Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
10.1109/ACCESS.2017.DOI

This work has been funded by the European Commission Horizon 2020 Project
DYNASAT (Dynamic Spectrum Sharing and Bandwidth-Efficient Techniques for High-
Throughput MIMO Satellite Systems) under Grant Agreement 101004145. The views
expressed are those of the authors and do not necessarily represent the
project. The Commission is not liable for any use that may be made of any of
the information contained therein.

Corresponding author: Alessandro Guidotti (e-mail: a.guidotti@unibo.it).

# Federated Cell-Free MIMO in Non-Terrestrial Networks: Architectures and
Performance

ALESSANDRO GUIDOTTI1  ALESSANDRO VANELLI-CORALLI2, , AND CARLA AMATETTI2
Consorzio Nazionale Interuniversitario per le Telecomunicazioni (CNIT), 43124
Parma, Italy, Research Unit of the University of Bologna, 40136 Bologna, Italy
(e-mail<EMAIL_ADDRESS> Department of Electrical, Electronic, and
Information Engineering, University of Bologna, 40136 Bologna, Italy (e-mail:
{alessandro.vanelli<EMAIL_ADDRESS>
###### Abstract

While 5G networks are being rolled out, the definition of the potential
5G-Advanced features and the identification of disruptive technologies for 6G
systems are being addressed by the scientific and academic communities to
tackle the challenges that 2030 communication systems will face, such as
terabit-capacity and always-on networks. In this framework, it is globally
recognised that Non-Terrestrial Networks (NTN) will play a fundamental role in
support to a fully connected world, in which physical, human, and digital
domains will converge. In this framework, one of the main challenges that NTN
have to address is the provision of the high throughput requested by the new
ecosystem. In this paper, we focus on Cell-Free Multiple Input Multiple Output
(CF-MIMO) algorithms for NTN. In particular: i) we discuss the architecture
design supporting centralised and federated CF-MIMO in NTN, with the latter
implementing distributed MIMO algorithms from multiple satellites in the same
formation (swarm); ii) propose a novel location-based CF-MIMO algorithm, which
does not require Channel State Information (CSI) at the transmitter; and iii)
design novel normalisation approaches for federated CF-MIMO in NTN, to cope
with the constraints on non-colocated radiating elements. The numerical
results substantiate the good performance of the proposed algorithm, also in
the presence of non-ideal information.

###### Index Terms:

Beamforming, Cell-Free MIMO, LEO, Location-based MIMO, NGSO, Non-Terrestrial
Networks, Satellite Communications

=-15pt

##  I Introduction

During the last years, telecommunication networks experienced an unprecedented
request for an ever increasing throughput, combined with the need to support
very diverse services with heterogeneous performance requirements in terms of
data rate and latency. While Ultra Reliable and Low Latency Communications,
massive Machine Type Communications, and enhanced Mobile Broadband 5G services
are being provided with global benefit for both economy and society, the
design of new features for 5G-Advanced (5G-A) and the research on 6G
technologies are already on-going, [1, 2, 3, 4]. Since 2021, ITU-R initiated
the development of the vision for IMT-2030 and beyond within Working Party
(WP) 5D, [5]; these activities are being performed in synergy with the ITU-T
Focus Group Technologies for Network 2030 (FG-NET-2030), which, between 2018
and 2020, identified a preliminary set of target services for 6G
communications, [6]. The envisioned 6G system will support a fully connected
world, characterised by the convergence of the physical, human, and digital
domains, [7, 8]. According to the 6G Infrastructure Association (6G-IA), three
broad classes of services can be foreseen, [9]: i) _digital twinning_ , of the
systems, with actuators and sensors tightly synchronising the above mentioned
domains to create digital twins of cities, factories, or even bodies; ii)
_connected intelligence_ , in which the network serves as the cornerstone
through which trusted Artificial Intelligence (AI) functions can manage the
virtual representations in the digital domain; and iii) _immersive
communications_ , in which high/ultra-high resolution visual/spatial,
tactile/haptic, and other sensory data can be exchanged to create a fully
immersive experience.

In the above context, service ubiquity and continuity are critical features
that only the full and seamless integration of terrestrial and Non-Terrestrial
Networks (NTN) can enable, [10, 11, 12, 13, 14, 15, 16]. The NTN segment
completes the overall system architecture by providing a ubiquitous,
continuous, flexible, and resilient infrastructure for: i) direct connectivity
to smartphones in outdoor and in light indoor/in-vehicle (emergency
communications) scenarios; ii) connectivity to mobile platforms (trains,
planes, ships, drones); iii) broadcast/multicast services; iv) low latency
communications to support vertical markets (_e.g._ , railway, automotive,
aeronautical); v) network-based positioning; and vi) Internet of Things (IoT)
applications. One of the key enablers of this communication infrastructure
will be the support for high throughput communications. Within NTN, current
Geostationary Earth Orbit (GEO) High Throughput Satellite (HTS) systems
provide hundred of Gbps through multi-beam coverage, [17, 18, 19]. These
systems are based on multi-colour, _e.g._ , 3 or 4 colours, frequency reuse
schemes in which the available bandwidth is split into multiple non-
overlapping spectrum chunks and assigned on a geographic basis to limit
interference. However, further improvements are needed to achieve the
envisioned Very High Throughput Satellites (VHTS) with terabit-capacity.
Several commercial endeavours are targeting the deployment of GEO systems with
thousands of spot-beams, [20]. Moreover, also Low Earth Orbit (LEO) mega-
constellations, which can significantly reduce the propagation delay, have
been receiving increasing interest and some of them have started the services
[21, 22]. Since current Physical layer (PHY) technologies already achieve a
spectral efficiency close to the theoretical Shannon limit, the emphasis for
future NTN systems is being placed on system design approaches aiming at
increasing the exploitation of the available spectrum. This can be achieved by
means of advanced spectrum usage paradigms, _e.g._ , Cognitive Radios, [23,
24], or Dynamic Spectrum Access, [11], or by decreasing the frequency reuse
factor down Full Frequency Reuse (FFR). Notably, the latter shall be combined
with effective interference management techniques, such as beamforming,
precoding, and Multiple Input Multiple Output (MIMO) to exploit the massive
generated co-channel interference.

In the past years, the implementation of beamforming techniques in NTN has
been extensively addressed, [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. These works, as
detailed in the literature review section, focused on the increase of the
system throughput in different scenarios, including unicast or multicast
transmissions, ideal and non-ideal Channel State Information (CSI) at the
transmitter, Geosynchronous (GSO) and, more recently, Non-GeoSynchronous Orbit
(NGSO) systems, and advanced Radio Resource Management (RRM) algorithms. In
this paper, we advance from that by addressing the design and performance of
Cell-Free (CF) MIMO in Non-Geosynchronous Orbit (NGSO) based NTN, considering
both centralised architectures, _i.e._ , MIMO with co-located radiating
elements on-board a single satellite, and federated solutions, _i.e._ , MIMO
with non co-located radiating elements on-board multiple satellites belonging
to the same formation.

###  I-A Literature review

The literature on the application of MIMO supported by precoding and digital
beamforming to NTN systems is extensive; in fact, based on the impressive
benefit brought by MIMO solutions to terrestrial communications, their
application to Satellite Communications has been one of the most discussed
research areas in the past years. Initially, the considered Multi-User MIMO
(MU-MIMO) techniques were based on the implementation of Zero Forcing (ZF) and
Minimum Mean Square Error (MMSE) MIMO in satellite scenarios, [28]; this work
showed that throughput gains in the order of $80\%$ could be obtained on both
the forward and the return links. In [29], the authors provide a detailed and
complete survey on the application of MIMO techniques over satellite channels;
both fixed and mobile satellite communications were addressed, also
identifying the most impacting channel impairments. In [30], the authors
discuss the availability of only partial, and not full, CSI at the transmitter
side, which is one of the most critical challenges in satellite-based MIMO. In
addition to this valuable insight, the authors also introduce a novel MIMO
scheme aimed at increasing the sum-rate and availability. Building on this
momentum, also several projects funded by the European Space Agency (ESA)
addressed the implementation of precoding to the DVB-S2X standard, [31]; more
specifically, the following practical challenges arising for MIMO in HTS
systems were discussed: framing issues, non-ideal phase estimates, non-ideal
CSI at the transmitter due to imperfect estimation at the user terminal, and
the impact of multiple gateways. The practical impairments in the application
of MIMO to DVB-S2X based systems were also discussed in [32] and the studies
in [33, 34] provides a thorough review of precoding solutions for multi-beam
satellite systems; in these latter works, the optimisation of the precoder
design with linear and non-linear power constraint is also discussed. In [35],
the non-linear Tomlinson-Harashima precoding is proposed. The performance of
linear precoding, when also taking into account the traffic demand, is
discussed in [36], where generic linear constraints were included in the
transmit covariance matrix, yielding to gains compared to traditional multi-
colour frequency reuse schemes as large as $170$%. On-board Beamforming (OBBF)
solutions for MIMO were discussed in [37] for multiple gateway systems, also
proposing potential solutions to mitigate the inter-beam and inter-forward
link interference. In [38], some of the authors of this paper provided a
detailed system design trade-off analysis for MMSE precoding with adaptive
antennas in terms of average spectral efficiency and outage probability. In
[26], the authors proposed a traffic-driven beam design combined with user
scheduling for precoding in GEO systems. The application of unsupervised
Machine Learning (ML) techniques for scheduling in precoded GEO systems is
discussed in [27].

More recently, multicast precoding has also been addressed. Initial studies
were mainly oriented towards regularised channel inversions in which the users
are served as a single terminal with an equivalent channel matrix equal to the
average of the single channel matrices, [39]. A pragmatic approach in which
the linear precoding and ground-based beamforming are jointly optimised and
computed at the ground segment is discussed in [40]. In [41], the precoding
matrix is computed through a Singular Value Decomposition (SVD). A preliminary
assessment of the challenges in optimally grouping the users in multicast
precoding is provided in [42, 43, 44, 45]. In [46], a robust multi-group
multicast precoding algorithm is proposed in the presence of outdated CSI.
Some of the authors of this paper proposed a thorough analysis of users
grouping in multicast precoding by modelling it as a clustering problem, [47];
in addition, novel clustering algorithms, both for variable and fixed cluster
sizes, are proposed showing significant performance improvements. The same
authors proposed a geographical scheduling for unicast and multicast
precoding, based on serving together only users that belong to the same zone
within the corresponding reference beam, [48].

Finally, it is worthwhile mentioning that, recently, the application of
precoding on the feeder link in the presence of multiple gateways has also
been addressed, [49, 50, 51].

Figure 1: System architecture for federated CF-MIMO in NGSO-based NTN with
regenerative payloads, functional split, and intra-/inter-swarm ISLs.

