# Robust Beamforming Design for Integrated Satellite-Terrestrial Maritime
Communications in the Presence of Wave Fluctuation

Kaiwei Xiong, Xiaoming Chen, and Ming Ying Kaiwei Xiong, Xiaoming Chen, and
Ming Ying are with the College of Information Science and Electronic
Engineering, Zhejiang University, Hangzhou 310027, China
(e-mail:{xiong_kaiwei, chen_xiaoming, and ming_ying}@zju.edu.cn).

###### Abstract

In order to provide wireless services for wide sea area, this paper designs an
integrated satellite-terrestrial maritime communication framework.
Specifically, the terrestrial base station (TBS) serves near-shore users,
while the low earth orbit (LEO) satellite communicates with off-shore users.
We aim to improve the overall performance of integrated satellite-terrestrial
maritime communication system. Thus, it makes sense to jointly optimize
transmit beamforming at the TBS and LEO satellite. Due to sea wave
fluctuation, the obtained channel state information (CSI) is often imperfect.
In this context, a robust beamforming design algorithm is proposed with the
goal of minimizing the total power consumption of integrated satellite-
terrestrial maritime communication system while satisfying quality of service
(QoS) requirements. Both theoretical analysis and simulation results confirm
the effectiveness of proposed algorithm in maritime communications.

###### Index Terms:

Maritime communication, integrated satellite-terrestrial communication system,
imperfect CSI, robust beamforming, LEO satellite.

##  I Introduction

Currently, oceanic activities such as offshore oil exploration, maritime
environmental monitoring, maritime fisheries, and maritime scientific research
are on the rise [1], [2]. These activities drive a growing demand for maritime
communications with reliable high-speed data rates and wide wireless coverage
[3]-[5]. For example, operational data and navigational information are
necessary for the majority of ships to navigate safely, and multimedia
communication services are essential for aboard personnel, passengers, and
fisherman. Similarly, offshore drilling platforms need to communicate
operational data in real time, and buoys are responsible for uploading
significant amounts of meteorological data. Moreover, in maritime rescue
operations, real-time video communication is often necessary to ensure
effective shore-to-vessel and vessel-to-vessel coordination. Therefore,
building the maritime communication network for the maritime Internet-of-
Things (IoT) [6] is of great significance for maritime transportation,
production safety and emergency rescue. In recent years, terrestrial
communication technology has witnessed remarkable advancements, offering
unprecedented data rates and a wide array of new applications. The upcoming
sixth generation (6G) era is expected to further enhance capacity by dozens of
times [7], [8]. Unfortunately, the maritime domain has lagged behind in this
communications revolution. Maritime communication still significantly lags
behind terrestrial communication.

Mobile terminals at sea primarily rely on maritime LEO satellites [9], [10] or
TBSs for connectivity. Hence, maritime communication systems can be broadly
categorized into two groups. The first category encompasses TBS-based
communication systems utilizing the ultra high frequency band and the very
high frequency band to cover near-shore areas. The second category involves
the maritime satellite communication systems operating in the S frequency band
and L frequency (VHF) band. Narrowband satellites primarily provide some low
communication rate services. However, there is a growing need for satellite
broadband communication services. Thus, numerous companies have introduced
some satellites with high throughput such as Starlink project and the
Jupiter-2. To expand the coverage of maritime communication systems, shore-
based and island-based TBSs [11] are also employed alongside maritime
satellites. There are some existing shore-based systems which can offer some
services with low speed data, such as the Automatic Identification System
(AIS) and Navigation Telex (NAVTEX) system [12]. However, efforts have still
been made by other companies to enhance communication data rates. These
include conducting tests on long distance transmission from shore to vessel by
using long term evolution (LTE) or global interoperability of microwave access
(WiMAX) networks [13]. However, the coverage provided by shore-based
communication systems is often limited, especially in the remote sea. The
dynamic nature of ships and maritime activities necessitates communication
systems capable of real-time communication with mobile vessels.

In maritime communications, well-equipped shore-based public communications
network offers the convenience of high-speed data services to maritime users
onboard ships. However, coverage is still a significant challenge to overcome.
In order to meet wide coverage, satellites have become a critical
consideration in the next generation of communication network. Through relay
cooperation, signal retransmission can be carried out for users who receive
poor satellite signals. Recent work has explored the integration of satellite
and ocean networks. In [14], stations based on island and shore are seen as
extensions of terrestrial networks to fill the gap in satellite transmission
at sea. In [15], the author studies cooperative transmission in satellite-
ground integrated networks and proposes the comprehensive framework for
cooperative transmission in satellite-ground systems in the future. While many
LEO satellites with high throughput can provide communication services of high
speed data and cover remote areas, satellite system resources remain
insufficient to meet the rapidly growing maritime service demands. In [16],
the authors analyze potential application of integrated satellite-ground
networks in maritime communications, and point out that existing satellite
systems have limited resources and cannot independently meet the increasing
bandwidth and reliability requirements of maritime services. Additionally, the
high cost of satellite communication is also an important problem that cannot
be ignored. Therefore, an efficient integrated satellite-terrestrial maritime
communication network is urgently needed. The integrated system can offer a
practical solution for maritime networks, which combines the wide coverage
advantages of satellites with the high capacity of shore-based systems. The
proposed integrated satellite-terrestrial maritime communication network can
transfer part of the traffic to the shore-based system, which reduces the
demand for satellite system resources, especially decreases the required
number of satellites. Therefore, the reduced number of satellites can
definitely decrease the overall deployment and operating costs of the entire
maritime communication system. Moreover, by combining the advantages of TBS
and LEO satellite, the integrated network can improve the performance of the
whole system and increase the resource utilization efficiency.

