Nevertheless, these gains come with challenges. First, the decomposition
lowers the degrees of freedom at the GW. In fact, instead of using all the
$N_{f}$ feeds to mitigate interference between the beams, the $N_{b}\times
N_{b}$ on-ground precoding matrix will act directly on the $N_{b}$ beams. This
can decline the final throughput by a percentage reaching 20% [199].
Additionally, the on-board analog processing reduces the overall system
capacity compared to the fully digital on-ground processing [221]. This is
mainly due to the lack of precision of analog processing on the one hand and
the quasi-static ergodic channel information used on-board on the other hand.
However, the trade-off between complexity and performance of the hybrid on-
board/on-ground architectures has been attracting increasing attention during
the last two decades, especially with the recent trends pushing towards more
flexibility and reconfigurability in the on-board payload [222].

The paper [223] addresses the problem of limited feeder link bandwidth by
introducing a hybrid precoding technique. This technique involves using fixed,
non-channel-adaptive beamforming for the satellite segment. Additionally, the
feed signals are compressed by projecting them onto a lower dimensional
subspace, and then preprocessed using linear MMSE with regularized inversion
at the GW. In [200], a fixed on-board beam generation pattern design was
proposed for a hybrid on-board/on-ground precoding system with low payload
complexity. Further, the linear MMSE with a regularized inversion is also
considered as in [223]. Then, the fixed beam generation process matrix is
obtained as an approximated solution of the sum mean squared error
minimization problem. Later on, the work in [200] has been extended to study
the return link in [221] and provide a more robust precoding design accounting
for the channel perturbations. In [224], reduced-rank on-ground precoding
adopting feed selection is proposed jointly with coarse on-board beamforming.
Further, the benefits of the proposed hybrid precoding scheme on the feeder
link requirement are shown compared to on-ground precoding.

In [225], the feasibility of massive MIMO for SatCom has been studied, and
efficient implementation via a hybrid analog/digital precoder was considered.
The design of the precoder is based on a two-stage approach. In the first
stage, the fully digital precoder is obtained using conventional precoding
techniques such as ZF. In the second stage, the optimal hybrid precoder is
derived as the minimizer of the Frobenius norm of the difference between the
hybrid and the fully digital precoders. In [211], a joint feed selection,
precoding, and signal switching algorithm for the on-board segment is proposed
to tackle the feeder link bandwidth scarcity while maintaining an affordable
complexity at the payload. Specifically, this work combines a multi-GW
architecture with a hybrid on-ground on-board precoding scheme to get both
their benefits on bandwidth requirements. In [226], constant-envelope
precoding is considered for hybrid on-board/on-ground architecture to offer
more robustness against non-linearities. Two designs are proposed considering
adaptive and fixed on-board beamforming. Designing the adaptive beamforming is
done jointly with the GW signals and results in non-convex optimization
problems, which are tacked using algorithmic solutions based on the saddle
point method.

The growing demand for broadband services has caused satellite systems to
shift from a single-beam to multi-beam coverage. Smart antenna technologies
are deployed on satellites to generate multiple beams for beamforming or
precoding. Conventionally, beam generation is implemented at the satellite
using analog or digital techniques. However, the beam pattern generated is
designed based on geographical considerations only, and does not account for
specific user positions. On-ground beamforming implementations are digital and
utilize Channel State Information (CSI) to compute the precoded signal.
Precoding optimizes the transmission to users and increases the system’s
flexibility and capacity. Future satellite communication systems are
anticipated to handle a large number of beams, necessitating precoding
matrices with high dimensions. Hybrid precoding schemes provide a trade-off
between performance and bandwidth requirements by decomposing the original
precoding matrix into two matrices. This approach decreases the number of
transmitted signals by the feeder link, but higher frequency bands or multi-GW
architectures can also be adopted to address bandwidth requirements.

###  IV-G Satellite Service: Mobile and Fixed

An essential aspect of satellite communication is the spectrum of operation,
according to which satellite services can be divided into two main categories:
fixed satellite services (FSS) and mobile satellite services (MSS). FSS
consists of communication between one or multiple space stations with fixed
earth stations. It also includes satellite-to-satellite communication links in
some cases [227]. On the other hand, MSS consists of radio-communication links
of one or multiple space stations with mobile earth stations [228]. FSS
operates on the Ku and Ka bands. Thus, it benefits from a larger available
bandwidth than MSS operating on the L and S bands. However, they suffer from
higher path loss and rain fading.

From a hardware perspective, FSS requires larger and more expensive antennas
offering more focused beams. On the other hand, MSS relies on cheaper and less
directive antennas to handle the non-stationary nature of its UTs. In
addition, FSS includes earth-exploration satellite services, such as
meteorological-satellite services, while MSS includes land, aeronautical, and
maritime mobile-satellite services. In the face of the growing demand for
higher data rates, up-to-date satellite systems rely on multi-beam coverage
with a frequency reuse strategy among the beams. This increases the overall
throughput with the frequency reuse factor at the cost of higher interference.
To push the throughput to its extreme, full frequency reuse with precoding
techniques has been proven very effective for FSS [80, 35, 40, 25], which have
been covered in the previous sections. However, few precoding studies have
been carried out for MSS.

Recently, with the increasing congestion of the L and S-bands and the need for
higher capacity in MSS, higher frequency reuse among beams has become an
interesting alternative [7]. Further, driven by the high gains of precoding
for FSS [80, 35, 40, 25], and considering the limited gains of multi-user
detection in mobile systems [229], precoding for MSS have been receiving
growing attention. Indeed, the mobility of the UT reduces the precoding gain
compared to the fixed scenario, as it decreases the accuracy of the CSI [230].
However, MSS can rely on a single GW to handle the total traffic, while FSS
needs multiple GWs to handle the high bandwidth requirement in practice.
Therefore, it is worth investigating the precoding gain in MSS [201, 204,
231].

When considering precoding for MSS, a crucial aspect is the time-varying
nature of the channel. In fact, the GW can only obtain delayed CSI, which can
constrain the precoding performance. A live precoding test for a mobile
satellite system has been reported in [232]. Later on, one of the first few
studies of precoding for MSS was carried out in [233]. A hot spot scenario is
considered where the problem of unbalanced service demand in the L-band is
addressed. In particular, Icolari et al. propose a channel assignment approach
that combines FDMA and SDMA, with both orthogonal channels and precoded
channels reusing the same frequency in multiple areas as in [234, 41]. In this
case, the total capacity is the sum of the two capacities resulting from the
two access schemes, and it is shown to be higher than a lower-frequency reuse
scenario. In [235], the impact of user mobility on the precoding performance
of satellite systems is investigated. In fact, channel models such as slow
nomadic, maritime and maritime low elevation are considered. In this context,
the time-varying nature of the MSS channel results in a delayed CSI at the GW,
which is proven of limited impact on the average throughput. Furthermore, both
the unicast and multicast scenarios are considered. A large throughput gain is
achieved for unicasting, and a sufficiently high gain is also attained for
multicasting if a proper scheduling strategy is employed. Another recent work
in the multi-beam mobile satellite context is done in [236, 237]. Therein,
various CSI feedback strategies aiming to maintain a lower CSI variation at
the GW are presented. In particular, two CSI feedback mechanisms are proposed,
one based only on the latest CSI estimation and the other based on all the
previous CSI estimations. Then, adapted precoding schemes for the proposed CSI
feedback mechanisms are developed and tested for a maritime communication
scenario to maintain a lower complexity of the proposed precoding schemes
compared to land-mobile satellite services.