###  I-B Paper contribution and organisation

To the best of our knowledge, the extensive and valuable studies in the
available literature focused on MIMO in NTN for CSI-based algorithms, with
centralised architectures, _i.e._ , all radiating elements colocated on the
same satellite; moreover, architecture aspects are seldom discussed. In this
work, inspired by the initial analyses that the authors reported in [25], and
moving from the current State-of-the-Art, we

  * •

design a novel location-based CF-MIMO algorithm for NTN NGSO constellations
which is completely user-centric, _i.e._ , tailored to the users’ and not
based predetermined beam lattices, and applicable in both centralised and
federated architectures;

  * •

provide a unified mathematical framework for both Cell-Free (user-centric) and
beam-based MIMO through federated NTN NGSO nodes;

  * •

design and thoroughly discuss the architecture design choices allowing the
implementation of federated Cell-Free and beam-based MIMO solutions in NTN
NGSO constellations. Both regenerative, with functional split options, and
transparent payloads are considered;

  * •

design novel power normalisation approaches for federated MIMO algorithms that
can be applied to swarms of NGSO nodes;

  * •

assess the performance taking into account different sources of non-ideal
knowledge at the transmitter for the computation of the beamforming matrices,
including non ideal location estimation and modelling errors of the radiation
pattern.

The remainder of this paper is as follows: i) in Section II, we discuss the
different architecture options and define the considered system architecture;
ii) in Section III-C we describe the system model and the main assumptions;
iii) in Section IV, the numerical assessment is provided with both ideal and
non-ideal knowledge of the required information at the transmitter side;
finally, Section V concludes this work.

###  I-C Notation

Throughout this paper, and if not otherwise specified, the following notation
is used: bold face lower case and bold face upper case characters denote
vectors and matrices, respectively. $\mathbf{a}_{i,:}$ and $\mathbf{a}_{:,i}$
denote the $i$-th row and the $i$-th column of matrix $\mathbf{A}$,
respectively. ${\left(\cdot\right)}^{-1}$ denotes the matrix inversion
operator. ${\left(\cdot\right)}^{T}$ denotes the matrix transposition
operator. ${\left(\cdot\right)}^{H}$ denotes the matrix conjugate
transposition operator. $\mathrm{diag}\left(\mathbf{a}\right)$ denotes a
diagonal matrix with the vector $\mathbf{a}$ on its main diagonal.
$\mathbf{I}_{K}$ denotes the identity matrix of order $K$.
$\mathrm{tr}\left(\mathrm{A}\right)$ denotes the trace of matrix $\mathrm{A}$.

TABLE I: Architecture options for MIMO in NGSO NTN. Payload type | Architecture option | Computation | Application |  $\Delta t$ factors
---|---|---|---|---
Transparent | OGC | OGBF | centralised | gNB (on-ground) | gNB (on-ground) | user+feeder(+ISLs)
OBBF | on-board
Regenerative | OGC | OGBF |  federated or centralised | gNB-CU (on-ground) | gNB-CU (on-ground) | user+feeder(+ISLs)
OBBF | gNB-DU (on-board)
OBC | OBBF | gNB-DU (on-board) | gNB-DU (on-board) | user(+ISLs)1
1: The ISLs with OBC solutions are intra-swarm and, thus, might be negligible in terms of additional latency. |

##  II System Architecture

In this Section, we provide a thorough description on the NTN architecture
design to support MIMO solutions. Notably, the implementation of beamforming
algorithms, detailed in Section III, is based on the knowledge of CSI or
location information at the transmitter side that shall be provided by the
user terminals. Different architecture options can be considered depending on
where the beamforming coefficients are computed and where they are applied to
the users’ signals, which is defined also based on the payload capabilities.
Section II-A describes the system architecture and the available design
options, while in Section II-B we discuss how they impact the beamforming
algorithms due to the different signalling latencies.

###  II-A Architecture design options

To support CF-MIMO in NTN, the satellite system architecture is impacted by
many design choices, such as: i) the type of satellite payload, _i.e._ ,
regenerative or transparent; ii) the type of functional split when
regenerative payloads are assumed, _i.e._ , which layers of the NR gNB are
implemented on-board in the Distributed Unit (gNB-DU) and which ones are
implemented on-ground in the Centralised Unit (gNB-CU); iii) the network
entity in which the beamforming coefficients are computed based on the
considered CF-MIMO algorithm; and iv) the network entity in which the
beamforming coefficients are applied to the signals, _i.e._ , On-Board
Beamforming (OBBF) or On-Ground Beamforming (OGBF). The system architecture is
represented in Figure 1 and it includes:

  * •

The ground segment, which includes $G$ on-ground gateways (GWs) providing NTN
access to the Terrestrial Network(s) (TN). In particular, the $G$ GWs provide
the connectivity between the NGSO nodes in the constellation, the gNBs, and
the Next Generation Core network (NGC). As for the latter, the ground segment
also includes the Operations Support Systems (OSS), which is in charge of
managing the overall satellite system. Depending on the type of payload on-
board the NTN nodes, different elements are needed in this segment: i) with
transparent payloads, the full gNB shall be implemented on-ground and the NTN
nodes basically act as relays; ii) with regenerative payloads and functional
split, as shown in Figure 1, the gNB-DUs can be located on-board, leaving the
gNB-CUs on-ground. In the latter case, it shall be mentioned that each gNB-CU
(full gNB with transparent payloads) can manage up to tens of connections;
assuming one connection per beam, depending on the total number of beams per
NTN node, multiple gNB-CUs (full gNBs) might be needed to manage all of the
connections supported by the gNB-DUs (single node).

  * •

The Non-Terrestrial (NT) access segment, which includes the NGSO nodes in the
constellation. We refer to nodes since the elements in the NGSO constellation
can be any type of platform on one or more NGSO orbits, _e.g._ , LEO
satellites organised in one or more sub-constellations at different altitudes,
or a High Altitude Platform Stations (HAPS) formation. As mentioned above,
these nodes can implement either a transparent or a regenerative payload,
depending on the cost and complexity of the target system. With respect to
coverage, in 3GPP beam-based coverage solutions we might have: i) Earth-fixed
beams, _i.e._ , through digitally steering of the signals, the coverage area
generated by each node is fixed on-ground independently of its position on the
orbit (as long as it falls in the node field of view); or ii) Earth-moving
beams, _i.e._ , the coverage area of each node is always centered around its
Sub Node Point (SNP) and, thus, the beams move on-ground along with the node
on its orbit. When Cell-Free approaches are considered, the very concept of
beams is not necessary anymore, as extensively discussed in Section III.

  * •

The on-ground user segment, composed by a potentially massive number of User
Equipments (UE), either fixed or moving. These directly connect to the serving
node(s) by means of the Uu air interface through the user access link. In this
work, we consider both handheld terminals and Very Small Aperture Terminals
(VSAT).

Based on the above observations, the selected functional split option, [52],
has an impact on where the users’ scheduling and beamforming coefficients are
computed; in particular, two architecture design options are possible: i) _On-
Ground beamforming Computation_ (OGC), where the scheduling and coefficients
computation is performed at the on-ground gNB-CU; or ii) _On-Board beamforming
Computation_ (OBC), where these operations are performed at the on-board gNB-
CUs. Moreover, with regenerative payloads, NGSO-based systems allow the
implementation of _federated_ (distributed) MIMO solutions, in which all of
the satellites in the same swarm (formation) cooperate to implement MIMO
transmissions.

###  II-B Centralised and Federated MIMO

Table I summarises the architecture options based on the payload type, where
the users’ scheduling and beamforming coefficients are computed (OGC or OBC),
where the beamforming coefficients are applied to the users’ signals (OGBF or
OBBF), and whether a federated MIMO solution is possible or not. Depending on
the selected option, the entity performing the different operations can be
identified.

With legacy transparent payloads, scheduling and beamforming are entirely
defined on-ground (OGC); then, the beamforming coefficients can be applied to
the users’ signals either on-board (OBBF) or on-ground (OGBF). In this
scenario, no federated solution is possible and each satellite in the
constellation operates as a standalone node; in fact, federated MIMO
architectures require a tight time and frequency synchronisation among the
cooperating satellites in the swarm, which can only be achieved by means of
intra-swarm Inter-Satellite Links (ISL) with regenerative payloads.

When considering future regenerative payloads, legacy centralised
architectures are clearly still possible. However, the exploitation of the
advanced on-board computational capabilities supports two additional MIMO
architecture options: i) OBC, in which the users’ scheduling and beamforming
coefficients are computed on-board; and ii) federated MIMO, in which multiple
satellites can tightly synchronise to realise a distributed MIMO system. OBC
solutions allow to perform all operations at the on-board gNB-DU: i)
computation of the users’ scheduling and beamforming coefficients; and ii)
application of the beamforming coefficients to the users’ signals. Such
advanced capabilities allow to implement either a centralised or a federated
MIMO algorithm. In the former case, each satellite in the NGSO constellation
can operate as a standalone NTN node, thus leading to a centralised
architecture; all of the UEs in the satellite’s service area send the required
ancillary information for the considered MIMO algorithm (CSI or location, as
discussed in Section III) on the return link and the satellite implements a
centralised MIMO algorithm. With federated MIMO, the synchronisation and
coordination among multiple NTN nodes in a single swarm is possible. In
particular: i) a master gNB-DU collects the ancillary information (CSI or
location) from the UEs, either directly or with the support of intra-swarm
ISLs; ii) the master gNB-DU computes the users’ scheduling and beamforming
coefficients and sends the beamformed signals to the other satellites in the
swarm; and iii) thanks to the intra-swarm ISLs, the satellites in the swarm
can tightly synchronise the transmission of the beamformed signals in the time
and frequency domains, leading federated MIMO via a distributed antenna
system. It is worthwhile highlighting that federated MIMO is also possible
with the on-ground computation of the users’ scheduling and beamforming
coefficients provided by OGC architectures; in this case, the NGSO satellites
in the swarm still realise a tightly synchronised distributed antenna system,
with the only difference being that the ancillary information from the UEs is
sent to the on-ground gNB-CU, which is in charge of all computations. Then,
the beamforming coefficients can be applied either on-ground (OGBF) or on-
board (OBBF).

On the one hand, the implementation of federated MIMO is challenging in terms
of increased system complexity, due to the need for regenerative payloads with
advanced on-board processors and tight intra-swarm synchronisation via ISLs.
On the other hand, the deployment of a flying distributed antenna system
allows to tackle the detrimental impact of harsh propagation environments,
thanks to the spatial diversity at the transmitter.

