
W(SPC)
p=√PtWp
√tr(WpWH
p)f
W(MPC)
p=√PtWp
qNmaxj[WpWH
p]j,j
(11)
3. USER SCHEDULING
We denote with G= (V,E)an undirected and unweighted
graph with vertex set Vand edge set E. A clique Qof Gis
a subset of the vertices, Q ⊆ V, such that every two distinct
vertices are adjacent, i.e., Qis a complete subgraph. In our
LEO NTN MIMO scenario, the set of vertices Vcoincides
with the set of users Uand the edge set is constructed based on
the channel Coefficient of Correlation (CoC) [11, 12], defined
as
[Ψ]i,j =hihH
j
∥hi∥∥hj∥(12)
where [Ψ]i,j ∈[0,1]. The set of edges Eof the Ggraph
is completely determined by its adjacency matrix A, whose
entries are defined as:
[A]i,j =(1,[Ψ]i,j ≤δth
0,[Ψ]i,j > δth
(13)
where δth denotes the graph threshold. An edge between Ui
and Ujimplies that their channels hiand hjare considered
nearly orthogonal and therefore they can be co-scheduled.
The graph threshold determines the density of the graph
D(Ψ, δth)which can be computed as
D(Ψ, δth) = 2|E|
|V|(|V| − 1) =1
2
K
X
i=1
K
X
j=1
[A]i,j (14)
Algorithm 1 Improved Max Clique scheduler with constant
graph density.
Require: Set of users U, channel CoC matrix Ψand target graph density ϵt
1: Initialize p= 1 and K=|U|
2: while U =∅do
3: δth = BISECTIONMET HO D(Ψ, 0, 1, ϵt,tol,Imax )
4: Compute Aas in (13)
5: Qmax = MAX CLI QU EDYN (A)
6: Cp← Qmax and Kp← |Cp|
7: Remove all rows and columns of Ψassociated with users in Qmax
8: U ← U − Qmax and K←K−Kp
9: p←p+ 1
10: end while
11: procedure BISECTIONMETHOD(Ψ,a,b,ϵt,tol,Imax)
12: c=a+b
2and i= 0
13: while |f(Ψ, c)|> tol &i<Imax do
14: if f(Ψ, b)·f(Ψ, c)<0then
15: a=c
16: else
17: b=c
18: end if
19: c=a+b
2
20: i→i+ 1
21: end while
22: return c
23: end procedure
As illustrated in Alg. 1, we design a greedy iterative al-
gorithm that aims at minimizing the total number of clus-
ters Pand maximizing the total sum-rate capacity. Simi-
larly to [9], the iterative procedure searches for the maximum
clique Qmax in the graph through the efficient MaxCliqueDyn
algorithm [15] and declares it as a cluster; at each step the
nodes in Qmax and any edges connected to them are removed.
The main novelty in the new algorithm is how the graph is
updated after each pruning. In [9], since a constant graph
threshold value is set at the beginning of the procedure, only
graph pruning is implemented at each step. Here instead we
first set a constant target graph density value ϵt, then at each
step (both at the beginning and after each pruning), we search
for the threshold δth such that D(Ψ, δth) = ϵt. To obtain
the respective threshold value for the required target density
ϵt, we aim at finding the root of the function f(Ψ, δth) =
D(Ψ, δth)−ϵtthrough the bisection method [16]. Within
this method, the root to be found is initially bounded between
0 and 1, the search interval is repeatedly halved until con-
vergence is reached through the parameter tol. Finally, the
parameter Imax limits iterations to prevent infinite loops.
Table 1: Simulation parameters.
Parameter Value
Carrier frequency 2 GHz
System band S (30 MHz)
Beamforming space feed
Receiver type VSAT
Receiver antenna gain 39.7 dBi
Noise figure 1.2 dB
Parameter Value
Propagation scenario Line of Sight
System scenario urban
Total on-board power density, Pt,dens 4 dBW/MHz
Coverage area 101077 km2
User density 0.05 user/km2
Number of transmitters N1024 (32 ×32 UPA)
4. SIMULATION SETUP AND RESULTS
In this section, we present the outcomes of the extensive nu-
merical simulations with the parameters specified as in Tab. 1.
The assessment is performed in full buffer conditions, i.e., in-
finite traffic demand. A single LEO satellite is considered
at a distance of 600 Km from the earth. On average the to-
tal number of users are K= 2850. The user terminals are
fixed VSAT and the propagation scenario is the line of sight
(LOS) model based on TR 38.821 and TR 38.811 [17, 18].
In all simulations, the performance of the improved maxi-
mum (Max) clique-based algorithm with constant graph den-
sity is compared against our original Max clique scheduler
and MADOC. Aiming at maximizing the total sum-rate ca-
Table 2: Simulation results for graph density optimization.
Parameters Original MADOC Improved
Max clique Max clique
Optimized threshold SPC 0.32 0.58 0.965
graph density MPC 0.27 0.51 0.957
Mean SPC 46.37 56.34 59.42
cluster size MPC 42.81 48.56 50.29
Sum-rate SPC 17.93 21.17 21.49
capacity (Gbps) MPC 15.82 18.19 18.64
pacity, we performed a heuristic optimization of the graph
density for the improved Max clique, and we obtained opti-
mized thresholds for the original Max clique and MADOC
for both MMSE-SPC and MMSE-MPC as shown in Tab. 2.
It is evident that the improved Max clique offers the high-
est average per-cluster sum-rate capacity (Gbps), and mean
cluster size as compared to the other schedulers, with a very
large improvements in both indicators w.r.t. the original Max
clique.
In Fig. 1, the Cumulative Distribution Function (CDF)
of the per-cluster sum-rate capacity for all the considered
schedulers is shown. Only 8% of the clusters experience less
than 15 Gbps with the improved Max clique for MMSE-SPC,
whereas the percentage increases to 28% with the original
max clique. Fig. 2 shows the CDF of the users’ through-
put. It can be noticed that on average the throughput with
the improved Max clique with MMSE-SPC is increased by
0.2 Mbps w.r.t. MADOC, and 1.4 Mbps w.r.t. the original
Max clique. To further validate the results, Fig. 3 shows the
histograms of the cluster size distributions for the improved
and the original Max clique schedulers. It is clear that Fig. 3b
Fig. 1: CDF of the per-cluster sum-rate capacity.
Fig. 2: CDF of the user throughput.
shows a significant reduction in the cluster size variance, as
opposed to Fig. 3a.
5. CONCLUSION
In this paper, we addressed the problem of user scheduling
for LEO MU-MIMO NTN systems by proposing a low-
complexity graph-based iterative procedure with constant
graph density with the goal of maximizing the per-cluster
sum-rate capacity. Results show that the proposed algorithm
significantly improves the sum-rate capacity, offers higher
throughput, and allows to significantly reduce the variance
of the cluster size distribution w.r.t. [9], thus improving the
fairness among users and the overall performance.
(a) Original Max Clique. (b) Improved Max Clique.
Fig. 3: Cluster size distribution for both original and im-
proved Max clique schedulers.
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