###  IV-H Summary & Lessons Learnt

This section presents the main contribution of this survey, which is a
classification of precoding techniques for HTS from two main levels: a problem
formulation level and a system design level. The references discussed in this
section are compared and summarized according to the various subcategories in
Table IV.

The problem formulation level is a classification considering the mathematical
aspects when designing the precoding scheme. The first aspect under this level
involves the scheduling strategy incorporated in the precoding scheme, which
we refer to as the precoding group. It can be unicast when one user per beam
is served in each time slot or multicast when more than one user is served in
each time slot. In the forward link of multi-beam satellites with full
frequency reuse, it is more practical to formulate the transmission as a
multicast problem. This raises the problem of user clustering, which addresses
the strategy of grouping users in each beam. The second aspect concerns the
optimization problem the precoding scheme is designed to solve. Therefore,
precoding schemes can be closed-form or numerical solutions to an optimization
problem. We classify the different optimization problems according to their
objective functions: spectral efficiency maximization, power minimization,
energy efficiency maximization, secrecy maximization, and in some cases, mixed
objectives where the precoding scheme is designed jointly with a MAC layer
protocol such as scheduling. The last aspect of the problem formulation level
is related to the precoding switching rate, which can be on a block level or a
symbol level. Specifically, precoding techniques relying only on CSI to
mitigate interference are updated at every coherence time of the channel.
Therefore, a block of several symbols can use the same precoding matrix, which
explains calling it block-level precoding. On the other hand, precoding
schemes that use both CSI and data information to mitigate interference are
updated with every new symbol, and therefore its called symbol-level
precoding.

The problem design level is a classification based on design aspects of the
satellite communication system that impact precoding. The first aspect under
this level concerns the number of GWs in the HTS system, and we refer to it as
system configuration. In fact, the feeder link bandwidth requirement becomes
too large when the number of users, beams, and polarizations increases.
Multiple GWs help to reduce the feeder link bandwidth requirements, processing
complexity, and manage gateway failures. However, multi-gateway systems face
challenges such as decreasing degrees of freedom for interference mitigation
and collective processing of precoding among the gateways. Different
cooperative and non-cooperative GWs strategies are proposed in this context,
depending on whether the CSI and DI are shared among the GWs or not, and more
robust designs are proposed to account for feeder link interference and
imperfect CSI. The second aspect is the precoding implementation, which can be
on-board the satellite, on-ground at the GW, or hybrid on-board on-ground.
Conventionally, beam generation is implemented at the satellite using analog
techniques. However, the beam pattern generated is designed based on
geographical considerations only and does not account for specific user
positions. On-ground precoding implementations are digital and utilize CSI to
compute the precoded signal. Therefore, they optimize the transmission to
users and increase the system’s flexibility and capacity at the cost of a high
feeder link bandwidth requirement compared to fixed on-board implementations.
Hybrid precoding schemes provide a trade-off between performance and bandwidth
requirements by decomposing the original precoding matrix into two matrices.
The last aspect under system design level is Satellite communication services
are categorized into FSS and MSS, which differ in their spectrum of operation,
hardware requirements, and service types. FSS operates on the Ku and Ka bands
with larger bandwidth but higher path loss and rain fading, while MSS operates
on the L and S bands with cheaper and less directive antennas. To improve
throughput, satellite systems use multi-beam coverage, full frequency reuse,
and precoding techniques for FSS. However, few precoding studies have been
done for MSS, where channel time variability limits the precoding performance
due to delayed CSI. Recent works have investigated some precoding aspects of
MSS, including a hot spot scenario, user mobility impact, and user scheduling
strategies, with proposed CSI feedback mechanisms and precoding schemes to
maintain lower complexity.

TABLE IV: Comparison of precoding schemes for HTS.