###  II-C Ancillary information aging: OGC and OBC

The choice between OGC and OBC is fundamental for MIMO in NTN. In fact, the
MIMO algorithms considered in this work (described in Section III) require
either the CSI vectors or the location estimated at the UEs’ locations to
build the beamforming matrix, denoted as _ancillary information_. The
ancillary information is obtained by the UEs at an _estimation time instant_
$t_{0}$ and then sent to the network element computing the coefficients: the
on-ground gNB(-CU) with OGC or the master on-board gNB-CU with OBC. The
transmission of the beamformed signals from either the federated gNB-CUs or
the standalone gNB-CU (centralised architectures) then happens at a
_transmission time instant_ $t_{1}>t_{0}$. During the _aging interval_ $\Delta
t=t_{1}-t_{0}$, both the NGSO nodes and the UEs have moved and, thus, there is
a misalignment between the actual channel encountered during the transmission
and the ancillary information used to compute the beamforming matrix. This is
represented in Figure 2, where only the swarm movement is depicted for the
sake of clarity. Notably, the MIMO performance is deeply impacted by any
misalignment between the actual channel and the beamforming matrix; thus, the
smaller the aging interval, the better the performance of federated MIMO.

When considering OGC, the users’ scheduling and coefficients are computed on-
ground and, thus, the aging interval can be computed as:

$\Delta
t_{OGC}=\tau_{user}+\tau_{feeder}^{(DL)}+\tau_{feeder}^{(UL)}+\tau_{p}+\tau_{rout}+\tau_{ad}$
(1)

where: i) $\tau_{user}$ is the latency on the user return link; ii)
$\tau_{feeder}^{(DL)}$ is the latency on the feeder downlink; iii)
$\tau_{feeder}^{(UL)}$ is the latency on the feeder uplink; iv) $\tau_{p}$ is
the processing delay to compute the users’ scheduling and beamforming
coefficients; v) $\tau_{rout}$ is the latency due to routing on the ISLs, if
present; and v) $\tau_{ad}$ includes any additional source of latency. When
OBC is considered, the aging interval is significantly reduced; in fact, all
computations are performed on-board and thus:

$\Delta t_{OBC}=\tau_{user}+\tau_{p}+\tau_{ad}$ (2)

Compared to $\Delta t_{OGC}$, $\Delta t_{OBC}$ only includes the over-the-air
latency on the user access link, in addition to the processing and additional
delays. Clearly, this advantage is lost if the functional split option does
not allow to perform the required computations on-board, _i.e._ , regenerative
payloads with a OGC approach. The factors impacting the aging interval $\Delta
t$ are summarised in Table I.

It is worthwhile highlighting that the advantage of OBC is not only related to
the reduction of the aging interval and, thus, improving the MIMO performance;
in fact, the signalling overhead on the feeder link, and on any inter-swarm
ISL that might be needed to connect the swarm with the serving gNB-CU, is
massively reduced. The reduction in signalling also includes all information
that is needed to implement the desired Radio Resource Management (RRM)
algorithm, which might include the UEs’ capacity request and type of traffic,
as well as the terminal class. With respect to the latter, it shall be
mentioned that this information might be classified by the manufacturers; in
this case, an estimate can be identified based on ancillary terminal
parameters/information. RRM aspects are not addressed in this work, without
impacting the generality of the proposed architectures or algorithms. Finally,
it is worthwhile highlighting that both the above advantages are achieved with
both centralised and federated MIMO architectures.

Figure 2: System architecture for federated CF-MIMO in NGSO-based NTN:
estimation and transmission phases.

##  III System Model

We consider a constellation with $M$ NGSO nodes providing connectivity to the
on-ground UEs in the service area. Notably, for a generic coverage area, only
a subset of nodes will be visible from all of the UEs, based on the nodes’
field of view and on minimum elevation angle requirements. In the following,
we assume that a single swarm of $N_{node}$ nodes is visible from all of the
$N_{ue}$ UEs in the considered area. Each node is equipped with a Uniform
Planar Array (UPA) with $N_{F}$ radiating elements; for the sake of
simplicity, and without affecting the generality of our work, we assume that
all of the nodes are at the same altitude and equipped with the same antenna
configuration (_i.e._ , number of radiating elements, UPA configuration,
element radiation pattern).

As discussed in Section II-C, the users estimate the ancillary information
required for the considered MIMO algorithm (CSI or location) at the estimation
time instant $t_{0}$; then, the actual transmission of the beamformed signals
occurs at the transmission time instant $t_{1}$. During the aging interval,
there are several sources of misalignment between the actual channel at
$t_{1}$ and the beamforming matrix based on information obtained at $t_{0}$:
i) the nodes moved along their orbits; ii) the UEs might have moved, depending
on the terminal type; and iii) different realisations of the stochastic terms
in the channel coefficients (_e.g._ , large scale losses, scintillation) are
present. As for the latter, the generic coefficient between the $i$-th on-
ground UE and the $n$-th radiating element of the UPA on-board the $s$-th node
at the generic time instant $t$ can be computed as:

$h_{i,n,s}^{(t)}=\frac{g_{i,n,s}^{(TX,t)}g_{i,n,s}^{(RX,t)}}{4\pi\frac{d_{i,s}^{(t)}}{\lambda}\sqrt{L_{i,s}^{(t)}\kappa
BT_{i}}}e^{-\jmath\frac{2\pi}{\lambda}d_{i,s}^{(t)}}e^{-\jmath\varphi_{i,s}^{(t)}}$
(3)

where: i) $d_{i,s}^{(t)}$ is the slant range between the $i$-th user and the
$s$-th node, which is assumed to be the same for all the co-located radiating
elements on-board the node; ii) $\lambda$ is the signal wavelength; iii)
$\kappa BT_{i}$ denotes the thermal noise power, with $B$ being the user
bandwidth (for simplicity assumed to be the same for all users) and $T_{i}$
the equivalent noise temperature of the $i$-th receiver; iv) $L_{i,s}^{(t)}$
represents the additional losses between the $s$-th node and the $i$-th user,
assumed to be the same for all the co-located radiating elements on-board the
node; v) $g_{i,n,s}^{(TX,t)}$ and $g_{i,n,s}^{(RX,t)}$ represent the
transmitting and receiving complex antenna patterns between the $i$-th user
and the $n$-th radiating element on-board the $s$-th node, respectively; and
vi) $\varphi_{i,s}^{(t)}$ is the phase misalignment that might be present
between different nodes due to non-ideal swarm synchronisation, modelled as a
Gaussian random variable (r.v.) $\mathcal{N}\left(0,2\pi\right)$. The
additional losses are computed based on TR 38.811, [53]:

$L_{i,s}^{(t)}=L_{i,s}^{(SHA,t)}+L_{i,s}^{(ATM,t)}+L_{i,s}^{(SCI,t)}+L_{i,s}^{(CL,t)}$
(4)

in which: i) $L_{i,s}^{(SHA,t)}$ denotes the log-normal shadowing loss with
standard deviation $\sigma_{SHA}$; ii) $L_{i,s}^{(ATM,t)}$ includes the
atmospheric loss due to gaseous absorption; iii) $L_{i,s}^{(SCI,t)}$ is the
scintillation loss; and iv) $L_{i,s}^{(CL,t)}$ is the Clutter Loss (CL), to be
included for UEs in Non-Line-of-Sight (NLOS) conditions. Referring to the 3GPP
channel model, the UE is defined to be in LOS or NLOS conditions with a
probability that is a function of the elevation angle and the propagation
environment (sub-urban, urban, dense-urban). In this context, we assume that a
UE that is LOS (NLOS) conditions during the estimation phase is still in LOS
(NLOS) conditions in the transmission phase. This assumption is motivated by
observing that the probability that the propagation conditions of the UE will
change from LOS (NLOS) to NLOS (LOS) after a few ms is negligible; in fact, as
shown in Figure 3, the values of the differential elevation angle between
$t_{0}$ and $t_{1}$ are negligible111These results where obtained with
$N_{node}=2$ and the system configuration described in Section IV.,
considering that the probabilities of LOS or NLOS conditions are provided with
a $10^{\circ}$ granularity in [53]. This assumption implies that the UE has
the same CL and $\sigma_{SHA}$ in both the estimation and transmission phases,
but the realisations of the log-normal r.v. modelling the shadowing are
different.

Figure 3: Probability density function (pdf) of the difference between the
elevation angle at $t_{1}$ and the elevation angle at $t_{1}$, $N_{node}=2$.

In the following, the estimated CSI vector between the $s$-th node and the
$i$-th user is represented by
$\mathbf{h}_{i,s}^{(t_{0})}=\left[h_{i,1,s}^{(t_{0})},\ldots,h_{i,N_{F},s}^{(t_{0})}\right]$.
For the generic $i$-th user, its overall channel signature can be obtained by
collecting the CSI vectors from all of the NGSO nodes into the
$N_{node}N_{F}$-dimensional
$\mathbf{h}_{i,:}^{(t_{0})}=\left[\mathbf{h}_{i,1}^{(t_{0})},\ldots,\mathbf{h}_{i,N_{node}}^{(t_{0})}\right]$.
Finally, the overall $N_{ue}\times\left(N_{node}N_{F}\right)$ channel matrix
at the estimation time $t_{0}$ is given by
$\mathbf{H}^{(t_{0})}_{sys}={\left[{\left(\mathbf{h}_{1,:}^{(t_{0})}\right)}^{T},\ldots,{\left(\mathbf{h}_{N_{node},:}^{(t_{0})}\right)}^{T}\right]}^{T}$.
For each time slot, the RRM scheduling function $\mathcal{S}$ provides a
subset of $K$ users to be served, leading to a
$K\times\left(N_{node}N_{F}\right)$ channel matrix
$\mathbf{H}^{(t_{0})}=\mathcal{S}\left(\mathbf{H}_{sys}^{(t_{0})}\right)\subseteq\mathbf{H}^{(t_{0})}_{sys}$.
Based on the channel matrix estimated at $t_{0}$, the beamforming algorithm
(detailed in the next sections) provides a $\left(N_{node}N_{F}\right)\times
K$ complex beamforming matrix
$\mathbf{W}^{(t_{0})}=\mathcal{B}\left(\mathbf{H}^{(t_{0})}\right)$, which
projects the $K$-dimensional column vector
$\mathbf{s}={\left[s_{1},\ldots,s_{N_{ue}}\right]}^{T}$ containing the unit-
variance user symbols into the $\left(N_{node}N_{F}\right)$-dimensional space
defined by all of the swarm radiating elements. The signal received by the
generic $i$-th UE in FFR is given by:

$y_{i}=\underbrace{\mathbf{h}_{i,:}^{(t_{1})}\mathbf{w}_{:,i}^{(t_{0})}s_{i}}_{\mathrm{intended}}+\underbrace{\sum_{\begin{subarray}{c}\ell=1\\\
\ell\neq
i\end{subarray}}^{K}\mathbf{h}_{i,:}^{(t_{1})}\mathbf{w}_{:,\ell}^{(t_{0})}s_{\ell}}_{\mathrm{interfering}}+z_{i}$
(5)

where $z_{i}$ is a circularly symmetric Gaussian r.v. with zero mean and unit
variance, which is licit observing that the channel coefficients in (3) are
normalised to the noise power. From (5), the $K$-dimensional vector of
received symbols is:

$\mathbf{y}=\mathbf{H}^{(t_{1})}\mathbf{W}^{(t_{0})}\mathbf{s}+\mathbf{z}$ (6)

As previously discussed, it can be noticed that there is a misalignment
between the estimated channel matrix exploited to compute
$\mathbf{W}^{(t_{0})}$, function of $\mathbf{H}^{(t_{0})}$, and the actual
channel in the transmission phase, $\mathbf{H}^{(t_{1})}$.