Energy efficiency plays a crucial role in maritime communication because of
the substantial loss experienced in maritime paths and the limited
availability of TBS. Maritime communication networks must cover a wide area
and often require the high-power TBS for remote transmission, which leads to
less efficient energy use. Hence, the design of efficient and energy-saving
allocation strategies holds great significance. At present, the resource
allocation technology of terrestrial network has been widely studied. In [17],
a cooperative power allocation and user scheduling scheme is proposed for
hybrid delay services, employing a cross-layer approach. Reference [18]
introduces a dynamic programming-based algorithm that combines power
allocation and user scheduling for multi-user multiple-input multiple-output
(MIMO) systems. More recently, the authors in [19] present a pilot and
scheduling allocation scheme for massive MIMO systems, aiming to maximize
users capacity while ensuring quality of service (QoS) requirements. Besides,
in [20], energy efficiency is improved by using specific characteristics of
sea lanes and user mobility. In particular, it is possible to estimate their
large-scale CSI by utilizing the channel information of maritime users.

However, the assumption behind these research is perfect CSI, which is
unfeasible in actual maritime communication. In general, CSI acquisition has
channel phase error due to the error and delay when the device conveys CSI to
the satellite through gateway. Radio communication at sea is subject to a
variety of interference, including signal scattering caused by rain, antenna
vibration caused by waves, and interference of mutual reflection. Thus, it
becomes necessary to take into account the influence of imperfect CSI. In
fact, several researches have been conducted to specifically investigate
multi-beam satellite communication in the presence of uncertain channel
conditions. The work [21] is devoted to solving the beamforming problem in
terrestrial-satellite integrated networks. By adopting beamforming technology
and considering the influence of channel uncertainty and interference, the
performance and reliability of the system are improved. In [22], it is
proposed a scheme for robust beamforming design to meet the requirements of
multi-group multicast communication and maintain the reliability of the system
under the interruption limitation. To maximize the total secrecy of the
system, robust beamforming schemes are proposed in [23] and [24], based on the
assumption of the additive norm bounded estimation error and the non-
orthogonal multi-access multi-beam access technique in satellite
communication. Furthermore, the work [25] is dedicated to optimizing the
resource utilization efficiency of satellite communication systems. By
introducing a robust beamforming technology and the reasonable resource
allocation strategy, the performance and reliability of the system are
improved. Although the above works have modeled the CSI error, these CSI error
models are not constructed for specific maritime scenarios and do not take
into account the characteristics of scenarios. Our work is specially designed
for maritime communications, which focuses on the characteristics of the
ocean, such as wave fluctuations. Therefore, the proposed CSI error model is
based on the characteristics of the ocean and is designed specially for the
maritime communication, which is applied to our proposed integrated satellite-
terrestrial maritime communication system. The integrated satellite-
terrestrial maritime communication network offers extensive coverage, stable
and reliable communication capabilities, and establishes a viable foundational
framework for the implementing of maritime IoT network framework. Furthermore,
the integrated satellite-terrestrial maritime communication network provides
connectivity and communication capabilities for the maritime IoT. In order to
provide extensive and seamless wireless network coverage and high-speed
communication services in the maritime environment, it is desired to increase
the transmit power, which also leads to severe co-channel interference.
Considering limited power resource, we advocate to jointly design the
beamforming at TBS and LEO satellite to minimize the transmit power of the
system.

This paper focuses on expanding the coverage of maritime communication network
and ensure communication quality for users. We propose an integrated
satellite-terrestrial maritime communication system, where TBS and LEO
satellites cooperatively provide wide-area wireless services. In addition, we
consider imperfect CSI of the maritime channels caused by wave fluctuation. To
reduce interference, and ensure the communication quality of users, we
investigate a robust beamforming design that combines LEO satellites and TBS.
The aim is to minimize the total transmit power while guaranteeing the minimum
data rate requirements for maritime users. The main contributions of this
paper can be summarized as following aspects.

  1. 1.

We propose an integrated satellite-terrestrial maritime communication system,
which combines a TBS with multiple antennas to provide network coverage for
near-shore users, and a maritime LEO satellite to provide network coverage for
off-shore users.

  2. 2.

We analyze the impacts of sea wave fluctuation on CSI acquisition, and
accurately model the CSI errors at the LEO satellite and TBS.

  3. 3.

We propose a robust beamforming design algorithm for integrated satellite-
terrestrial maritime communications based on the obtained CSI. Both
theoretical analysis and simulation results confirm the effectiveness of the
proposed algorithm.

The rest of this article is organized as follows. Section II introduces an
integrated satellite-terrestrial maritime communication system. In section
III, the transmit beamforming at the TBS and LEO satellite is jointly designed
to minimize the overall power consumption while satisfying QoS requirements.
Section IV provides simulation results to testify the effectiveness of
proposed algorithm. Finally, section V concludes this paper.

_Notations_ : This article represents matrices and vectors by bold letters,
$\|\cdot\|$, $(\cdot)^{H}$, $|\cdot|$, $\|\cdot\|_{F}$ to denote $L_{2}$-norm
of a vector, conjugate transpose, absolute value and the Frobenius norm of a
matrix, respectively. $J_{0}(\cdot)$, $J_{1}(\cdot)$, and $J_{3}(\cdot)$ are
employed to represent the zeroth, first and third order of the first-kind of
Bessel function, respectively. $\text{Rank}(\cdot)$ and $\text{tr}(\cdot)$
denote the rank and trace of a matrix, and $\odot$ denotes the Hadamard
product. ${{\mathbb{C}}^{m\times n}}$ denotes the set of $m\times n$
dimensional complex matrices, $[\cdot]_{n,m}$ denotes the $[n,m]$th element of
a matrix, $\mathcal{CN}(\mu,\sigma^{2})$ and $\mathcal{N}(\mu,\sigma^{2})$ are
denote the complex Gaussian distribution and the Gaussian distribution with
mean $\mu$ and variance $\sigma^{2}$, respectively.