| Problem formulation | System design |  |
---|---|---|---|---
|  | Objective |  |  |  |  |  |
Paper | Group | Closed form | Opt. Problem | Level | Configuration | Implementation | Service | Robustness | Further remarks
[40] | Multicast | RZF/MMSE |  | SE (sum rate
---
maximisation)
Block | Single/Multiple GW | On-ground | Fixed |  | Outdated CSI.
---
No data sharing.
Cooperative
[233] | Unicast | LMMSE | × | Block | Single GW | Hybrid | Mobile | Perfect CSI. | -
[210] | Multicast | × |  | SE (sum rate
---
maximisation)
Block | Multiple GW | On-ground | Fixed |  | Perfect CSI.
---
Feeder link interference.
Cooperative
[43] | Multicast | MMSE | × | Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[199] | Unicast | RCI/ZF | × | Block | Single GW |  | On-ground/On-board/
---
Hybrid
Fixed | Imperfect CSI. | -
[237] | Unicast | MMSE/ZF | ✓ | Block | Single GW | On-ground | Mobile | Outdated CSI. | -
[78] | Unicast | × |  | EE/SE (sum rate
---
maximisation)
Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[200] | Unicast |  | Worst case MMSE/
---
LMMSE
× | Block | Single GW | Hybrid | Fixed |  | Perfect/
---
Imperfect CSI.
-
[205] | Unicast | ZF | × | Block | Single/Multiple GW | On-ground/Hybrid | Fixed | Perfect CSI. | Cooperative
[132] | Unicast | × |  | Secrecy (sum secrecy
---
rate maximisation)
Block | Single GW | On-ground | Fixed | Imperfect CSI. | -
[73] | Unicast | MMSE/ZF | SE | Block | Multiple GW | On-ground | Fixed |  | Imperfect CSI.
---
Non-linearity.
-
[127] | Unicast | ZF |  | Mixed (secrecy+
---
Precoding)
Block | Single GW | On-board/Hybrid | Fixed | Imperfect CSI. | -
[190] | Unicast | × | Peak power minimsation | Symbol | Single GW | On-ground | Fixed |  | Perfect CSI.
---
Non-linear.
distortions.
| M-PSK/
---
Multi-level
constellation
[90] | Multicast | × | SE (max-min fair) | Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[208] | Multicast | × | SE (max-min fair) | Block | Multiple GW | On-ground | Fixed |  | Perfect CSI.
---
Data sharing.
Cooperative
[79] | Unicast | × |  | SE (rate matching,
---
sum rate maximisation,
rate balancing)
Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[226] | Unicast | × | ✓ | Symbol | Single GW | Hybrid | Fixed | Perfect CSI. | -
[91] | Multicast | × | SE (max-min fair) | Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[77] | Unicast | × | THP | - | Single GW | On-ground | Fixed | Perfect CSI. | -
[133] | Unicast | × | Secrecy (EE) | Block | Single GW | On-ground | Fixed |  | Imperfect CSI of
---
eavesdroppers.
-
[128] | Unicast | × |  | Secrecy (sum power
---
minimsation)
Block | Single GW | On-ground | Fixed |  | Imperfect CSI
---
of eavesdropper.
-
[42] | Multicast | MMSE | × | Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[204] | Unicast | RZF |  | SE (sum rate
---
maximisation)
Block | Multiple GW | On-ground | Fixed |  | Partial CSI.
---
Partial data sharing.
| Cooperative/
---
Non-cooperative
[8] | Unicast | ZF | × | Block | Single GW | On-ground | Fixed | Partial CSI. | -
[225] | Unicast | ZF | ✓ | Block | Single GW | Hybrid | Fixed | Perfect CSI. | -
[221] | Unicast | Worst case MMSE | × | Block | Single GW | On-board | Fixed | Perfect CSI. | -
[206] | Unicast | ZF | SE (max-min fair) | Block | Multiple GW | On-ground | Fixed |  | Perfect CSI.
---
No data sharing.
Cooperative
[207] | Unicast | ZF/MMSE | × | Block | Multiple GW | On-ground | Fixed |  | Perfect CSI.
---
Feeder link interference.
Cooperative
[75] | Unicast | × | ✓ | Block | Single GW | On-ground | Fixed | Partial CSI. | -
[188] | Unicast | × | Min-max fair | Symbol | Single GW | On-ground | Fixed | Perfect CSI. | M-PSK
[136] | Unicast | × |  | Secrecy (worst secrecy
---
rate maximisation)
Block | Single GW | On-ground | Fixed |  | Imperfect CSI of
---
eavesdroppers.
-
[96] | Multicast | × | SE (max-min fair) | Block | Single GW | On-ground | Fixed | Outdated CSI. | -
[105] | Unicast | × | SE (rate balancing) | Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[134] | Unicast | × |  | Secrecy (worst secrecy
---
rate maximisation)
Block | Single GW | On-ground | Fixed |  | Imperfect CSI of
---
eavesdroppers.
-
[236] | Unicast | ✓ | × | Block | Single GW | On-ground | Mobile | Outdated CSI. | -
[223] | Unicast | RZF | × | Block | Single GW | Hybrid | Fixed | Perfect CSI. | -
[220] | Unicast | RCI | × | Block | Single GW | On-board/Hybrid | Fixed | Imperfect CSI. | -
[26] | Unicast | × | EE | Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[25] | Multicast | × |  | SE (sum rate
---
maximisation)
Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[116] | Multicast | × | EE | Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[76] | Unicast | × | THP | - | Multiple GW | On-ground | Fixed | Perfect CSI. | -
[235] | Multicast | MMSE | × | Block | Single GW | On-ground | Mobile | Outdated CSI. | -
[202] | Unicast | RCI/ZF/MMSE | × | Block | Single GW | On-ground/Hybrid | Fixed | Imperfect CSI. | -
[115] | Unicast | × |  | Sum power
---
minimisation
Block | Single GW | On-ground | Fixed |  | Imperfect CSI.
---
Phase uncertainty.
-
[203] | Multicast | MMSE |  | SE (sum rate
---
maximisation)
Block | Multiple GW | On-ground/On-board | Fixed |  | Perfect CSI.
---
Feeder link interference.
| Cooperative/
---
Limited cooperation
[41] | Multicast | × | ✓ | Block | Single GW | On-ground | Fixed |  | Imperfect CSI.
---
Outdated CSI.
Phase offset.
-
[189] | Unicast | × |  | Spatial peak-to-average
---
power ratio minimization
Symbol | Single GW | On-ground | Fixed |  | Perfect CSI.
---
Non-linear distortions.
M-PSK
[51] | Multicast | MMSE | × | Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[224] | Unicast | MMSE | ✓ | Block | Single GW | Hybrid | Mobile | Statistical CSI. | -
[92] | Multicast | × | SE (max-min fair) | Block | Single GW | On-ground | Fixed | Imperfect CSI. | -
[201] | Unicast | ZF/LMMSE | × | Block | Multiple GW | On-ground | Fixed |  | Imperfect CSI.
---
Reduced data sharing.
Feeder link interference.
Cooperative
[135] | Unicast | × |  | Secrecy (worst secrecy
---
capacity maximisation)
Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[155] | Multicast | × |  | Mixed (scheduling+
---
Precoding)
Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[211] | Unicast | RZF | × | Block | Multiple GW | On-Board | Fixed |  | Imperfect CSI.
---
No data sharing.
Non-cooperative
[13] | Unicast | × |  | SE (sum rate
---
maximisation)
Block | Single GW | On-ground | Fixed | Perfect CSI. | -
[209] | Unicast | MMSE | × | Block | Multiple GW | On-board/Hybrid | Fixed |  | Channel statistics.
---
No data sharing.
Limited cooperation

##  V Precoding in Practice

The majority of the precoding literature for multi-beam satellites assumes
ideal transmission conditions that are not achievable in practical systems. In
fact, Satcoms suffer from various practical impairments that can heavily
impact the overall system performance. These impairments can be classified
into channel-induced impairments and transceiver and payload-related
impairments. The latter includes non-linearity effects of high power
amplifiers (HPA), multi-carrier distortion, phase noise, sampling clock
offset, and I/Q imbalance [16]. On the other hand, channel-induced impairments
are related to channel estimation errors, and propagation delays [16].

###  V-A Transceiver and Payload Impairments

####  V-A1 Non-linearity

High power amplifiers (HPA) are an integral part of satellites. However, they
are inherently non-linear. A simple solution to avoid the non-linear effects
of HPAs is to reduce the input power, ensuring a more linear operation regime.
However, this will cause a loss of a large portion of power in the form of
heat. Therefore, HPAs are generally operated close to their saturation point
to address the trade-off between energy efficiency and non-linear effects.
Nevertheless, the latter operation mode still can introduce non-linearity
effects related to the amplitude and phase modulation responses because most
of HPAs for satellites utilize the traveling-wave tube amplifier (TWTA)
technology [238]. In this context, pre-distortion [239, 240, 187] and/or
equalization [241] solutions are employed to address distortions while
operating the HPA near its saturation. This is also of paramount relevance in
the context of inter-modulation distortion. In fact, the satellite transponder
jointly amplifies multiple signal carriers simultaneously using a single HPA
for cost-effectiveness. However, this generates non-linear inter-modulation
distortions in the amplified signals. In this context, the channel becomes
non-linear, and Volterra series represent a common mathematical framework to
model such non-linearity. [241, 242].