The Signal-to-Interference-plus-Noise Ratio (SINR) of the generic $i$-th UE
can be obtained as:

$\gamma_{i}=\frac{{\left|\mathbf{h}_{i,:}^{(t_{1})}\mathbf{w}_{:,i}^{(t_{0})}\right|}^{2}}{1+\sum_{\begin{subarray}{c}\ell=1\\\
\ell\neq
i\end{subarray}}^{K}{\left|\mathbf{h}_{i,:}^{(t_{1})}\mathbf{w}_{:,\ell}^{(t_{0})}\right|}^{2}}$
(7)

From the above SINR, the rate achieved by the $i$-th user can be evaluated
either from the Shannon bound formula or from the adopted Modulation and
Coding scheme (MCS). In this framework, 3GPP TR 38.803 reports that the
spectral efficiency for system-level simulations can be obtained through the
following truncated form of the Shannon bound, [54]:

$\eta_{i}=\begin{cases}0,\ \gamma_{i}<\gamma_{min}\\\
\varepsilon\cdot\log_{2}\left(1+\gamma_{i}\right),\
\gamma_{min}\leq\gamma_{i}<\gamma_{max}\\\
\varepsilon\cdot\log_{2}\left(1+\gamma_{max}\right),\
\gamma_{i}\geq\gamma_{max}\end{cases}$ (8)

where $\gamma_{min}=-10$ dB and $\gamma_{max}=30$ dB are the minimum and
maximum SINR of the MCS, respectively, and $\varepsilon$ is an attenuation
factor representing the implementation loss. Since the attenuation factor is a
multiplicative term outside of the Shannon formula, in the following we assume
$\varepsilon=1$ since a different value only acts as a scaling factor on all
of the results discussed below, _i.e._ , it does not impact the relationship
among the different techniques and scenarios and the general trends.

###  III-A CSI-based CF-MIMO

CSI-based techniques require each UE to estimate the CSI vector
$\mathbf{h}_{i,:}^{(t_{0})}$ during the estimation phase and to report it to
the network element in charge of the computation of $\mathbf{W}^{(t_{0})}$
(_i.e._ , gNB-DU with OGC or gNB-DU with OBC). Notably, among these, Minimum
Mean Square Error (MMSE) beamforming is the best algorithm in the sense of
SINR maximisation; as such, it is considered as the upper-bound performance
benchmark222This is a an equivalent and more computationally efficient
formulation of the MMSE beamformer, [55].:

$\mathbf{W}_{MMSE}^{(t_{0})}={\mathbf{H}}^{H}\left({\mathbf{H}}{\mathbf{H}}^{H}+\mathrm{diag}(\bm{\alpha})I_{K}\right)^{-1}$
(9)

where, for the sake of clarity, we dropped the time instant $t_{0}$ from the
channel matrix. In the above equation, $\mathrm{diag}(\bm{\alpha})$ is a
vector of $K$ regularisation factors; since the channel coefficients are
normalised to the noise power, the optimal value is given by
$\bm{\alpha}=N_{F}/P_{t}$, [56], where $P_{t}$ is the available power per node
in the swarm.

TABLE II: Summary of the considered beamforming algorithms. Algorithm | Channel coefficient | Beamforming matrix | Information | Errors
---|---|---|---|---
MMSE | $h_{i,n,s}^{(t)}=\frac{g_{i,n,s}^{(TX,t)}g_{i,n,s}^{(RX,t)}}{4\pi\frac{d_{i,s}^{(t)}}{\lambda}\sqrt{L_{i,s}^{(t)}\kappa BT_{i}}}e^{-\jmath\frac{2\pi}{\lambda}d_{i,s}^{(t)}}e^{-\jmath\varphi_{i,s}^{(t)}}$ | ${\mathbf{H}}^{H}\left({\mathbf{H}}{\mathbf{H}}^{H}+\mathrm{diag}(\bm{\alpha})I_{K}\right)^{-1}$ | CSI |  CSI estimation Low-SINR estimation Air interface adj. UE/node movement
LB-MMSE | $\widetilde{h}_{i,n,s}^{(t)}=\frac{\widetilde{g}_{i,n,s}^{(TX,t)}\widetilde{g}_{i,n,s}^{(RX,t)}}{4\pi\frac{\widetilde{d}_{i,s}^{(t)}}{\lambda}\sqrt{\kappa BT_{i}}}e^{-\jmath\frac{2\pi}{\lambda}\widetilde{d}_{i,s}^{(t)}}$ | ${\widetilde{\mathbf{H}}}^{H}\left(\widetilde{\mathbf{H}}{\widetilde{\mathbf{H}}}^{H}+\mathrm{diag}(\bm{\alpha})I_{K}\right)^{-1}$ | Location |  Location estimation Radiation pattern model UE/node movement
SS-MMSE | $\widetilde{h}_{i,n,s}^{(BC,t)}=\frac{\widetilde{g}_{i,n,s}^{(BC,TX,t)}\widetilde{g}_{i,n,s}^{(BC,RX,t)}}{4\pi\frac{\widetilde{d}_{i,s}^{(BC,t)}}{\lambda}\sqrt{\kappa BT_{i}}}e^{-\jmath\frac{2\pi}{\lambda}\widetilde{d}_{BC,i,s}^{(t)}}$ | ${\widetilde{\mathbf{H}}}^{H}_{BC}\left(\widetilde{\mathbf{H}}_{BC}{\widetilde{\mathbf{H}}}^{H}_{BC}+\mathrm{diag}(\bm{\alpha})I_{K}\right)^{-1}$ | Location |  Location estimation Radiation pattern model Approx. location UE/node movement
MB | $b_{n,\ell}=\frac{1}{\sqrt{N_{F}}}e^{-\jmath k_{0}\mathbf{r}_{n}\cdot\mathbf{c}_{\ell}}$ | ${\left[b_{n,\ell}\right]}_{\begin{subarray}{c}n=1,\ldots,N_{F}\\\ \ell=1,\ldots,N_{B}\end{subarray}}$ | Location |  Location estimation Approx. location UE/node movement

###  III-B Location-based CF-MIMO

Inspired by the Spatially Sampled (SS-MMSE) algorithm proposed in [25], we
design a new location-based CF-MIMO solution. In particular, when the UEs are
equipped with GNSS capabilities, they can estimate their locations and provide
them to the gNB-CU (OGC) or gNB-DU (OBC). This information, combined with the
knowledge of the swarm ephemeris, can be exploited to infer the channel
coefficients between the UE and the NGSO nodes. In particular, all terms in
(3) can be estimated, exception made for the additional losses and the phase
misalignment, which are stochastic terms:

$\widetilde{h}_{i,n,s}^{(t)}=\frac{\widetilde{g}_{i,n,s}^{(TX,t)}\widetilde{g}_{i,n,s}^{(RX,t)}}{4\pi\frac{\widetilde{d}_{i,s}^{(t)}}{\lambda}\sqrt{\kappa
BT_{i}}}e^{-\jmath\frac{2\pi}{\lambda}\widetilde{d}_{i,s}^{(t)}}$ (10)

where the tilde denotes that the terms are actually deduced from the relative
positions of the UE and the swarm nodes. Clearly, the accuracy of the
estimated channel coefficient is directly impacted by the accuracy of the
location estimate performed by the UE and by the accuracy of the ephemeris
data. In addition, as extensively discussed in Section IV-C,
$\widetilde{g}_{i,n,s}^{(TX,t)}$ and $\widetilde{g}_{i,n,s}^{(RX,t)}$ might
also be impacted by a non-ideal knowledge of the radiation pattern, _i.e._ ,
the antenna mathematical model used to compute the antenna pattern might not
be flawlessly representing the actual radiation. From (10), we can compute the
beamforming matrix for the proposed _Location-Based MMSE_ (LB-MMSE) as:

$\mathbf{W}_{LB-
MMSE}^{(t_{0})}={\widetilde{\mathbf{H}}}^{H}\left(\widetilde{\mathbf{H}}{\widetilde{\mathbf{H}}}^{H}+\mathrm{diag}(\bm{\alpha})I_{K}\right)^{-1}$
(11)

Compared to CSI-based techniques, LB-MMSE has the advantage of not requiring
the UEs to estimate the channel coefficients, with manifold benefits. Firstly,
the signalling overhead is significantly reduced, as the CSI vectors are not
needed at the gNB-CU (OGC) or gNB-DU (OBC) and the UEs’ positions require much
smaller data packets to be transmitted333Assuming $32$ bits for floating point
values, $64$ bits are needed per channel coefficient per user; this leads to
$64\cdot N_{F}$ bits per user. For the location, each coordinate in the Global
Positioning System (GPS) requires $24$ bits, leading to $48$ bits per user..
Moreover, it shall be noticed that the estimation of the channel coefficients
is typically assuming the presence of beams, each of which is associated to a
known Data/Pilot sequence on which the estimation is performed; when moving to
CF solutions, the concept of beam is lost and, thus, adjustments to the air
interfaces would be needed. Finally, this approach also avoids the well-known
issue of estimating the channel coefficients in low Signal-to-Interference
Ratio (SIR) conditions. In fact, some of the known Data/Pilot sequences might
be received with a significantly lower power compared to others. In such
conditions, the UE is not able to estimate the channel coefficient, which
shall then be either inferred at the gNB-CU/gNB-DU through more complex
approaches, _e.g._ , Machine Learning, or reported as a null in the channel
matrix, which might lead to sparse ill-conditioned matrices.