##  II System Model

###  II-A System Description

Consider an integrated satellite-terrestrial maritime communication system
shown in Fig. 1, where a LEO satellite and a shore-based TBS cooperatively
provide communication services to various maritime users. In general, maritime
communication systems work in very high frequency (VHF) band from 30 to 300
MHz. Thus, we consider that the LEO satellite and TBS work over the same
spectrum in the VHF band [26]. Considering the obstruction of sea surface due
to the curvature of the earth, we use the trigonometric function to calculate
the maximum line of sight (LoS) transmission distance between TBS and maritime
user as follows

$L_{\max}=\sqrt{H_{t}^{2}+2H_{t}R_{e}}+\sqrt{H_{r}^{2}+2H_{r}R_{e}},$ (1)

where $H_{t}$ is the height of TBS, $H_{r}$ denotes the effective antennas
height of user, and $R_{e}$ is the Earth radius. For clarity, we define the
near-shore users as those whose distance to the TBS is less than $L_{\max}$
and off-shore users as those whose distance to the TBS is more than
$L_{\max}$. The near-shore users communicate with the TBS, while the off-shore
users communicate with the maritime LEO satellite. The TBS on the shore
equipped with $M_{1}$ antennas transmits the desired signals to $K_{1}$ near-
shore users, and the LEO satellite equipped with $M_{2}$ antennas serves
$K_{2}$ off-shore users. For simplicity, we assume that all the maritime
terminals in the system have a single antenna each.

Figure 1: Illustration of an integrated satellite-terrestrial communication
system for maritime purposes, where the TBS and LEO satellite provide wide-
area services in a coordinated manner.

###  II-B Channel Model

In general, maritime wireless channel is location independent due to the lack
of obstacles and the sparsity of scatters. Thus, maritime channel is mainly
composed of reflected path on the ocean surface and LoS path, namely two-ray
channel model [27]. As the scattering effects are significantly weaker
compared to the first two main paths in the two-ray model, we ignore the
impacts resulted by the diffusion paths of the rough ocean surface. In
general, the communication distance will affect the transmit power, and the
maritime path loss also affects the energy efficiency directly. The channel
gain from the TBS to near-shore user $i$ can be modeled as

$\displaystyle\mathbf{h}_{1,i}=\bigg{[}\frac{\lambda}{2\pi
d_{i}}\sin(\frac{2\pi H_{t}H_{u}}{\lambda
d_{i}})\bigg{]}\mathbf{a}(\theta_{1}),$ (2)

where $d_{i}$ is the distance between TBS and the user $i$, $H_{u}$ is the
effective antenna height of users, and $\mathbf{a}(\theta_{1})$ denotes the
antenna array response of the link from TBS to the user $i$, which can be
expressed as

$\displaystyle\mathbf{a}(\theta_{1})=[1,e^{-j(\frac{2\pi
b}{\lambda}\cos\theta_{1})},\dots,e^{-j(\frac{2\pi
b(M_{1}-1)}{\lambda}\cos\theta_{1})}],$ (3)

where $\theta_{1}$ denotes the angle of departure, and $b$ is the distance
between two adjacent antennas at the TBS. Since the distance between the
transmitter and receiver is large enough, we assume that the departure angles
of all links are the same. On the other hand, considering the electromagnetic
propagation characteristics from the LEO satellite to the off-shore users
[27]-[29], the space-to-sea channel associated with off-shore user $m$ can be
modeled as

$\displaystyle\mathbf{h}_{2,m}=g_{m}\sqrt{P_{m}}\bigg{(}\sqrt{\frac{1}{1+K}}\mathbf{h}_{2,m}^{\text{NLoS}}+\sqrt{\frac{K}{1+K}}\mathbf{h}_{2,m}^{\text{LoS}}\bigg{)},$
(4)

where $g_{m}$ is the large-scale fading factor, which is given by

$\displaystyle g_{m}=\mathbf{r}_{m}\sqrt{r_{\text{FSL}}G_{m}},$ (5)

where $G_{m}$ is the receive antenna gain of off-shore user $m$, and
$r_{\text{FSL}}$ denotes the free space loss (FSL) as follows

$\displaystyle\ r_{\text{FSL}}=\frac{\lambda^{2}}{(4\pi)^{2}D_{m}},$ (6)

where $D_{m}$ is the distance between LEO satellite and the user $m$.
Furthermore, $\mathbf{r}_{m}$ is the rain attenuation coefficient vector. In
the form of dB, the rain attenuation model
$r_{i}^{\text{dB}}(k)=20\log_{10}r_{i}(k)$ follows a log-normal distribution
with
$\ln(r_{i}^{\text{dB}})\sim\mathcal{C}\mathcal{N}(\mu_{r},\sigma_{r}^{2})$
[30], where $\sigma_{r}^{2}$ and $\mu_{r}$ denote the scale parameter and log-
normal location, respectively. Moreover, $K$ denotes the Rician fading factor
of the space-to-sea channel, $\mathbf{h}_{2,m}^{\text{NLoS}}$ is the non line
of sight (NLoS) component of $\mathbf{h}_{2,m}$, which is an identically and
independently distributed variable following distribution
$\mathcal{C}\mathcal{N}(0,1)$, and $\mathbf{h}_{2,m}^{\text{LoS}}$ is the LoS
component of $\mathbf{h}_{2,m}$, which is given by

$\displaystyle\mathbf{h}_{2,m}^{\text{LoS}}=[1,e^{-j(\frac{2\pi
b}{\lambda}\cos\theta_{2})},\dots,e^{-j(\frac{2\pi
b(M_{2}-1)}{\lambda}\cos\theta_{2})}],$ (7)

where $\theta_{2}$ is the angle of departure of the link from the LEO
satellite to the off-shore user $m$. Besides, the LEO satellite transmit
antenna gain $P_{m}$ can be expressed as [30]

$\displaystyle
P_{m}=G_{\max}\bigg{(}\frac{J_{1}(\psi_{m})}{2\psi_{m}}+36\frac{J_{3}(\psi_{m})}{\psi_{m}^{3}}\bigg{)}^{2},$
(8)

where $G_{\max}$ is the maximal LEO satellite antenna gain, $J_{3}(\cdot)$ and
$J_{1}(\cdot)$ denote the third and first orders of the first-kind of Bessel
functions, respectively, and
$\psi_{m}=2.07123\sin{\varphi_{\text{sm}}}/\sin{\varphi_{\text{3dB},m}}$ with
$\varphi_{\text{sm}}$ being the angle of the satellite beam boresight to the
user $m$, and $\varphi_{\text{3dB},m}$ being the 3dB angle.