####  V-A2 Phase Perturbations

The satellite payload introduces phase noise that rotates the phase of the
channel. This can be caused by the satellite’s motion within its station and
thermal noise. Another dominant form of phase noise originates from the
instabilities of oscillators on-board the satellite and low noise block down
converters (LNBs) at the UTs. In fact, frequency offsets appear during down-
conversion, when the frequency of the oscillator at the receiver side is not
identical to the carrier frequency of the received signal. To model this
effect, the received signal after down-conversion can be written as [16]

$y(t)=x(t)e^{-j2\pi f_{n}t+\phi_{n}},$ (27)

where $y(t)$ and $x(t)$ are the base-band received and transmitted signals
respectively, and $\phi_{n}$ and $f_{n}$ are the phase and the frequency
offsets respectively.

Oscillator-induced frequency offsets always occur in practice because the
transmitter and the receiver are independent and cannot have oscillators at
the exact same frequency. However, this phenomenon is further accentuated when
the transmitted signal experiences frequency shifts during propagation. For
instance, when there is a relative velocity between the satellite and the UTs,
Doppler frequency shifts occur. The Doppler effect is not apparent in GEO
satellite systems and fixed UTs. Nevertheless, this effect has to be accounted
for in LEO satellite systems and mobile ground terminals. In fact, there have
been a lot of recent works accounting for Doppler effects in precoding for LEO
satellites [243, 157, 244].

###  V-B Channel Impairments

Unlike its terrestrial counterpart, SatCom channels can have long round trip
delays (RTD) (0.5 s for GSO satellites), time-varying characteristics, and
significant Doppler effects depending on the constellation and type of
service. Therefore, the common hypothesis of perfect CSI is highly unrealistic
because of channel impairments such as latency, and estimation errors. This
can be accounted for by writing the estimated channel matrix
$\hat{\textbf{H}}$ as[16]

$\hat{\textbf{H}}=\textbf{H}+\textbf{E},$ (28)

where H is the true channel matrix and E is the error channel error matrix,
whose elements can be modeled as independent and identically distributed
Gaussian following on the classical Clarke’s isotropic scattering model [245,
16].

The long RTD effect should also be considered for a more accurate channel
model. In fact, the CSI received at the GW is not instantaneous but somewhat
outdated. In general, this can be modeled as[16]

$\hat{\textbf{H}}[n]=\textbf{H}[n-D]+\textbf{E}[n],$ (29)

where $D$ denotes the delay and $\hat{\textbf{H}}[n]$, $\textbf{H}[n]$ and
$\textbf{E}[n]$ denote the estimated channel, true channel and error matrices
at the $n$-th time instance, respectively.

The long RTD is generally assumed to affect the channel’s phase, while its
effect on the magnitude of the channel can be negligible during the feedback
interval [246]. The main component influencing the magnitude of the channel is
rain attenuation, which induces slow variations and can be easily estimated
[247]. Therefore, the effects of outdated CSI can be considered phase offsets
in a static environment due to the propagation path as in [41].

###  V-C Designs Accounting for Phase Perturbations

Phase uncertainties are the dominant factor influencing the accuracy of the
CSI, and their presence increases are the center frequency increases, as for
beyond Ka-band frequencies with fixed users [60]. These perturbations arise
from the nature of the satellite channel, as well as the satellite transceiver
and payload. However, the quality of the CSI has a significant influence on
precoding performance. Therefore, a realistic assessment of the precoding
performance necessitates accounting for imperfect and outdated CSI. This is
done by adding randomness to the estimated channel to represent the phase
noise process. Various distributions considering apriori knowledge are
assigned to this process, such as Tikhonov, uniform, and normal [248]. In this
context, the SINR is a random variable, and common robust precoding
problems[60] can be formulated under power constraints as

##### Outage Minimization

$\min_{\textbf{w}_{1},\ldots,\textbf{w}_{N_{b}}}\max_{k}~{}\mathrm{P}(\mathrm{SINR_{k}}\leq\epsilon),$
(30)

where $\epsilon$ is the outage threshold.

##### Average SINR

$\max_{\textbf{w}_{1},\ldots,\textbf{w}_{N_{b}},k}~{}\mathrm{E}(\mathrm{SINR_{k}}).$
(31)

Unlike the deterministic formulations that assume a perfect and instantaneous
CSI, the above formulations are based on stochastic uncertainty models that
consider the impact of phase noise. The model in (30) is probabilistic, and
the precoding scheme is designed to guarantee a certain outage level, while
the model in (31) is expectation-based and aims to optimize the average
performance. This consideration is often neglected in the literature. However,
there have been several works proposing robust precoding schemes in this
context.

In [41], three sources of phase noise are considered. Namely, the phase
offsets originating from the signal propagation path, the LNBs, and the
satellite on-board oscillators. This is modeled by adding the sum of three
phase perturbations to each element of the channel matrix. Each phase is
normally distributed with standard deviations reported by the ESA in [249].
The effect of imperfect CSI estimation is also considered, and outdated CSI is
accounted for as a signal propagation phase offset. Then, the performance of a
linear precoding scheme based on [202] is assessed considering user clustering
under the aforementioned practical impairments, and system trade-offs are
discussed. The work in [246] proposes a robust precoding design in the face of
long RTD in multi-beam satellite systems. The amplitude of the channel is
assumed to be constant during the feedback interval, while the phase is
assumed to be rapidly varying, causing an outdated CSI at the GW. Therefore,
the channel phase used for precoding at an instant $t_{1}$ is modeled as the
sum of the channel phase at $t_{0}=t_{1}-250ms$, and a normally distributed
error. Then, the precoding weights are derived as the solution of the
minimization of the transmit power under per antenna power constraints and the
maximum probability of outage constraints. The numerical results demonstrate
the considerable gain of the robust precoding scheme in terms of availability
and transmit power.

However, the probability of outage constraints arising from the randomness of
SINR complicates the computation of precoding weights. This issue has been
tackled in [115], where the problem of transmit power minimization under
outage probability constraints is further investigated for a more flexible
representation. Therein, the authors derived a conservative approximation of
the probability of outage based on Taylor series expansion and large deviation
inequality to provide a tractable solution to the power minimization problem
in the face of phase perturbation. The numerical results demonstrate a
superior power performance of the proposed algorithm compared to [246].
However, the power gain comes at the cost of a higher probability of outage
compared to [246].