###  III-C Beam-based MIMO

Aiming at providing a complete performance comparison for the newly designed
LB-MMSE algorithm, we also consider two beam-based solutions. The first one is
a system implementing a _phase-only_ digital beam steering. In this case, an
on-ground hexagonal beam lattice is defined based on the desired number of
tiers around the center of the coverage area and on the 3 dB angular
beamwidth, $\vartheta_{3dB}$, [57]. Denoting as $\mathbf{c}_{\ell,s}$ the
$(u,v)$ coordinates of the generic $\ell$-th beam center from the $s$-th
node444The $(u,v)$ system is centered at the satellite and, thus, the
coordinates depend on the considered node, [55]., its $n$-th beamforming
coefficient is:

$b_{n,\ell,s}=\frac{1}{\sqrt{N_{F}}}e^{-\jmath
k_{0}\mathbf{r}_{n,s}\cdot\mathbf{c}_{\ell,s}}$ (12)

where $k_{0}=2\pi/\lambda$ is the wave number and $\mathbf{r}_{n,s}$ the
position of the $n$-th radiating element on the $s$-th on-board UPA. Assuming
that in each time slot one user per beam is served, the beamforming matrix of
the $s$-th node is fixed and it is given by
$\mathbf{W}_{PO,s}={\left[b_{n,\ell,s}\right]}_{\begin{subarray}{c}n=1,\ldots,N_{F}\\\
\ell=1,\ldots,N_{B}\end{subarray}}$, where $N_{B}$ is the number of beams. The
overall $\left(N_{node}N_{F}\right)\times N_{B}$ beamforming matrix
$\mathbf{W}_{PO}$ is obtained by vertically concatenating the $N_{F}\times
N_{B}$ $\mathbf{W}_{PO,s}$, $s=1,\ldots,N_{node}$, node matrices. It shall be
noticed that, when advanced scheduling algorithms are implemented and not all
of the beams are illuminated in all time slots, the Switchable Multi-Beam (MB)
MIMO algorithm proposed in [55] is obtained. In the following, we refer to MB
beamforming also with all beams being illuminated.

The second beam-based approach is the SS-MMSE algorithm, [25]. In this case,
the generic $i$-th user is associated to the closest beam center, thus leading
to a beam-based solution; then, the channel coefficients are estimated based
on (10), in which each term is computed based on the corresponding beam center
location, and not the estimated UE location:

$\widetilde{h}_{i,n,s}^{(BC,t)}=\frac{\widetilde{g}_{i,n,s}^{(BC,TX,t)}\widetilde{g}_{i,n,s}^{(BC,RX,t)}}{4\pi\frac{\widetilde{d}_{i,s}^{(BC,t)}}{\lambda}\sqrt{\kappa
BT_{i}}}e^{-\jmath\frac{2\pi}{\lambda}\widetilde{d}_{BC,i,s}^{(t)}}$ (13)

where $BC$ indicates that the terms shall be computed at the corresponding
beam center. From this estimated channel matrix, the MMSE equation is again
applied to obtain $\mathbf{W}_{SS-MMSE}$. Thus, this approach is different
from LB-MMSE since is is beam-based and it approximates the UEs’ locations to
the those of closest beam centers.

Table II summarises the different beamforming techniques, reporting the
required information, the beamforming and channel coefficient equations, and
the potential sources of misalignment between the channel matrices at $t_{0}$
and $t_{1}$. It shall be noticed that, for SS-MMSE,
$\widetilde{\mathbf{H}}_{BC}$ denotes the beam centers channel matrix
estimated based on (13).

###  III-D Power normalisation

The normalisation of the beamforming matrix is a fundamental operation, as
extensively discussed in [25, 38]. In fact, the Frobenius norm of the
beamforming matrix, ${\left\|\mathbf{W}\right\|}_{F}^{2}$, represents the
total emitted power and, by applying the MMSE formula, there is no guarantee
that such power will be upper-bounded so as to not exceed the total available
power. First considering a single node in the swarm, for the sake of
simplicity, this means that there might be situations in which
${\left\|\mathbf{W}\right\|}_{F}^{2}>P_{t}$, where $P_{t}$ denotes the total
available power on-board. Aiming at addressing this issue, several
normalisations have been considered in the literature:

  * •

Sum Power Constraint (SPC):

$\widetilde{\mathbf{W}}=\sqrt{\frac{P_{t}}{{\left\|\mathbf{W}\right\|}_{F}^{2}}}\mathbf{W}=\sqrt{\frac{P_{t}}{\mathrm{tr}\left(\mathbf{W}\mathbf{W}^{H}\right)}}\mathbf{W}$
(14)

This normalisation guarantees that: i) the overall power allocated by the
beamforming matrix is equal to that actually available, _i.e._ ,
${\left\|\mathbf{W}\right\|}_{F}^{2}=P_{t}$; ii) preserves the orthogonality
among the beamforming matrix columns, _i.e._ , it does not disrupt the optimal
MMSE solution since all columns are normalised by the same scalar quantity.
However, SPC does not control the power emitted per radiating element, which
might lead to a performance degradation due to driving the on-board High Power
Amplifiers (HPAs) close to or above the saturation level, thus introducing
undesired non-linear effects.

  * •

Maximum Power Constraint (MPC)

$\widetilde{\mathbf{W}}=\sqrt{\frac{P_{t}}{K\max_{k=1,\ldots,K}{\left\|\mathbf{w}_{:,k}\right\|}^{2}}}\mathbf{W}$
(15)

In this case, the normalisation is similar to SPC, with the only difference
being that only one radiating element is emitting the maximum allowed power,
while all of the others emit a lower power level. Thus: i) the overall emitted
power is below the available power, _i.e._ ,
${\left\|\mathbf{W}\right\|}_{F}^{2}<P_{t}$; ii) the orthogonality among the
beamforming matrix columns is preserved; and iii) the emitted power is limited
per radiating element. However, since only one radiating element is emitting
the maximum power, while the others are significantly limited in their
emissions, this approach might lead to a degradation of the SNR and, in
general, to a performance loss.

  * •

Per Antenna Constraint (PAC)

$\widetilde{\mathbf{W}}=\sqrt{\frac{P_{t}}{K}}\mathrm{diag}\left(\frac{1}{\left\|\mathbf{w}_{1,:}\right\|},\ldots,\frac{1}{\left\|\mathbf{w}_{K,:}\right\|}\right)\mathbf{W}$
(16)

With this approach, each radiating element transmits at an equal power level,
thus ensuring that the overall available power is not exceeded, _i.e._ ,
${\left\|\mathbf{W}\right\|}_{F}^{2}=P_{t}$. However, since each row of the
beamforming matrix is normalised independently from each other, the
orthogonality of the beamformer columns are disrupted and potentially large
performance degradation is introduced.

When multiple NGSO nodes are considered, some further considerations are
needed. In this case, the beamforming matrix includes the power emitted by
radiating elements on-board different nodes. Consequently, the normalisations
shall be adjusted to satisfy the power constraints per node. Assuming that
each node has the same available on-board power, $P_{t,node}$, we introduce
the following normalisations:

  * •

swarm SPC (sSPC): directly applying the SPC normalisation with a total power
$N_{node}P_{t,node}$ guarantees that this power is not exceeded at swarm
level, _i.e._ , ${\left\|\mathbf{W}\right\|}_{F}^{2}=N_{node}P_{t,node}$;
however, a single node might be required to emit more power than available. To
circumvent this issue, the swarm-based sSPC normalisation is introduced based
on the observation that the overall $\left(N_{node}N_{F}\right)\times K$
beamforming matrix can be divided in blocks corresponding to the single nodes
beamforming matrices, _i.e._ :

$\mathbf{W}=\begin{bmatrix}\mathbf{W}_{1}\\\ \vdots\\\
\mathbf{W}_{N_{node}}\end{bmatrix}$ (17)

with $\mathbf{W}_{s}$ denoting the $N_{F}\times K$ beamforming matrix of the
$s$-th NGSO node. Each node beamforming matrix can be normalised with the SPC
approach as a standalone matrix, guaranteeing that: i) the overall emitted
power satisfies ${\left\|\mathbf{W}\right\|}_{F}^{2}=N_{node}P_{t,node}$; and
ii) each satellite emits a power
${\left\|\mathbf{W}_{s}\right\|}_{F}^{2}=P_{t,node}$, $s=1,\ldots,N_{node}$.
Clearly, this approach leads to a slight degradation in the performance,
because a normalisation that is not scalar for the entire beamforming matrix
$\mathbf{W}$ leads to a loss of orthogonality in the beamforming matrix
columns, _i.e._ , to a loss in its interference cancellation capabilities.

  * •

swarm MPC (sMPC): in this case, there are two options. In fact, if the
objective is to preserve the orthogonality in the beamforming matrix columns,
then (15) can be directly applied since only one radiating element will emit
its maximum power; this guarantees the preservation of the orthogonality, but
actually leads to lower emitted power levels, since only a single element from
a single node in the swarm will transmit the maximum power. Another
possibility is to better exploit the available power, by normalising with the
MPC approach each node matrix, as in the sSPC solution; this guarantees that
the overall emitted power is still satisfying the condition
${\left\|\mathbf{W}\right\|}_{F}^{2}<N_{node}P_{t,node}$ and that each node
emits a power ${\left\|\mathbf{W}_{s}\right\|}_{F}^{2}<P_{t,node}$,
$s=1,\ldots,N_{node}$. Only a single element per node emits its maximum power.
In the following, we assume the latter solution is implemented.

As for PAC, from (16) it can be noticed that no issue arises since each row of
the beamforming matrix is individually normalised to guarantee that each
radiating element emits the same power level. Thus, as previously discussed,
we are ensuring that each nodes emits a total power $P_{t,node}$ and that the
swarm collectively emits a power $N_{node}P_{t,node}$, with a performance loss
related to the disrupted orthogonality in the beamforming matrix columns on a
node basis.

In the following, we consider the (s)SPC and (s)MPC normalisations. It was
observed that, as expected, the PAC approach has a poor interference
cancellation performance and, as such, it usually leads to very low SINRs. The
(s)MPC solution is a viable normalisation approach since it satisfies all
power constraints and avoids operating the on-board HPAs in the non-linear
region. The (s)SPC normalisation is retained as it provides an upper-bound
performance, even though not practically implementable due to the possibility
that a node is required to transmit more power than available.

TABLE III: System configuration parameters for the numerical assessment. Parameter | Value
---|---
Transmission EIRP |  $0,4,8,12$ dBW/MHz
Channel model | clear-sky, NLOS
Propagation environment | dense-urban
Carrier frequency | S-band ($2$ GHz)
User bandwidth |  $30$ MHz
Receiver type |  fixed VSAT/handheld (antenna parameters in [57])
$N_{F}$ |  $1024$ ($32\times 32$ UPA) $0.55\lambda$ spacing
Nodes per swarm $N_{node}$ | $1,2$
Altitude |  $600$ km
Beams per node $N_{B,node}$ |  $91(N_{node}=1)$ $61(N_{node}=2)$
User density |  $0.5$ users/km2
Aging interval $\Delta t$ |  $16.7$ ms
Scheduling | random

##  IV Numerical Assessment

In this Section, we discuss the extensive numerical results obtained through
Monte Carlo simulations with the MMSE, LB-MMSE, SS-MMSE, and MB algorithms.
The system configuration parameters for the numerical assessment are reported
in Table III. It is worthwhile noticing that we consider with $\Delta t=16.7$
ms, computed based on OGC; clearly, lower values will results in a better
performance, while larger values in worse results due to the reduced/increased
information aging at the transmitter. However, the general trends discussed
below still hold. Moreover, for the following system-level analyses, only the
value of $\Delta t$ impacts the results and not the specific choice of a OGC
or OBC design.