###  II-C Signal Model

In order to improve the overall performance of maritime communications, the
TBS and the LEO satellite coordinate their transmit signals via spatial
beamforming. Specifically, the transmit signals $\mathbf{x}_{1}$ for near-
shore users and $\mathbf{x}_{2}$ for off-shore users can be expressed as

$\displaystyle\mathbf{x}_{1}=\sum_{i=1}^{K_{1}}\mathbf{w}_{i}s_{1,i},$ (9)

$\displaystyle\mathbf{x}_{2}=\sum_{m=1}^{K_{2}}\mathbf{v}_{m}s_{2,m},$ (10)

where $\mathbf{w}_{i}$ and $\mathbf{v}_{m}$ are the designed beams for the
near-shore user $i$ and the off-shore user $m$, respectively. The $s_{1,i}$
and $s_{2,m}$ denote the signals for the near-shore user $i$ and the off-shore
user $m$ with unit norm, respectively. For the near-shore users, they receive
the desired signals from TBS, and also suffer the interference from the LEO
satellite due to wide space-to-sea beam. Thus, the received signal at the
near-shore user $i$ is formulated as

$\displaystyle y_{1,i}$
$\displaystyle=\mathbf{h}_{1,i}\mathbf{x}_{1}+\hat{\mathbf{h}}_{2,i}\mathbf{x}_{2}+n_{1,i}$
(11)
$\displaystyle=\underbrace{\mathbf{h}_{1,i}\mathbf{w}_{i}s_{1,i}}_{\text{Desired
Signal}}+\underbrace{\sum_{j=1,j\neq
i}^{K_{1}}\mathbf{h}_{1,i}\mathbf{w}_{j}s_{1,j}}_{\text{Inter-device
Interference}}+$
$\displaystyle\underbrace{\sum_{m=1}^{K_{2}}\hat{\mathbf{h}}_{2,i}\mathbf{v}_{m}s_{2,m}}_{\text{Intra-
device Interference}}+\underbrace{n_{1,i}}_{\text{AWGN}},$

where $\hat{\mathbf{h}}_{2,i}$ denotes the space-to-sea channel gain from LEO
satellite to the near-shore user $i$, and $n_{1,i}$ is the additive white
Gaussian noise (AWGN) at the near-shore user $i$ with variance
$\sigma_{1,i}^{2}$. Hence the signal to interference plus noise ratio (SINR)
at the near-shore user $i$ can be computed as

$\displaystyle\text{SINR}_{1,i}=\frac{|\mathbf{h}_{1,i}\mathbf{w}_{i}|^{2}}{\sum_{j=1,j\neq
i}^{K_{1}}|\mathbf{h}_{1,i}\mathbf{w}_{j}|^{2}+\sum_{m=1}^{K_{2}}|\hat{\mathbf{h}}_{2,i}\mathbf{v}_{m}|^{2}+\sigma_{1,i}^{2}}{\color[rgb]{0,0,1}.}$
(12)

Accordingly, the achievable date rate of near-shore user $i$ can be expressed
as

$\displaystyle R_{1,i}=\log_{2}(1+\text{SINR}_{1,i}){\color[rgb]{0,0,1}.}$
(13)

Due to the curvature of the earth, the off-shore users can not receive the
signals from the TBS, and only from the LEO satellite. Thus, the received
signal at the off-shore user $m$ is formulated as

$\displaystyle y_{2,m}$ $\displaystyle=\mathbf{h}_{2,m}\mathbf{x}_{2}+n_{2,m}$
(14)
$\displaystyle=\underbrace{\mathbf{h}_{2,m}\mathbf{v}_{m}s_{2,m}}_{\text{Desired
Signal}}+\underbrace{\sum_{k=1,k\neq
m}^{K_{2}}\mathbf{h}_{2,m}\mathbf{v}_{k}s_{2,k}}_{\text{Inter-device
Interference}}+\underbrace{n_{2,m}}_{\text{AWGN}},$

where $n_{2,m}$ is the AWGN at the off-shore user $m$ with variance
$\sigma_{2,m}^{2}$. Then the SINR at the off-shore user $m$ can be computed as

$\displaystyle\text{SINR}_{2,m}=\frac{|\mathbf{h}_{2,m}\mathbf{v}_{m}|^{2}}{\sum_{k=1,k\neq
m}^{K_{2}}|\mathbf{h}_{2,m}\mathbf{v}_{k}|^{2}+\sigma_{2,m}^{2}}{\color[rgb]{0,0,1}.}$
(15)

Similarly, the achievable date rate of off-shore user $m$ can be expressed as

$\displaystyle R_{2,m}=\log_{2}(1+\text{SINR}_{2,m}){\color[rgb]{0,0,1}.}$
(16)

It is not difficult to find that the beam $\mathbf{w}_{i}$ and
$\mathbf{v}_{m}$ have a great impact on the overall performance of the
integrated satellite-terrestrial maritime communication system. In the
maritime communication system, the power of LEO satellite and TBS is often
limited. Thus, we aim to design the beamforming in the next section to
minimize the total power at the transmitter while ensuring the communication
quality of maritime users.