In [40], a robust multicast multigroup precoding is proposed, where the CSI
corruption at the GW is modeled as a general perturbation matrix using the
first order perturbation theory of eigenvectors and eigenvalues [250].
Robustness in the context of multigroup multicast precoding is also studied in
[96]. Therein, a channel model considering phase noise is presented. Then, two
optimization problems are investigated, and the relationship between them is
established: the maximization of the minimum average SINR and the power
minimization with partial CSI. Further, authors in [96] propose a low
complexity and robust clustering strategy that only necessitate partial CSI.
Their results highlight the performance gain of the robust precoding in terms
of SINR distribution and per-user rate, especially when the variance of the
phase noise increases.

Nevertheless, all the aforementioned works address robust precoding in the
context of fixed satellite services. In [237], robust precoding is studied for
mobile satellite services. In fact, MSS also suffer from degraded CSI that is
not only due to the long RTD and payload impairments but also to the mobility
of users. Authors in [237] assume a perfectly calibrated payload and
investigate the impact of users’ mobility and long RTD on precoding. They
propose a robust precoding scheme based on a reliable CSI feedback mechanism
that exploits the past CSI measurements to provide the current CSI estimate.
Then, they compare the sum-rate performance considering both outdated and
instantaneous CSI to show that their proposed robust precoding scheme provides
a large gain in throughput in spite of the degradation with respect to the
real-time CSI case.

Most of the aforementioned contributions deal with phase noise in precoding by
representing the phase uncertainty as a Gaussian process [41][246]. However,
additional practical phenomena affecting oscillators should be taken into
account. Namely, near-carrier power spectral density [251, 252]. In fact, the
output signal of an oscillator can not be a unique spectral tone but rather
dispersed over a small band, which makes the synchronization between multiple
geographically distant antennas a challenging task. In this context, the
impact of synchronization among transmitted waveforms is addressed in [248].
Therein, the authors model the phase noise of the satellite oscillator and
show the degradation of precoding performance due to the independence of
oscillators in a realistic setting.

Although various theoretical works in the literature investigate the practical
impairments in precoding for satellites, testbeds are still limited. In this
context, a field trial of precoding for two co-located satellites has been
carried out in [253]. The effects of carrier offset and oscillator phase noise
are accurately modeled and compensated at GW. Then, ZF precoding is
implemented based on the proposed robust CSI estimation strategy. The
numerical results reported by the testbed show rate and modulation error ratio
enhancements.

###  V-D Designs Accounting for Non-linearity

Most precoding literature for satellites assumes a linear channel and assesses
precoding performance without accounting for non-linearity effects introduced
by HPAs. These effects are even more dominant when peak to average power
ratios (PAPR) are high for signals, namely, those with spectral-efficient
modulation such as 19/32 point multi-ring constellations. Therefore, the
impact of HPAs has to be considered to guarantee an accurate evaluation of
precoding gain, especially since precoding increases the PAPR of the input
signal by modifying its distribution [254, 184]. The first step towards a more
robust precoding design in the face of non-linearity consists of including
these effects in the received signal model. Various models in the literature
capture non-linearity effect [242], commonly, Volterra series-based models as
in [254]. Therein, the signal received by the $i$-th user can be written as

$y_{i}=g_{1,i}\textbf{h}_{i}^{H}\textbf{W}\textbf{s}+g_{3,i}\textbf{h}_{i}^{H}((\textbf{W}\textbf{s})\odot(\textbf{W}\textbf{s})\odot(\textbf{W}\textbf{s})^{*})+n_{i},$
(32)

where $\odot$ is the Hadamard product, $g_{1,i}$ and $g_{3,i}$ are the first
and third Volterra coefficients respectively.

From (32), we can see that the received signal is a non-linear mapping of the
precoding matrix, which calls for different signal processing frameworks. One
strategy imposes constraints on the maximum signal amplitude, forcing HAPs to
operate in their linear regime [255]. In this case, the received signal in
(32) can be approximated as (3), and the previously covered precoding
techniques can be applied. The other strategy solves the precoding problem in
the general framework of non-linear signal pre-distortion (SPD) [5]. In fact,
there has been an intensified attention, in the SatCom community, toward SPD
as a technique to mitigate non-linearity impairments [187, 256, 5]. This
technique consists of compensating the non-linear effects of HPAs by a non-
linear transformation of the waveform. Traditionally, the processing related
to such techniques happens in the GW to exploit the powerful computational
capacity on the ground. However, the on-board implementation has also been
attracting growing attention, as it circumvents the limitations of tight
emission masks [5].

There have been several works proposing joint designs of precoding and pre-
distortion. For instance, authors in [257] combine THP with a fractional pre-
distorter based on a gain-based Look-Up Table. In [258], codebook-based
precoding is combined with a linear adaptive pre-distortion technique. The
power minimization problem is addressed, where the codebook-based precoding
and power assignment weights are jointly optimized with the pre-distortion
matrix. In [254], multistream Crest factor reduction (CFR) and SPD are
designed in a cascade to mitigate non-linearities of HPAs as well as
interference. However, unlike the two aforementioned works, where either
single channel systems [257] or linear processing of the channel [258] are
considered, the work in [254] proposes a non-linear processing at the GW, with
multistream CFR and SPD. This approach provides various performance gains and
opens the door for a joint pre-distortion and precoding framework.

Constant-envelope precoding has also been proposed as a robust precoding
technique in the face of non-linear distortions of amplifiers. In this
technique, the precoded signals are designed to have a constant amplitude, and
it has been well studied in the literature [259, 260, 261, 262, 263, 181].
Non-linear distortions on the transmitted signals depend on their
instantaneous transmitted power, which worsens with high variations [255,
189]. Therefore, maintaining a constant amplitude of the transmitted signal
across the satellite antennas increases the robustness against TWTAs’ non-
linearities. Constant-envelope precoding for multi-beam satellites has been
proposed in [226]. Therein, an on-ground precoding scheme is designed
considering a fixed on-board beamformer. Then an on-ground precoding scheme is
designed jointly with an adaptive on-board beamformer. The proposed constant-
envelope precoding schemes are compared with linear MMSE precoding, and
numerical results show their effectiveness and robustness.

In [192], a real-time demonstration of precoding for a multi-beam satellite
system is reported, where a proper collection of CSI is considered, and HPAs
non-linearity is modeled. The results using the hardware demonstrator prove
the feasibility of precoding using the super-framing structure of DVB-S2X.
Another hardware precoding demonstration is reported in [264] and experimental
verification of Volterra-based distortion models for multicarrier satellite
systems is reported in [242].