###  IV-A Simulation scenarios, assumptions, and metrics

The users are uniformly distributed in the coverage area. Based on the user
density reported in Table III, approximately $30000$ users are considered. The
UEs are assumed to be fixed; in fact, by means of extensive numerical
simulations, it was observed that for users moving at up to $250$ km/h the
performance loss was in the order of $10^{-4}$ bit/s/Hz in terms of spectral
efficiency, _i.e._ , negligible. To provide connectivity to the uniformly
distributed users, the UPA on-board each node is designed as a square lattice
with $N_{F}=1024$ radiating elements; the spacing between adjacent elements is
fixed at $0.55\lambda$. Notably, the radiation pattern from the $n$-th element
on the $s$-th node in the direction of the $i$-th user, identified by the
direction cosines coordinates
$\mathbf{p}_{i,s}^{(t)}=\left[u_{i,s}^{(t)},v_{i,s}^{(t)}\right]$ at the
$t$-th time instant, is obtained as, [55]:

$g^{(TX,t)}_{i,n,s}=g_{E,i}^{(t)}e^{\jmath
k_{0}\mathbf{r}_{n,s}\cdot\mathbf{p}_{i,s}^{(t)}}$ (18)

where $g_{E,i}$ is the radiation pattern of the single radiating element in
the direction of the $i$-th user $\mathbf{r}_{n,s}$ the position of the $n$-th
element on the $s$-th UPA. The element pattern $g_{E,i}$ is assumed to be the
same for all radiating elements and computed as in [58, Section 5.1].

Figure 4: Centralised ($N_{node}=1$) scenario.

We consider both a centralised ($N_{node}=1$) and federated ($N_{node}=2$)
configurations. In order to provide a fair comparison between Cell-Free (MMSE,
LB-MMSE) and beam-based (SS-MMSE, MB) algorithms, each node in the swarm
generates $N_{B,node}$ beams organised in an hexagonal lattice on-ground. The
number of beams per node is different in the two scenarios so as to obtain a
similar number of beams per swarm $N_{B}$ ($91$ with $N_{node}=1$ and $122$
with $N_{node}=2$). Moreover, it shall be noticed that, with multiple nodes,
each node in the swarm generates its corresponding lattice, but then it covers
all of the beams created by the swarm. Figures 4 and 5 show the on-ground beam
footprints with $N_{node}=1$ and $N_{node}=2$, respectively. With multiple
nodes, the fact that each node is initially generating a single beam lattice
leads to beams that significantly overlap at the border between the two
lattices, _i.e._ , there are beams that have their centers inside other beams
boundaries at less than $-3$ dB. Notably, when implementing MIMO techniques,
this might lead to a performance loss, [38, 48]; in particular, two UEs
selected from two of such beams might be scheduled in the same time slot. In
this case, the two users have very similar channel signatures (CSI
coefficients) and, thus, the channel matrix to be inverted in MMSE-like
solutions might be ill-conditioned. To circumvent this issue, proper
scheduling algorithms might be implemented; in the following, we assume that
the RRM algorithm avoids such situations by activating only one beam among
those in which the relative distance among the beam centers does not guarantee
a $3$ dB separation. An example is shown in Figure 6. Consequently, to provide
a fair comparison between the centralised and federated scenarios, the
available power per node is scaled to guarantee that the same average power
per beam is available:

Figure 5: Federated ($N_{node}=2$) scenario. Blue lines represent the
footprint of satellite 1 and magenta lines those for satellite 2. Figure 6:
Example of RRM algorithm activating a subset of beams (green) to avoid an ill-
conditioned channel matrix due to beam proximity issues.

$P_{t,node}=P_{t}\frac{N_{B,multi}}{N_{node}N_{B,single}}$ (19)

where $P_{t,node}$ is the available power per node and $N_{B,single}$,
$N_{B,multi}$ the number of active beams with $N_{node}=1$ and $N_{node}=2$,
respectively. It can be noticed that when all beams are active in the multiple
nodes case (_i.e._ , there is no beam proximity issue): i) if
$N_{B,multi}=N_{node}N_{B,single}$, _i.e._ each node in the swarm generates
the same amount of beams as in the centralised case, then $P_{t,node}=P_{t}$;
and ii) if $N_{B,multi}=N_{B,single}$, _i.e._ the same number of beams is
generated independently of the number of nodes, $P_{t,node}=P_{t}/N_{node}$.
Referring to Figure 5 and 6, there are $118$ active beams compared to $91$
with a single node; a transmission power density of $4$ dBW/MHz leads to
$P_{t}=18.77$ dBW over a $30$ MHz bandwidth and to $P_{t,node}=16.89$ dBW/MHz.

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 7: Average spectral efficiency in clear-sky with $N_{node}=1$ (left)
and $N_{node}=2$ (right).

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 8: Percentage of unserved users in clear-sky with $N_{node}=1$ (left)
and $N_{node}=2$ (right).

With respect to scheduling, a random algorithm is implemented, [25, 38]: in
each time slot, one UE per beam is randomly selected and the total number of
time slots is computed so as to guarantee that all UEs are served at least
once. The coverage is based on Earth moving beams, _i.e._ , the beams move
together with the NGSO nodes along their orbits. It is worthwhile highlighting
that a beam lattice is only needed for the MB and SS-MMSE solutions; MMSE and
LB-MMSE allow to implement CF-MIMO, as long as an advanced scheduler operating
exlusively on the location or CSI vectors is implemented. Thus, the concept of
beams for MMSE and LB-MMSE is only exploited for the random scheduler and to
have a comparison with MB and SS-MMSE.

Two channels are considered: i) clear-sky, in which no additional loss is
present, _i.e._ , $L_{i,s}^{(t)}=1$, $\forall i,s,t$; and ii) NLOS, modelled
as described in Section III-C assuming a dense-urban environment, _i.e._ , the
worst conditions in terms of clutter loss and $\sigma_{SHA}$. In the latter
scenario each UE is in LOS or NLOS conditions according to the probabilities
reported in [53] as a function of its elevation angle.

Below, we discuss the numerical results obtained with both ideal and non-ideal
information to compute the beamforming matrix. The performance is provided in
terms of average spectral efficiency in [bit/s/Hz] and percentage of unserved
users. To this aim, it shall be mentioned that, based on the truncated Shannon
bound in (8), the average spectral efficiency is computed only on the UEs
which are served by the system, _i.e._ , those for which the SINR is above
$\gamma_{min}$. The percentage of unserved users is computed as the percentage
of users with SINR below $\gamma_{min}$.

###  IV-B Ideal information

In this Section, we discuss the numerical results under ideal conditions,
_i.e._ , ideal CSI estimation (MMSE), ideal location estimation (LB-MMSE, SS-
MMSE, MB), ideal knowledge of the radiation pattern (LB-MMSE and SS-MMSE). It
is worthwhile mentioning that, despite the ideal conditions, there is still a
misalignment between the beamforming matrix $\mathbf{W}^{(t_{0})}$ and the
channel matrix $\mathbf{H}^{(t_{1})}$ because of the movement of the swarm
nodes on their orbits.

Figure 7 shows the average spectral efficiency in clear-sky conditions with
$N_{node}=1$ and $N_{node}=2$. The following general trends can be observed:

  * •

MMSE and LB-MMSE provide the best performance for all transmission power
densities and terminal types. Moreover, they provide the same performance;
this is motivated by observing that, in the absence of additional losses and
with ideal estimations, the channel coefficients in (3) and (10) are
identical. The performance of the SS-MMSE and MB solutions are significantly
worse, with losses in the order of $6$ and $4$ bit/s/Hz with VSATs in the
centralised and federated scenarios.

  * •

As expected, the (s)SPC normalisation provides the best performance. However,
this solution provides a theoretical upper-bound but it is not feasible. With
(s)MPC, the performance is worse in particular in systems with large received
power (VSAT and large power density); this is motivated by the fact that the
(s)MPC normalisation does not exploit the entire available power.

  * •

Handheld terminals have a worse performance compared to VSATs, due to the
omni-directional antennas. The different techniques and normalisations provide
a similar performance with low EIRP (in this case, the performance is more
noise limited than interference limited), while with larger transmission power
levels interference increases and the impact of a different technique or
normalisation becomes slightly more evident.

(a) SNR

(b) SIR

Figure 9: CDF of the SNR (left) and SIR (right) with MMSE beamforming and
VSATs. For the centralised case, SPC is considered.

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 10: CDF of the SIR in the centralised (left) and federated (right)
scenarios with MMSE beamforming and handheld terminals.

Comparing the performance between the centralised and federated scenarios,
interestingly, it can be noticed that:

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 11: Average spectral efficiency in NLOS dense-urban conditions with
$N_{node}=1$ (left) and $N_{node}=2$ (right).

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 12: Percentage of unserved users in NLOS dense-urban conditions with
$N_{node}=1$ (left) and $N_{node}=2$ (right).

  * •

With both handheld and VSAT terminals, the spectral efficiency is larger for
increasing values of the power density. However, with multiple nodes and
VSATs, the performance tends to saturate with large power levels. Figure 9
shows the CDF of the SNR and SIR with MMSE beamforming. It can be observed
that, clearly, the SNR is always improved by increasing the power density;
however, the SIR is slightly worse with larger power with $N_{node}=2$, while
with $N_{node}=1$ it is improved. Thus, with multiple nodes in the swarm
interference cancellation is critical. This is motivated by observing that,
with $N_{node}$ nodes there are $N_{nodes}N_{F}$ radiating elements; this
leads to an increased sensitivity to any misalignment between
$\mathbf{W}^{(t_{0})}$ and $\mathbf{H}^{(t_{1})}$. Moreover, also the block
sSPC and sMPC normalisations with multiple nodes increase such misalignment.

  * •

The performance with VSATs and $N_{node}=2$ is beneficial only at low power
and with (s)MPC. In all other cases, _i.e._ , when the interference to be
dealt with by the beamformer is increased, the centralised scenario provides a
better result.