##  III Robust Beamforming Design

In this section, we first model the acquisition of CSI in the context of
maritime communication, and then design transmit beams for TBS and LEO
satellite based on the obtained CSI.

###  III-A CSI Acquisition

In general, the maritime electromagnetic propagation is influenced by a
variety of time varying factors such as atmospheric conditions and sea
surface. The height of sea surface is affected by tidal movement and wave
fluctuation. Wherein, the wave fluctuation causes the change of azimuth angle
of maritime wireless channel. As a result, there exists angle mismatch between
the actual CSI and the obtained CSI, which is given by

$\displaystyle\theta_{1}=\bar{\theta}_{1}+\Delta\theta_{1},$ (17)
$\displaystyle\theta_{2}=\bar{\theta}_{2}+\Delta\theta_{2},$

where $\theta_{i}$ is the actual angle of departure, $\bar{\theta}_{i}$ is the
obtained angle of departure, and $\Delta\theta_{i}$ is the error of angle of
departure, which is constrained within a hyper-spherical region centered at
the origin with radii of $\varepsilon_{i}$. Mathematically, the error of angle
of departure are bounded as

$\displaystyle\parallel\Delta\theta_{1}\parallel_{2}\leq\varepsilon_{1},$ (18)
$\displaystyle\parallel\Delta\theta_{2}\parallel_{2}\leq\varepsilon_{2}.$

We assume that the angle errors are unknown but the corresponding bounds
$\varepsilon_{1}$ and $\varepsilon_{2}$ are available. Hence, the relationship
between actual CSI and obtained CSI can be expressed as

$\displaystyle\mathbf{h}_{1,i}=\bar{\mathbf{h}}_{1,i}+\Delta\mathbf{h}_{1,i},$
(19)

$\displaystyle\mathbf{h}_{2,m}=\bar{\mathbf{h}}_{2,m}+\Delta\mathbf{h}_{2,m},$
(20)

where $\bar{\mathbf{h}}_{1,i}$ and $\bar{\mathbf{h}}_{2,m}$ denote the
obtained value of $\mathbf{h}_{1,i}$ and $\mathbf{h}_{2,m}$, respectively, and
they can be expressed as

$\displaystyle\bar{\mathbf{h}}_{1,i}=\bigg{[}\frac{\lambda}{2\pi
d_{i}}\sin(\frac{2\pi H_{t}H_{u}}{\lambda
d_{i}})\bigg{]}\bar{\mathbf{a}}(\theta_{1}),$ (21)

$\displaystyle\bar{\mathbf{h}}_{2,m}=g_{m}\sqrt{P_{m}}\bigg{(}\sqrt{\frac{1}{1+K}}\mathbf{h}_{2,m}^{\text{NLoS}}+\sqrt{\frac{K}{1+K}}\bar{\mathbf{h}}_{2,m}^{\text{LoS}}\bigg{)}.$
(22)

To derive the CSI error $\Delta{\mathbf{h}_{1,i}}$, according to the angle
mismatch in (17), we take the first-order Taylor expansion on the $k$-th
element of antenna array response in (3). Then, we have

$\displaystyle\exp\bigg{(}-j\frac{2\pi
bk}{\lambda}\cos(\bar{\theta}_{1}+\Delta\theta_{1})\bigg{)}$ (23)
$\displaystyle\approx$ $\displaystyle\underbrace{\exp\bigg{(}-j\frac{2\pi
bk}{\lambda}\cos(\bar{\theta}_{1})\bigg{)}}_{\text{$\bar{\mathbf{A}_{k}}(\theta_{1})$}}+$
$\displaystyle\underbrace{\bigg{(}j\frac{2\pi
bk}{\lambda}\sin(\bar{\theta}_{1})\Delta\theta_{1}\bigg{)}\times\exp\bigg{(}-j\frac{2\pi
bk}{\lambda}\cos(\bar{\theta}_{1})\bigg{)}}_{\text{$\Delta\mathbf{a}_{k}(\theta_{1})$}}.$

Further we redefine $\Delta\mathbf{a}(\theta_{1})$ as

$\displaystyle\Delta\mathbf{a}(\theta_{1})\triangleq\bar{\mathbf{a}}(\theta_{1})\odot\sin(\bar{\theta}_{1})[1,\dots,j\frac{2\pi
b}{\lambda}(k-1)]\Delta\theta_{1}{\color[rgb]{0,0,1}.}$ (24)

Taking each element of the antenna array response matrix applied by (24) and
plugging it into channel, we can get the CSI error $\Delta\mathbf{h}_{1,i}$ as
follows

$\displaystyle\Delta\mathbf{h}_{1,i}$
$\displaystyle=\bar{\mathbf{h}}_{1,i}\Delta\mathbf{a}(\theta_{1})$ (25)
$\displaystyle=\bar{\mathbf{h}}_{1,i}\bar{\mathbf{a}}(\theta_{1})\odot\sin(\bar{\theta}_{1})[1,\dots,j\frac{2\pi
b}{\lambda}(M_{1}-1)]\Delta\theta_{1}.$

By introducing a new variable
$\mathbf{n}_{1}=\bar{\mathbf{h}}_{1,i}\bar{\mathbf{a}}(\theta_{1})\odot\sin(\bar{\theta}_{1})[1,\dots,j\frac{2\pi
b}{\lambda}(M_{1}-1)]$, where $\mathbf{n}_{1}$ is constrained within
$\xi_{1}$, the CSI error can be bounded as

$\displaystyle\parallel\Delta\mathbf{h}_{1,i}\parallel_{2}=\parallel\mathbf{n}_{1}\Delta\theta_{1}\parallel_{2}\leq\xi_{1}\varepsilon_{1}.$
(26)