###  V-E Summary & Lessons Learnt

Satellite communication systems face several impairments that affect the
quality of communication. These impairments result mainly from payloads,
transceivers, and the nature of the satellite communication channel. One major
impairment is the non-linearity caused by HPAs. These non-linear distortions
can be modeled using the Volterra series framework, and solutions in this
context include pre-distortion and equalization techniques. Another important
impairment to take into account while designing precoding for satellites is
phase perturbations. Specifically, phase perturbations are caused by
oscillators on-board the satellite and LNB converters at the UTs. Frequency
offsets appear during down-conversion, and Doppler frequency shifts occur when
there is a relative velocity between the satellite and the UTs. In addition to
the transceiver and payload impairments, SatCom channels can have long round
trip delays (RTD), time-varying characteristics, and significant Doppler
effects depending on the constellation and type of service. Therefore, the
common hypothesis of perfect CSI is unrealistic because of channel impairments
such as latency and estimation errors. To account for phase noise, various
stochastic uncertainty models, such as Tikhonov, uniform, and normal
distributions, can be considered. In this context, the SINR becomes a random
variable. Therefore, the precoding scheme can be designed to minimize the
probability of a certain outage level or to maximize the expected SINR.

##  VI Trends in Precoding for HTS

Figure 8:  Trends and challenges in precoding for HTS.

In this section we discuss the trends in precoding for HTS to inspire further
research in this field. A summary of the discussed research trends is
illustrated in Fig. 8.

###  VI-A Flexibility

Most in-orbit satellite communication systems are based on a regular beam
layout plan, where a four-frequency reuse pattern is employed to mitigate
inter-beam interference, and power is allocated uniformly across all the
beams. This leads to a quasi-static beam pattern that does not necessarily
adapt to the heterogeneous nature of traffic. In fact, demand generally varies
across the beams during the day and along the different seasons. Therefore, a
quasi-static beam layout plan can cause a mismatch between the demand and the
offered capacity. Some beams can be experiencing a hot-spot scenario and
exhausting the offered bandwidth without properly satisfying the demand. On
the other hand, some other beams can be under-used and waste spectral
resources. This calls for more traffic-aware radio resource management and
transmission strategies.

Demand-based optimization in the forward link of multi-beam satellite
communication systems has been receiving intensified attention thanks to the
advances in on-board processing and large-scale active antennas [265]. Dynamic
beamforming is investigated in [266] to provide an adaptive beam layout plan
using directive and irregular beams. Further, precoding is employed to combat
the inter-beam interference from overlapping beams. Numerical results show
that the proposed adaptive beam layout plan considerably reduces capacity
mismatches across the beams compared to a regular beam layout plan. The hot-
spot problem has been addressed using flexible precoding for mobile satellite
systems in [233] and for fixed satellite systems in [267]. Overall, demand-
aware optimization can be introduced on various levels. It can be applied for
scheduling as in [268], where a demand-based scheduling strategy is
investigated for precoded HTS. It also can be applied for bandwidth and/or
power allocation as in [269, 270]. In particular, [269] shows that the
combination of flexible bandwidth allocation with linear precoding guarantees
higher efficiency in demand satisfaction and bandwidth than benchmark schemes.
Another strategy to design demand-aware transmission relies on re-configurable
beam positions and sizes to spatially adapt the demand [266, 271, 272].
Finally, beam hopping allows for a demand-aware coverage in the time domain
[273], and its combination with precoding has been investigated in [51].

###  VI-B Massive MIMO

Massive MIMO technology has been widely applied for terrestrial systems thanks
to its capability to enhance data rates and spectral efficiency [274].
Therefore, it is natural to investigate the extension of its performance
benefits to satellite communication systems. However, deploying this
technology for satellite communication systems faces various challenges.
First, massive MIMO performance greatly depends on accurate instantaneous CSI.
Nonetheless, as it is described in section V, satellite communication systems
suffer from various practical impairments that hinder the acquisition of
accurate instantaneous CSI. In fact, the CSI obtained at the satellite is
outdated and imperfect due to the long round trip delay and channel estimation
error. Therefore, integrating massive MIMO into satellite communication
systems requires the development of appropriate CSI feedback mechanisms that
bypass the performance degradation introduced by inaccurate channel
estimation. Second, massive MIMO communication requires massive antenna arrays
whose deployment can be quite challenging considering the satellite’s payload
limitations. In fact, satellites have space restrictions that can prevent
deploying the many antennas required for massive MIMO. This means that the
design of the satellite and its antennas must be carefully optimized to
maximize the benefits of massive MIMO while minimizing the space it occupies
on-board the satellite. Further, satellite communication systems typically
have strict power constraints, making it challenging to provide the power
needed for a large number of antennas. Therefore, power-efficient antenna
designs and signal processing are needed to mitigate this issue. Third, the
widely adopted four frequency reuse schemes for in-orbit satellites are
incompatible with massive MIMO technology, which calls for new considerations
in the design of frequency reuse strategies. Finally, the absence of multi-
path fading in the satellite channel significantly impacts the performance of
massive MIMO. Thus it should be taken into consideration to analyze the
potential benefits of this technology. These challenges represent key problems
to be addressed in order to integrate massive MIMO technology into satellite
communication systems.

Recently, motivated by the developments in multi-beam payloads incorporating
active antennas with a large number of radiating elements, massive MIMO in
satellites has been receiving increased attention, and precoding emerges as a
prolific technique to exploit the benefits of this technology fully. In
particular, there have been several recent works proposing massive MIMO for
LEO satellite communication systems [275, 244, 276]. This has also been
motivated by the attention that LEO satellites are receiving thanks to their
lower deployment and launching costs, as well as their reduced transmission
delays. Massive MIMO is proposed to be implemented for LEO satellites equipped
with uniform planar arrays and using statistical CSI instead of instantaneous
CSI. In particular, [275] shows that the performance of a statistical CSI
precoder approaches that of an instantaneous CSI precoder asymptotically.
Authors in [276] investigate a pragmatic approach for massive MIMO
implementation at a system and satellite payload level, which they extended in
[277] to a heuristic and computationally efficient solution. In [278], broad
coverage transmission is investigated for massive MIMO satellite systems, and
a precoder design is proposed based on the min-max criterion. Further, Hybrid
beamforming is investigated in [279] for LEO massive MIMO satellite systems to
design an energy-efficient scheme that is robust CSI errors.