  * •

With handheld terminals, the average spectral efficiency is typically slightly
better ($0.05-0.1$ bit/s/Hz) with $N_{node}=2$. The only exception is when the
power density is large and (s)SPC normalisations are considered. However, in
this case we should also observe the percentage of unserved users, reported in
Figure 8. In the centralised case, there are no users in outage with MMSE and
LB-MMSE; with SS-MMSE, based on an approximation of the UEs’ locations, the
impact of a larger transmission power is more detrimental for the interference
cancellation capability and the outage increases. With two nodes, the
situation is worse. In particular:

    * –

with SS-MMSE there are many unserved users with both terminal types (up to
$22\%$). This is motivated by the much larger number of radiating elements and
the consequent impact on any misalignment between $\mathbf{W}^{(t_{0})}$ and
$\mathbf{H}^{(t_{1})}$, more evident in this case due to the further
approximation in the UEs’ locations;

    * –

there is a slight outage also with MMSE/LB-MMSE and VSATs, always below
$0.7\%$, motivated as above. With VSATs, the impact of such misalignment is
limited by the presence of directive antennas;

    * –

with MMSE/LB-MMSE and handheld terminals, the outage is significantly higher
(up to $19\%$ in some cases) and it tends to decrease for increasing power
levels. This trend is motivated by the omni-directional antennas at the
receiver. In fact, VSATs are assumed to be ideally pointing and tracking the
node providing the best radiation pattern, thus limiting the residual intra-
swarm interference. Handheld terminals do not have directive radiation
patterns and, thus, they are significantly more subject to the residual intra-
swarm interference. This behaviour is shown in Figure 10.

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 13: Loss in the average spectral efficiency in clear-sky conditions
with $N_{node}=1$ (left) and $N_{node}=2$ (right).

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 14: Loss in the average spectral efficiency in NLOS dense-urban
conditions with $N_{node}=1$ (left) and $N_{node}=2$ (right).

Figures 11 and 12 show the performance on the NLOS dense-urban channel. In
general, we can observe that the performance is worse compared to clear-sky
conditions in all cases, as expected. Moreover:

  * •

The LB-MMSE algorithm has a slightly worse performance compared to MMSE. Since
this channel includes the additional losses, which are not taken into account
by LB-MMSE, the misalignment between $\mathbf{W}^{(t_{0})}$ and
$\mathbf{H}^{(t_{1})}$ is larger with this algorithm. However, considering all
of the advantages that location-based solutions provide, extensively discussed
in Section III-B, this algorithm is indeed a viable and effective approach to
implement CF-MIMO in NTN systems. The performance with SS-MMSE and MB is still
much worse compared to MMSE and LB-MMSE, with a loss in the spectral
efficiency up to $4$ bit/s/Hz with VSAT receivers.

  * •

With respect to the normalisations, as in clear-sky conditions, (s)SPC
performs better than (s)MPC, which however is the practically implementable
solution.

  * •

Comparing the centralised and federated scenarios, we can observe that the
spectral efficiency is always better with a single node. In particular, with
large transmission power and VSAT receivers, its gain can be as large as $2$
bit/s/Hz. However, observing the percentage of unserved users, it can be
noticed that, with MMSE and LB-MMSE, the exploitation of a second node in the
swarm is significantly beneficial with: i) low transmission power and VSAT;
and ii) large transmission power and handheld. Thus, the exploitation of
additional nodes is beneficial in harsh propagation conditions, since it
introduces a gain in terms of path diversity. In dense-urban environments, the
advantage of a second node in terms of outage can be: i) with low transmission
power, as large as $6.1\%$ for MMSE-MPC with VSAT terminals; and ii) with
large transmission power, as large as $5.7\%$ for MMSE-SPC with handheld
terminals. In these conditions, despite the lower spectral efficiency, the
system would be able to serve many more users with two nodes.

###  IV-C Non-ideal information

In this Section, we discuss the performance obtained when non-ideal
information on the UEs’ position and the radiation pattern model is considered
for LB-MMSE and SS-MMSE.

#### Positioning

In this case, we assume that each user estimates its position with a uniformly
distributed error of $\mathcal{U}\left[0,10\right)$ meters, in a random
direction $\mathcal{U}\left[0,2\pi\right)$, with respect to its correct
location. It shall be noticed that the maximum location error is significantly
larger compared to the accuracy that current receivers equipped with GNSS can
achieve. This is aimed at showing the significant robustness of the proposed
LB-MMSE algorithm to a non-ideal estimation of the location.
Figures 13 and 14 show the loss in the average spectral efficiency in clear-
sky and NLOS dense-urban conditions, respectively. At most, the performance is
degraded by $0.04$ bit/s/Hz. This is in line with the previous statement
related to the movement of the users (_i.e._ , even when moving at $250$ km/h,
the performance loss is in the order of $10^{-4}$ bit/s/Hz). In terms of the
percentage of unserved users, the loss was observed to be negligible. Thus,
LB-MMSE and SS-MMSE are significantly robust to positioning errors.

#### Radiation pattern model

When computing the LB-MMSE and SS-MMSE channel coefficients based on the UEs’
location, another source of error is the non-ideal representativeness of the
radiation pattern model. To model this impairment, we focus on the on-board
radiation pattern model. In particular, we assume that the estimated antenna
radiation between the $n$-th element on-board the $s$-th satellite and the
$i$-th user at the $t$-th time instant, $\widehat{g}_{i,n,s}^{(TX,t)}$, is
given by:

$\widehat{g}_{i,n,s}^{(TX,t)}=g_{i,n,s}^{(TX,t)}+\Delta g_{i,n,s}^{(TX,t)}$
(20)

where $\Delta g_{i,n,s}^{(TX,t)}=\left|\Delta
g_{i,n,s}^{(TX,t)}\right|e^{-\jmath\angle\Delta g_{i,n,s}^{(TX,t)}}$ is a r.v.
modelling the error on the radiation pattern with the following amplitude and
phase statistics:

$\displaystyle\left|\Delta g_{i,n,s}^{(TX,t)}\right|$
$\displaystyle\sim\mathcal{N}\left(0,{\left|g_{i,n,s}^{(TX,t)}\right|}^{2}\varepsilon_{rp}\right)$
(21) $\displaystyle\angle\Delta g_{i,n,s}^{(TX,t)}$
$\displaystyle\sim\mathcal{N}\left(0,{\left|\angle
g_{i,n,s}^{(TX,t)}\right|}^{2}\varepsilon_{rp}\right)$ (22)

The coefficient $\varepsilon_{rp}$ allows to adjust the variance of the
amplitude and phase errors. In the following, we assume
$\varepsilon_{rp}=0.05$, _i.e._ , a $5\%$ error on the correct amplitude and
phase of the radiation pattern.

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 15: Average spectral efficiency in clear-sky with $N_{node}=1$ (left)
and $N_{node}=2$ (right) and non-ideal radiation pattern.

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 16: Average spectral efficiency in NLOS dense-urban conditions with
$N_{node}=1$ (left) and $N_{node}=2$ (right) and non-ideal radiation pattern.

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 17: Reduction in the percentage of unserved users in clear-sky with
$N_{node}=1$ (left) and $N_{node}=2$ (right) and $\varepsilon_{rp}=0.05$.

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 18: Reduction in the percentage of unserved users in NLOS dense-urban
conditions with $N_{node}=1$ (left) and $N_{node}=2$ (right) and
$\varepsilon_{rp}=0.05$.

Figures 15 and 16 show the average spectral efficiency for the LB-MMSE and SS-
MMSE algorithms and $\varepsilon_{rp}=0.05$. It can be noticed that both
algorithms are particularly impacted by a non-ideal knowledge. The loss with
VSATs in the centralised scenario can be as large as $4.5$ bit/s/Hz and $3.5$
bit/s/Hz in clear-sky and NLOS dense-urban conditions, respectively; with
federated solutions and two nodes, the performance is less degraded, since the
system benefits from the transmission from distributed sources, and the loss
is in the order of $1-1.5$ bit/s/Hz. With handheld terminals, the performance
with two nodes is still better, but the advantage is more limited compared to
the centralised case.
In terms of unserved users, some interesting behaviours arise. In particular,
with VSAT terminals the performance with an error on the radiation pattern
model is actually improved in terms of outage, while this phenomenon is absent
for handheld terminals. Figures 17 and 18 show the outage reduction
representing such gain; it can be noticed that in the centralised scenario
this phenomenon is more relevant. To understand the motivation for this
behaviour, let us focus on the centralised case in NLOS dense-urban
conditions, where it is more evident. In particular, Figures 19 and 20 show
the geographical distribution of the SINR with $\varepsilon_{rp}=0$ (ideal),
and $\varepsilon_{rp}=0.05$, respectively. It can be noticed that:

(a) LB-MMSE

(b) SS-MMSE

Figure 19: Geographical distribution of the SINR over $10$ time slots with LB-
MMSE (left) and SS-MMSE (right) in NLOS dense-urban conditions with
$\varepsilon_{rp}=0$ and a centralised node.

(a) LB-MMSE

(b) SS-MMSE

Figure 20: Geographical distribution of the SINR over $10$ time slots with LB-
MMSE (left) and SS-MMSE (right) in NLOS dense-urban conditions with
$\varepsilon_{rp}=0.05$ and a centralised node.

  * •

With LB-MMSE and $\varepsilon_{rp}=0$, the users in the outer part of the
coverage are those in the best channel conditions; the uniformly distributed
users that are experiencing a bad SINR are those in NLOS conditions, which
include the clutter loss and a harsh standard deviation of the shadow fading.
The best behaviour at the coverage edge is due to the combination of the
reduced interference in the external areas of the coverage region and the
nature of LB-MMSE, which is a user-centric technique in which the CSI vectors
are identified specifically per user without approximations, compared to SS-
MMSE. When $\varepsilon_{rp}=0.05$, the performance at the edge of the
coverage is still better compared to the inner area, but in general the SINR
is significantly lower and much more uniformly distributed, leading to a loss
in the spectral efficiency and a gain in the outage, since all users are now
above the SINR threshold. The error on the radiation pattern is making the CSI
vectors more uniform from the beamformer perspective and all users are served
with a similar SINR (as also demonstrated by the constant performance in
Figures 17 and 18 for increasing values of the power density). This is further
substantiated in Figure 21, which shows the power allocated to the users in a
time slot with LB-MMSE: i) in ideal conditions, the users at the coverage edge
are allocated less power because they experience less interference and also
have large antenna gains; ii) when the error is introduced, the power tends to
be uniformly distributed across the beams, since the users’ CSI vectors tend
to be more and more similar.