Similar to $\Delta\mathbf{h}_{1,i}$, $\Delta\mathbf{h}_{2,m}$ can be
approximated as

$\displaystyle\Delta\mathbf{h}_{2,m}=g_{m}\sqrt{P_{m}}\sqrt{\frac{K}{1+K}}\Delta\mathbf{h}_{2,m}^{\text{LoS}},$
(27)

where
$\Delta\mathbf{h}_{2,m}^{\text{LoS}}=\bar{\mathbf{h}}_{2,m}^{\text{LoS}}\bar{\mathbf{a}}(\theta_{2})\odot\sin(\bar{\theta}_{2})[1,\dots,j\frac{2\pi
b}{\lambda}(M_{2}-1)]\Delta\theta_{2}$. Also by introducing a new variable
$\mathbf{n}_{2}=\bar{\mathbf{h}}_{2,m}^{\text{LoS}}\bar{\mathbf{a}}(\theta_{2})\odot\sin(\bar{\theta}_{2})[1,\dots,j\frac{2\pi
b}{\lambda}(k-1)]$, the CSI error can be bounded as

$\displaystyle\parallel\Delta\mathbf{h}_{2,m}\parallel_{2}=\bigg{|}\bigg{|}g_{m}\sqrt{P_{m}}\sqrt{\frac{K}{1+K}}\mathbf{a}_{2}\Delta\theta_{2}\bigg{|}\bigg{|}_{2}\leq\xi_{2}\varepsilon_{2}.$
(28)

###  III-B Robust Design

Due to wave fluctuation, the obtained CSI is usually imperfect in maritime
communications. In this context, we attempt to design a robust beamforming
scheme to guarantee QoS of near-shore and off-shore users in the worst case.
Specifically, the optimization problem can be formulated as

$\displaystyle\mathcal{Q}1:\underset{\mathbf{w}_{i},\mathbf{v}_{m}}{\mathop{\text{min}}}\,\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\sum_{i=1}^{K_{1}}\alpha_{1,i}\parallel\mathbf{w}_{i}\parallel^{2}+\sum_{m=1}^{K_{2}}\alpha_{2,m}\parallel\mathbf{v}_{m}\parallel^{2}$
(29a) s.t.
$\displaystyle\\!\\!\underset{\parallel\Delta\mathbf{h}_{1,i}\parallel\leq\epsilon_{1}}{\min}R_{1,i}\geq\gamma_{1},$
(29e)
$\displaystyle\\!\\!\underset{\parallel\Delta\mathbf{h}_{2,m}\parallel\leq\epsilon_{2}}{\min}R_{2,m}\geq\gamma_{2},$
$\displaystyle\\!\\!\sum_{i=1}^{K_{1}}\parallel\mathbf{w}_{i}\parallel^{2}\leq
P_{T},$
$\displaystyle\\!\\!\bigg{[}\sum_{m=1}^{K_{2}}\mathbf{v}_{m}\mathbf{v}_{m}^{H}\bigg{]}_{k,k}\leq
P_{S},\quad\forall k$

where $\alpha_{1,i}$ and $\alpha_{2,m}$ are the weights of the near-shore
users and off-shore users, $\gamma_{1}$ and $\gamma_{2}$ are the minimum data
transmission rates required by near-shore users and off-shore users, and
$P_{T}$ and $P_{S}$ are the maximum transmit power of TBS and a single antenna
of LEO satellite, respectively. Obviously, the problem ($\mathcal{Q}1$) is
non-convex, and it is difficult to get the optimal solution directly. On the
one hand, the constraints (29e) and (29e) involve an infinite number of
inequality constraints, as these constraints must be set up in any possible
realizations of $\Delta\mathbf{h}_{1,i}$ and $\Delta\mathbf{h}_{2,m}$. On the
other hand, the optimization variables $\mathbf{w}_{i}$ and $\mathbf{v}_{m}$
are highly coupled in the expressions of $R_{1,i}$ and $R_{2,m}$, which are
difficult to deal with. We take some transformations for the optimization
problem to make it more tractable. Firstly, we introduce auxiliary variables
$\mathbf{W}_{i}=\mathbf{w}_{i}\mathbf{w}_{i}^{H}\in\mathbb{C}^{M_{1}\times
M_{1}}$, and
$\mathbf{V}_{m}=\mathbf{v}_{m}\mathbf{v}_{m}^{H}\in\mathbb{C}^{M_{2}\times
M_{2}}$, then the problem is equivalent to the following form

$\displaystyle\mathcal{Q}2:\underset{\mathbf{W}_{i},\mathbf{V}_{m}}{\mathop{\text{min}}}\,\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\sum_{i=1}^{K_{1}}\alpha_{1,i}\text{tr}(\mathbf{W}_{i})+\sum_{m=1}^{K_{2}}\alpha_{2,m}\text{tr}(\mathbf{V}_{m})$
(30a) s.t.
$\displaystyle\\!\\!\underset{\parallel\Delta\mathbf{h}_{1,i}\parallel\leq\epsilon_{1}}{\min}R_{1,i}\geq\gamma_{1},$
(30g)
$\displaystyle\\!\\!\underset{\parallel\Delta\mathbf{h}_{2,m}\parallel\leq\epsilon_{2}}{\min}R_{2,m}\geq\gamma_{2},$
$\displaystyle\\!\\!\sum_{i=1}^{K_{1}}\parallel\mathbf{w}_{i}\parallel^{2}\leq
P_{T},$
$\displaystyle\\!\\!\bigg{[}\sum_{m=1}^{K_{2}}\mathbf{V}_{m}\bigg{]}_{k,k}\leq
P_{S},\quad\forall k$ $\displaystyle\\!\\!\mathbf{W}_{i}\succeq
0,\text{rank}(\mathbf{W}_{i})=1,\forall i,$
$\displaystyle\\!\\!\mathbf{V}_{m}\succeq
0,\text{rank}(\mathbf{V}_{m})=1,\forall m$