###  VI-C Reconfigurable Intelligent Surfaces

Reconfigurable intelligent surfaces (RIS) are emerging as one of the most
potential technologies for beyond 5G and 6G communication systems, promising
higher spectral and energy efficiency at a low cost and low hardware
complexity. The main idea behind RIS consists of using thin materials equipped
with low-cost passive antennas to reflect signals and shift their phases
toward desired users. RIS can be deployed on walls or ceilings and controlled
on an electromagnetic level using a software controller. This technology has
been receiving intensified attention over the last decade. It has been widely
studied for terrestrial systems in contexts like MIMO [280], millimeter wave
communication [281], and cognitive radio [282]. Studies of RIS for terrestrial
systems highlight the capability of this technology to enhance data security
and user fairness in a cost-effective manner.  Thus, integrating this
technology into satellite communication systems can have attractive benefits.
In fact, satellite communication systems can suffer performance degradation
caused by shadowing in the UT environment. In this case, RIS mounted on a high
building can be exploited to enhance the effective gain of the signal
transmitted from the satellite to the ground terminal by shifting the signal’s
phase toward the UT. RIS can also be used to enhance security in the downlink
of satellite communication networks. In fact, the nature of wireless channels
and the large coverage areas of satellites increase the vulnerability of these
systems to eavesdropping. A RIS-assisted link can enhance the signal at the UT
while degrading it at the eavesdropper. In the context of full frequency
reuse, problems that jointly optimize precoding weights at the satellite and
phase shifts are the RIS can be investigated. Furthermore, RIS can be
exploited to enhance inter-satellite links. In fact, These links suffer from
misalignment fading and path loss, which can be compensated for using RIS
mounted on the satellites to enhance the signals’ propagation.

There have been several recent studies investigating RIS-assisted networks for
satellite communication systems. In particular, cognitive integrated
satellite-terrestrial networks have been investigated in [283], where the
issue of downlink security was addressed using a RIS-assisted link. In [284],
RIS has been considered to enhance the link budget between the user terminals
and a LEO satellite. It has also been considered to enhance LEO inter-
satellite links in the terahertz band in [285]. Further, RIS-assisted networks
have been studied for GEO satellites in [286], where the RIS adjusts the phase
of the signal from the satellite toward the ground user to boost the effective
gain of the system. Therein, the transmit power of the satellite is jointly
optimized with the phase shifts applied at the RIS to maximize the channel
capacity. Another recent work in [287] investigates RIS for hybrid satellite-
terrestrial relay networks where the beamforming weights’ optimization for the
satellite and the base station is performed jointly with the optimization of
the RIS’s phase shifts to minimize the total transmit power. Thus, this opens
the door to investigating coupled problems where RIS phase shifts are
optimized jointly with precoding/beamforming weights at the satellite.

###  VI-D Non-orthogonal Multiple Access

Protocols such as TDMA, FDMA, CDMA, and their variants fall under orthogonal
multiple access (OMA) protocols, in contrast to non-orthogonal multiple access
(NOMA) transmission techniques [288]. NOMA is primarily a physical layer
technique that allows different users to use the same radio resources to
transmit their signals. Then, multi-user detection and successive interference
cancellation are employed at the receiver side to recover the individual data
streams. However, some variants of NOMA incorporate MAC layer techniques such
as scheduling to optimize the performance and allocate resources dynamically.
Over the last few years, NOMA has been actively researched for various
wireless communication systems, and it has demonstrated outstanding results in
terms of capacity, user fairness, and latency for terrestrial communication
systems[288]. Therefore, it is only natural to investigate its application to
satellite communication systems.

In multi-beam satellite communication systems, NOMA can be applied in the
return or the forward links. Although NOMA has been widely studied for the
return link of multi-beam satellites, more research needs to be dedicated to
its application to the forward link [289]. In fact, the forward link of a
multi-beam satellite has different characteristics than that of terrestrial
systems, which makes applying NOMA, in this case, less of straight forward
extension of its terrestrial counterpart. Multi-beam satellite systems enable
frequency reuse strategies across the different beams. In non-aggressive reuse
schemes such as four or three colors, UTs can use single user-decoding thanks
to the spectrum isolation between the different beams. However, in more
aggressive frequency reuse strategies, precoding becomes a more interesting
way to cope with increased inter-beam interference. Therefore, the application
of NOMA to the forward link of multi-beam satellites depends on the
implemented frequency reuse scheme. In fact, in non-aggressive frequency
reuse, NOMA can be used within a single beam to mitigate intra-beam
interference, thanks to the low level of inter-beam interference, as long as
there is a considerable SNR asymmetry among users. Nevertheless, when
considering more aggressive frequency reuse strategies, inter-beam
interference becomes significant, which deteriorates the performance of NOMA
in single beams. In this scenario, strategies to integrate NOMA in the forward
link transmission have to be investigated. In [289], beam cooperation is
discussed as a solution to bypass the performance degradation of NOMA due to
inter-beam interference. In other words, frames corresponding to users from
different beams should be jointly processed, which is far from the standard
NOMA and more compatible with precoding.

The combination of NOMA with precoding techniques in the forward link of
multi-beam satellites further pushes the spectral efficiency of full-frequency
reuse strategies to its extreme. In a simple full-frequency reuse unicast
scenario, $N_{b}$ users (with one user per beam) are scheduled in each time
slot to be served simultaneously. Then, the precoding matrix is designed to
mitigate inter-beam interference between the scheduled users. Incorporating
NOMA into this setting allows the satellite to serve $LN_{b}$ users
simultaneously instead of $N_{b}$, with $L$ being the number of non-orthogonal
frames per beam. In other words, the scheduler selects a total of $LN_{b}$
users, with $L$ users per beam. Then, the satellite applies precoding to
mitigate inter-beam interference, while NOMA is applied in each beam to
mitigate intra-beam interference. However, to guarantee the benefits of NOMA,
the selected users within the same beam should have a significant SNR
imbalance. In this context, authors in [290] show that MMSE precoding in the
case of L=2 reduces inter-beam interference. It is also worth mentioning that
in realistic multi-beam satellite systems, more than one user is served by the
same physical frame. Therefore, the scheduler has to group users with similar
channel conditions in each beam as one multicast group to guarantee a good
precoding performance, which contradicts the scheduling required by NOMA based
on SNR imbalance. Therefore, combining NOMA with precoding in a multicast
scenario calls for a scheduling strategy that provides a trade-off between the
two criteria [291].

NOMA with precoding for the forward link of multi-beam satellites has been
receiving increased attention during recent years [289]. For instance, authors
in [292] investigate sub-optimal user-scheduling strategies to incorporate in
NOMA-based multi-beam satellites. Furthermore, [293] addresses the challenge
of aligning the offered capacity with traffic demands and proposes a solution
incorporating NOMA to alleviate intra-beam interference and allow for
aggressive frequency reuse. Nevertheless, numerous aspects remain to be
further investigated for NOMA in multi-beam satellites. For instance, it is
imperative to study the performance of NOMA under practical satellite
impairments such as non-linearity, channel estimation errors, and imperfect
interference cancellation. Moreover, NOMA has to be studied in conjunction
with flexible resource allocation to adapt to dynamic traffic changes.
Finally, physical layer security in NOMA-based multi-beam satellites is
another essential aspect to be studied by future research.