  * •

SS-MMSE is a beam-based algorithm, as each user is approximated with the
closest beam center. Thus, the performance is driven by the distance from the
associated beam center: the larger this distance, the larger the approximation
and, consequently, the worst the performance. In addition to this aspect, also
with SS-MMSE the external beams are allocated a lower power; however,
differently from LB-MMSE, the reduced interference in the external tiers is
not sufficient to cope with both the lower allocated power and the
approximation in the channel coefficients. Thus, the external tiers experience
a worse performance compared to the inner ones. When $\varepsilon_{rp}=0.05$,
the same phenomenon described above for LB-MMSE arises. The CSI vectors tend
to become more similar from the beamformer perspective and a uniform
performance is achieved across the coverage area. This is substantiated in
Figure 22, with a very limited variability in the power allocations to the
beams when $\varepsilon_{rp}=0.05$. As observed above, the consequence is that
the spectral efficiency is worse compared to the ideal scenario, while the
outage is improved since users at the coverage edge that were below the SINR
threshold now experience an SINR close to that of all users.

(a) $\varepsilon_{rp}=0$

(b) $\varepsilon_{rp}=0.05$

Figure 21: Power allocation per served user with LB-MMSE with centralised
MIMO.

(a) $\varepsilon_{rp}=0$

(b) $\varepsilon_{rp}=0.05$

Figure 22: Power allocation per served user (beam) with SS-MMSE with
centralised MIMO.

The above phenomenon is still present for SS-MMSE in clear-sky, while for the
LB-MMSE it is not. In clear-sky conditions, no user is experiencing the shadow
fading or clutter loss; thus, with an algorithm in which the actual locations
are used (and not an approximation), no user is in outage at the coverage edge
and only the loss in the spectral efficiency is present. Similar
considerations and observations can be made for the federated scenario.

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 23: Average capacity in clear-sky with $N_{node}=1$ (left) and
$N_{node}=2$ (right).

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 24: Average capacity in NLOS dense-urban with $N_{node}=1$ (left) and
$N_{node}=2$ (right).

###  IV-D Frequency reuse

To conclude the extensive numerical assessment of the considered CF-MIMO
techniques, we now discuss the comparison with legacy frequency reuse schemes
with 3 and 4 colours (FR3 and FR4) implemented with a single node. To this
aim, we consider the (s)MPC normalisation only. Figures 23 and 24 show the
average capacity in clear-sky and NLOS dense-urban conditions, respectively.
It can be noticed that the advantage of using a FFR scheme is significant: i)
for VSAT receivers, the gain is in the order of $100$ Mbps in NLOS dense-urban
conditions and even larger in clear-sky; ii) with handheld terminals, the gain
is in the order of $15-20$ Mbps and $20-35$ Mbps for clear-sky and NLOS dense-
urban environments, respectively. In terms of outage:

(a) $N_{node}=1$

(b) $N_{node}=2$

Figure 25: Average capacity in NLOS dense-urban with $N_{node}=1$ (left) and
$N_{node}=2$ (right).

  * •

It was observed that no unserved users are present with 3 and 4 colours in
clear-sky conditions; the outage performance of CF-MIMO on this channel
(Figure 8) showed that for the (s)MPC normalisation a significant amount of
unserved users was present only with two nodes and handheld terminals. Thus,
for VSATs and handheld terminals in the centralised scenario the advantage of
FFR with CF-MIMO techniques is significant.

  * •

Figure 25 shows the unserved users in NLOS dense-urban conditions. As for
VSATs, it can be observed that: i) with large transmission power, the MMSE
algorithm shows a limited outage and, thus, the advantage of CF-MIMO is clear;
ii) with low transmission power, the outage is a bit larger, but it might
still be acceptable considering the gain in terms of capacity. As for handheld
terminals, there is a significant amount of unserved users also with FR3 and
FR4: i) in the centralised scenario, the outage is slightly below that with
all CF-MIMO solutions, but considering the advantage in terms of capacity, the
latter shall be selected; and ii) in the federated scenario, the path
diversity introduced by the additional nodes provides an advantage over FR3
and FR4 also in terms of outage.

In conclusion, the capacity is significantly improved with CF-MIMO and FFR.
However, by also taking into account the percentage of unserved users, there
are some scenarios in which legacy systems might still be better, _e.g._ ,
with VSAT terminals and low transmission power. In all other cases, the CF-
MIMO approach is by far the best system-level solution.

###  IV-E Recommendations

Focusing on the (s)MPC normalisation, which is the feasible solution avoiding
non-linear effects in the HPAs and limiting the power per node, the following
considerations hold:

  * •

The proposed LB-MMSE provides a performance identical to MMSE in clear-sky.
The loss when taking into account additional stochastic losses, which cannot
be _a priori_ estimated, is negligible even in harsh conditions, such as the
considered NLOS dense-urban case.

  * •

A federated Cell-Free solution shall be selected with VSAT terminals and lower
transmission power (_i.e._ , up to $0-2$ dBW/MHz). In this case, the
achievable spectral efficiency is larger and the increase in the outage is
negligible ($0.3-2.1\%$) in clear-sky conditions, while it is significantly
lower in NLOS dense-urban conditions (up to $6.1\%$). With larger transmission
power, a centralised solution shall be preferred.

  * •

With handheld terminals, centralised beamforming is the best option based on
the outage probability (a $0.1-0.2$ bit/s/Hz increase in the spectral
efficiency is not enough to justify a $25\%$ outage) in clear/sky conditions,
while in NLOS dense-urban environments federated solutions with multiple nodes
provide an advantage in both the spectral efficiency ($0.02-0.2$ bit/s/Hz) and
outage (up to $5.7\%$), thanks to the spatial diversity at the transmitter.

  * •

Compared to FR3 and FR4 schemes, the capacity is significantly better with CF-
MIMO solutions. When the outage is taken into account, CF-MIMO shall still be
selected in the vast majority of the scenarios, with some limited exceptions
(_e.g._ , VSAT receivers and low transmission power).

Finally, when considering non-ideal information at the transmitter, it was
observed that the proposed LB-MMSE algorithm: i) is particularly robust to
positioning errors; ii) is significantly impacted by errors on the radiation
pattern model. As for the latter, it is worthwhile highlighting that in real
deployment scenarios the variance of the amplitude and phase errors on the
antenna pattern coefficients is expected to be much lower than $5\%$ as
assumed in the above analyses.

##  V Conclusions

In this paper, we provided a detailed discussion on the design choices
allowing the implementation of both federated and centralised Cell-Free or
beam-based MIMO solutions in NTN NGSO constellations; both regenerative, with
functional split options, and transparent payloads were addressed, as well as
OGBF and OBBF solutions. A detailed description of the architecture options,
with the related challenges and benefits, has been reported. Then, we
designed: i) a novel location-based CF-MIMO algorithm for NTN NGSO
constellations which is completely user-centric, _i.e._ , tailored to the
actual users’ locations.; and ii) novel power normalisation approaches for
federated MIMO algorithms that can be applied to swarms of NGSO nodes. The
outcomes of the extensive numerical results showed that: i) the proposed LB-
MMSE algorithm provides a performance close to the CSI-based MMSE, but with a
significantly reduced overhead and complexity at the terminal side; ii)
federated CF-MIMO solutions over NGSO swarms provide benefits with VSAT
terminals and low transmission power, and with handheld terminals in NLOS
dense-urban conditions. The performance was assessed also considering 3 and 4
colours frequency reuse schemes; it was shown that, apart from VSAT terminals
and low transmission power, the CF-MIMO paradigm provides a significant
performance improvement. Finally, the robustness of the considered location-
based algorithms was assessed with non-ideal location estimation and non-ideal
knowledge of the actual radiation of the on-board UPA elements, with respect
to their mathematical model. It was shown that the former does not pose
issues, while the latter is more critical. Future developments of this work
include the analysis of the tight synchronisation among the swarm nodes for
federated CF-MIMO and the performance assessment with variable UPA
configurations and more than two nodes per swarm.

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|  Alessandro Guidotti (Member, IEEE) received the master degree (magna cum
laude) in telecommunications engineering and the Ph.D. degree in electronics,
computer science, and telecommunications from the University of Bologna,
Italy, in 2008 and 2012, respectively. From 2009 to 2011, he was a
representative for the Italian Administration within CEPT SE43. In 2011 and
2012, he was a Visiting Researcher at SUPELEC, Paris, France. From 2014 to
2021, hew was a Research Associate with the Department of Electrical,
Electronic, and Information Engineering “Guglielmo Marconi,” University of
Bologna. From 2021, he is a Researcher with the Consorzio Inter-Universitario
delle Telecomunicazioni (CNIT), located at the Research Unit of the University
of Bologna. He is active in national and international research projects on
wireless and satellite communication systems in several European Space Agency
and European Commission funded projects. He is a member of the Editorial Board
as Review Editor of the Aerial and Space Networks journal for Frontiers in
Space Technologies. He has been serving as TPC and Publication Co-Chair at the
ASMS/SPSC Conference since 2018. His research interests include wireless
communication systems, spectrum management, cognitive radios, interference
management, 5G, and Machine Learning.
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|  Alessandro Vanelli-Coralli (Senior Member, IEEE) received the Dr.Ing.
degree in electronics engineer- ing and the Ph.D. degree in electronics and
computer science from the University of Bologna, Italy, in 1991 and 1996,
respectively. In 1996, he joined the University of Bologna, where he is
currently an Associate Professor. He chaired the Ph.D. Board, Electronics,
Telecommunications and Information Technologies from 2013 to 2018. From 2003
to 2005, he was a Visiting Scientist with Qualcomm Inc., San Diego, CA, USA.
He participates in national and international research projects on wireless
and satellite communication systems and he has been a Project Coordinator and
scientific responsible for several European Space Agency and European
Commission funded projects. He is currently the Responsible for the Vision and
Research Strategy task force of the Networld2020 SatCom Working Group. He is a
member of the Editorial Board of the Wiley InterScience Journal on Satellite
Communications and Networks and Associate Editor of the Editorial Board of
Aerial and Space Networks Frontiers in Space Technologies. Dr. Vanelli-Coralli
has served in the organisation committees of scientific conferences and since
2010 he is the general CoChairman of the IEEE ASMS Conference. He is co-
recipient of several the Best Paper Awards and he is the recipient of the 2019
IEEE Satellite Communications Technical Recognition Award.
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|  Carla Amatetti obtained the Master Degree in Communications and Computer
Networks Engineering in 2018 from the Politecnico di Torino, Italy. From
January to June 2019, she worked for the Fiat Research Center in Torino,
contributing to the application of 5G to vehicular communications, for
emergency scenarios. From November 2019, she is purchasing a PhD at the
University of Bologna in the Department of Electrical, Electronic and
Information Engineering “Guglielmo Marconi” (DEI). From January to July 2022
she was a visiting researcher at the Interdisciplinary Centre for Security,
Reliability, and Trust (SnT) within the University of Luxembourg. From July
2019 to November 2019, she received a research grant about “5G Non-Terrestrial
Networks: physical layer design and assessment.” Her research work is focused
on the study of solutions and techniques for the integration of Non
Terrestrial and Terrestrial communication systems, with particular emphasis on
Narrowband Internet of Things (NB-IoT) technology.
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