$\underset{\parallel\Delta\mathbf{h}_{1,i}\parallel\leq\epsilon_{1}}{\min}\frac{\mathbf{h}_{1,i}^{H}\mathbf{W}_{i}\mathbf{h}_{1,i}}{\sum_{j=1,j\neq
i}^{K_{1}}\mathbf{h}_{1,i}^{H}\mathbf{W}_{j}\mathbf{h}_{1,i}+\sum_{m=1}^{K_{2}}\hat{\mathbf{h}}_{2,i}^{H}\mathbf{V}_{m}\hat{\mathbf{h}}_{2,i}+\sigma_{1,i}^{2}}\geq
2^{\gamma_{1}}-1$ (31)

$\underset{\parallel\Delta\mathbf{h}_{2,m}\parallel\leq\epsilon_{2}}{\min}\frac{\mathbf{h}_{2,m}^{H}\mathbf{V}_{m}\mathbf{h}_{2,m}}{\sum_{k=1,k\neq
m}^{K_{2}}\mathbf{h}_{2,m}^{H}\mathbf{V}_{k}\mathbf{h}_{2,m}+\sigma_{2,m}^{2}}\geq
2^{\gamma_{2}}-1$ (32)

Despite the transformation mentioned above, the problem ($\mathcal{Q}2$)
remains intractable, primarily due to the inclusion of an limitless number of
constraints in (30g) – (30g). To convert these two constraints into an
equivalent problem with a finite number of constraints, we need to use the
following lemma.

_Lemma 1_ (S-procedure): We define the following quadratic function as

$\displaystyle
f_{k}(\mathbf{x})=\mathbf{x}^{H}\mathbf{A}_{k}\mathbf{x}+2\text{Re}\\{\mathbf{B}_{k}^{H}\mathbf{x}\\}+c_{k},k=1,2,\dots,n$
(33)

where $\mathbf{x}\in\mathbb{C}^{N\times 1}$ is variable,
$\mathbf{A}_{k}=\mathbf{A}_{k}^{H}\in\mathbb{C}^{N\times N}$ can be any
Hermitian matrix, vector $\mathbf{b}_{k}\in\mathbb{C}^{N\times 1}$, and scalar
$c_{k}$, $k=1,2,\dots,n$. Only if there exists $\mu_{k}\geq 0,k=1,2,\dots,n$,
we can get the condition $f_{i}(\mathbf{x})\geq 0\Rightarrow
f_{0}(\mathbf{x})\geq 0$, such that

$\displaystyle\sum_{k=1}^{n}\mu_{k}{\left[\begin{array}[]{ccc}\mathbf{A}_{k}&\mathbf{b}_{k}\\\
\mathbf{b}_{k}^{H}&c_{k}\end{array}\right]}-{\left[\begin{array}[]{ccc}\mathbf{A}_{i}&\mathbf{b}_{i}\\\
\mathbf{b}_{i}^{H}&c_{i}\end{array}\right]}\succeq 0.$ (34)

Note that the current constraints (30g) and (30g) cannot be applied directly
to the S-procedure, we perform some matrix transformations and equality
transformations [23], [31]. By substituting (12) - (13) and (15) - (16) into
constraints (30g) and (30g), the constraints are transformed as (31) and (32)
at the bottom of the next page. The essence of constraints (31) and (32) is
that the two inequality always hold when all channel CSI errors are satisfied,
i.e., both inequalities hold in all realizable cases at the same time. Hence,
the constraint (31) can be rewritten as

$\displaystyle\mathbf{h}_{1,i}^{H}\mathbf{W}_{i}\mathbf{h}_{1,i}-(2^{\gamma}_{1}-1)\bigg{(}\sum_{j=1,j\neq
i}^{K_{1}}\mathbf{h}_{1,i}^{H}\mathbf{W}_{j}\mathbf{h}_{1,i}+$ (35)
$\displaystyle\sum_{m=1}^{K_{2}}\hat{\mathbf{h}}_{2,i}^{H}\mathbf{V}_{m}\hat{\mathbf{h}}_{2,i}+\sigma_{1,i}^{2}\bigg{)}\geq
0,$

$\displaystyle\Delta\mathbf{h}_{1,i}^{H}\times(-\mathbf{I}_{N})\times\Delta\mathbf{h}_{1,i}\geq
0,\forall\Delta\mathbf{h}_{1,i}.$ (36)

Then we combine the like terms and introduce auxiliary variable
$\mathbf{N}_{1}=\mathbf{W}_{i}-(2^{\gamma}_{1}-1)\sum_{j=1,j\neq
i}^{K_{1}}\mathbf{W}_{j}$. Moreover, by substituting (19) to (35), the
constraint (31) can be transformed as

$\displaystyle\Delta\mathbf{h}_{1,i}^{H}\mathbf{N}_{1}\Delta\mathbf{h}_{1,i}+2Re\\{\Delta\mathbf{h}_{1,i}\mathbf{N}_{1}\bar{\mathbf{h}}_{1,i}^{H}\\}+\bar{\mathbf{h}}_{1,i}^{H}\mathbf{N}_{1}\bar{\mathbf{h}}_{1,i}-$
(37)
$\displaystyle(2^{\gamma}_{1}-1)\bigg{(}\sigma_{1,i}^{2}+\sum_{m=1}^{K_{2}}\hat{\mathbf{h}}_{2,i}^{H}\mathbf{V}_{m}\hat{\mathbf{h}}_{2,i}\bigg{)}\geq
0.$

By introducing a new variable $\lambda_{1,i}$, we can transform the constraint
(30g) into a linear matrix inequality (LMI) form with Lemma 1, which can be
expressed as