###  VI-E Artificial Intelligence

Artificial intelligence techniques such as machine learning (ML), deep
learning (DL), and reinforcement learning (RL) have shown outstanding results
in automating processes in various fields. Although their extension to
satellite communication and wireless communication, in general, is still at
its beginning, there are various applications where these techniques can be
highly beneficial. ML techniques can assist SatCom systems in tasks such as
interference detection, flexible payload configuration, and congestion
prediction. Current interference detection strategies in satellite control
centers rely on human expertise to analyze the spectrum and identify an
abnormal error rate rise or a QoS deterioration. However, this task can be
automated using unsupervised ML as discussed in [294]. Flexible payload
configuration is another interesting use case of ML models. In fact, the
current satellite communication systems require human assistance to determine
payload configuration, such as frequency and power allocation, as well as the
beam pattern. However, future satellite communication systems should be able
to adapt quickly and flexibly to changes in traffic and interference levels.
Therefore, flexible resource allocation can be an interesting application of
ML models in the payload configuration of satellites. In fact, as proposed in
[294], sub-carrier allocation can be determined by an ML model exploiting data
about the CSI, the interference levels, and the capacity requested by UTs.
Moreover, ML models can be highly beneficial in predicting congestion in
traffic. In fact, in satellite communication, traffic can be quite irregular
due to the geographical distribution of users, which induces some areas to
have high demand while others have low demand. Generally, these congestion
events are predicted based on traffic patterns from the preceding days.
Nevertheless, incorporating ML into this process can increase the accuracy and
reactivity time to enhance the system’s adaptation and therefore improve
performance [294].

In precoding-related applications, ML mechanisms can be used for resource
management, particularly for user scheduling. In fact, as discussed in section
IV-C4, the precoding performance significantly depends on the underlying
scheduling strategy. Therefore, optimal performance is obtained by jointly
optimizing scheduling and precoding parameters, which leads to a highly
complex optimization problem. One way to enhance the performance of the
scheduling strategy while maintaining a relatively lower complexity can be to
apply a learning technique for clustering. For instance, authors in [295]
proposed a scheduling strategy based on unsupervised learning techniques,
namely, K-means, hierarchical clustering, and self organization. The
clustering exploits both the CSI of users and their geographical positions to
determine the multicast groups to be served simultaneously, and it is done in
conjunction with RZF precoding. The results are encouraging and demonstrate an
increased average performance per beam compared to geographical clustering.
This opens the door for further research to investigate the benefits of
unsupervised learning on scheduling and precoding performance.

In addition to unsupervised learning, supervised learning can be exploited for
resource management of satellite communication systems. For instance, recent
work in [296] shows the potential benefit of using neural networks in
bandwidth and power management of HTS. In fact, a neural network classifier is
considered to choose an optimal resource allocation configuration from a
required traffic per beam. Another recent work in [297] utilizes deep
reinforcement learning to perform dynamic channel allocation for efficient use
of the radio resources. Moreover, ML techniques can be used to generate more
robust precoding schemes. These models can be trained to compensate for the
non-linearity and estimation errors of the satellite channel.

###  VI-F Summary & Lessons Learnt

This section covers some catalyzing wireless communication technologies
driving the research trends in precoding for HTS. These technologies include
flexibility, massive MIMO, RIS, NOMA, and ML.

The trend towards flexibility promotes adaptive demand-based optimization in
multi-beam satellite systems to reduce mismatches between the traffic demand
and the offered capacity. Flexible precoding can be used to address hot-spot
scenarios, and demand-aware optimization can be applied to scheduling,
bandwidth, and power allocation. On the other hand, the extension of various
successful terrestrial technologies, such as massive MIMO, RIS, and NOMA, is
at the center of the current research trends. The application of massive MIMO
to HTS faces various challenges, such as channel estimation errors and
satellite payload limitations. However, recent research focuses on LEO
satellites with uniform planar arrays and statistical CSI and employs
precoding techniques to exploit the benefits of massive MIMO. Recent research
also explores RIS-assisted networks in satellite systems, addressing issues
such as downlink security and link budget enhancement, which can be
investigated using joint optimization of RIS phase shifts and satellite
beamforming/precoding weights.

Furthermore, the application of NOMA in the forward link of multi-beam
satellite communication systems presents unique challenges and opportunities.
In a full-frequency reuse scenario, combining NOMA with precoding allows
serving a larger number of users simultaneously, with precoding mitigating
inter-beam interference and NOMA addressing intra-beam interference. Research
in this area can further investigate practical impairments, flexible resource
allocation, and physical layer security. In addition to NOMA, ML is another
technology that holds great potential for revolutionizing satellite
communication systems. ML can automate tasks like interference detection,
flexible payload configuration, and congestion prediction. In precoding-
related applications, ML mechanisms can be used for resource management,
particularly user scheduling. It can also enhance the performance of precoding
schemes by compensating for non-linearity and estimation errors in the
satellite channel. The application of these techniques in satellite
communication systems is still in its early stages but shows promising
benefits for improving performance and automation.

##  VII Conclusion

Satellite communication systems are witnessing various technological
breakthroughs making them an attractive solution for future communication
systems. Specifically, the next generation of satellite communications is
anticipated to handle high data rates by exploiting multi-beam and full
frequency reuse technologies. Therefore, precoding emerges as a prolific
interference management technique, enabling high spectral and energy
efficiency. Given the versatile nature of precoding design, this survey
presents a classification of the recent precoding techniques for HTS
communication.

The survey starts by providing an overview of the physical and modeling
aspects of the satellite communication system. Then, a brief overview of the
satellite PHY and MAC layers techniques is presented to serve as a reference
in the precoding analysis. However, the main contribution of this work is a
comprehensive classification of the precoding literature for multi-beam
satellites considering two different perspectives. The first perspective
considers the precoding problem formulation to classify the schemes based on
the scheduling strategy (i.e. precoding group), the the precoding optimization
problem , and the switching rate (i.e. precoding level). The second
perspective categorizes precoding techniques based on the satellite system
design, which includes precoding for single-GW or multi-GW systems (i.e.
system architecture), precoding for on-board, on-ground, or hybrid on-
board/on-ground implementations (i.e. precoding implementation), and precoding
for fixed or mobile satellites (i.e. satellite services). This classification
provides a canvas of the recent precoding schemes and covers the trending
direction in the literature. However, while the majority of the works in the
literature consider perfect channel and transmission conditions, real
satellite communication systems are subject to various channel and transceiver
impairments. Therefore, this survey highlights HTS systems’ practical
impairments and presents robust precoding techniques in the literature.
Finally, research trends in precoding for HTS are presented before concluding.

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