$\displaystyle{\left[\begin{array}[]{ccc}\mathbf{N}_{1}+\lambda_{1,i}\mathbf{I}_{N}&\mathbf{N}_{1}\bar{\mathbf{h}}_{1,i}\\\
\bar{\mathbf{h}}_{1,i}^{H}\mathbf{N}_{1}&\bar{\mathbf{h}}_{1,i}^{H}\mathbf{N}_{1}\bar{\mathbf{h}}_{1,i}-(2^{\gamma_{1}}-1)c_{1}-\lambda_{1,i}\epsilon_{1}^{2}\end{array}\right]}\succeq
0,$ (38)

where
$c_{1}=\sum_{m=1}^{K_{2}}\hat{\mathbf{h}}_{2,i}^{H}\mathbf{V}_{m}\hat{\mathbf{h}}_{2,i}+\sigma_{1,i}^{2}$,
and $\lambda_{1,i}$ is the slack variable with $\lambda_{1,i}\geq 0$.
Likewise, the constraint (30g) is transformed into LMI constraint as follows

$\displaystyle{\left[\begin{array}[]{ccc}\\!\\!\mathbf{N}_{2}+\lambda_{2,m}\mathbf{I}_{N}&\\!\\!\\!\\!\mathbf{N}_{2}\bar{\mathbf{h}}_{2,m}\\\
\\!\\!\\!\\!\\!\\!\\!\\!\bar{\mathbf{h}}_{2,m}^{H}\mathbf{N}_{2}&\\!\\!\\!\\!\\!\\!\\!\\!\bar{\mathbf{h}}_{2,m}^{H}\mathbf{N}_{2}\bar{\mathbf{h}}_{2,m}-(2^{\gamma_{2}}-1)\sigma_{2,m}^{2}-\lambda_{2,m}\epsilon_{2}^{2}\end{array}\right]}\succeq
0,$ (39)

where $\mathbf{N}_{2}=\mathbf{V}_{m}-(2^{\gamma}_{2}-1)\sum_{k=1,k\neq
m}^{K_{2}}\mathbf{V}_{k}$, and $\lambda_{2,m}$ is a new introduced slack
variable with $\lambda_{2,m}\geq 0$. Based on the above transformations, the
optimization problem can be rewritten as

$\displaystyle\mathcal{Q}3:\\!\\!\\!\\!\\!\\!\underset{\mathbf{W}_{i},\mathbf{V}_{m},\lambda_{1,i},\lambda_{2,m}}{\mathop{\text{min}}}\,\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!\sum_{i=1}^{K_{1}}\alpha_{1,i}\text{tr}(\mathbf{W}_{i})+\sum_{m=1}^{K_{2}}\alpha_{2,m}\text{tr}(\mathbf{V}_{m})$
(40a) s.t. $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\ref{1}),$ (40g)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\ref{2}),$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{i=1}^{K_{1}}\parallel\mathbf{w}_{i}\parallel^{2}\leq
P_{T},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\bigg{[}\sum_{m=1}^{K_{2}}\mathbf{v}_{m}\mathbf{v}_{m}^{H}\bigg{]}_{k,k}\leq
P_{S},\quad\forall k$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{W}_{i}\succeq
0,\text{rank}(\mathbf{W}_{i})=1,\forall i,$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{V}_{m}\succeq
0,\text{rank}(\mathbf{V}_{m})=1,\forall m$

Problem ($\mathcal{Q}3$) is still nonconvex due to the rank-1 constraints
[32]. To effectively tackle the rank-1 constraints (40g) and (40g) and obtain
a high-quality solution for problem ($\mathcal{Q}3$), we propose a penalty-
based algorithm. To facilitate the algorithm based on penalty, we need the
following lemma.

$\displaystyle\mathcal{Q}4:\\!\\!\\!\\!\\!\\!\underset{\mathbf{W}_{i},\mathbf{V}_{m},\lambda_{1,i},\lambda_{2,m}}{\mathop{\text{min}}}\,\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!\sum_{i=1}^{K_{1}}\alpha_{1,i}\text{tr}(\mathbf{W}_{i})+\sum_{m=1}^{K_{2}}\alpha_{2,m}\text{tr}(\mathbf{V}_{m})+\rho_{1}\bigg{(}\sum_{i=1}^{K_{1}}\big{(}\text{tr}(\mathbf{W}_{i})-\lambda_{1,i,\max}\big{)}+\sum_{m=1}^{K_{2}}\big{(}\text{tr}(\mathbf{V}_{m})-\lambda_{2,m,\max}\big{)}\bigg{)}$
(41a) s.t. $\displaystyle(\ref{1}),(\ref{2}),(\ref{OP5st3}),(\ref{OP5st4}),$
(41d) $\displaystyle\mathbf{W}_{i}\succeq 0,\forall i,$
$\displaystyle\mathbf{V}_{m}\succeq 0,\forall m$

_Lemma 2:_ The rank-1 condition $\text{rank}(\mathbf{M})=1$ for any positive
semi-definite $\mathbf{M}\in\mathbb{C}^{N\times N}$ can be expressed as the
following condition:

$\displaystyle\text{tr}(\mathbf{M})=\parallel\mathbf{M}\parallel_{2},$ (42)

where $\parallel\mathbf{M}\parallel_{2}$ is the spectral norm of $\mathbf{M}$,
$\text{tr}(\mathbf{M})=\sum_{i=1}^{N}\lambda_{n}$, and $\lambda_{n}$ is the
$n$-th largest eigenvalue of matrix $\mathbf{M}$. Note that $\mathbf{W}_{i}$
and $\mathbf{V}_{m}$ are all positive semidefinite matrix, which means that
each eigenvalue of $\mathbf{W}_{i}$ and $\mathbf{V}_{m}$ satisfied
$\lambda_{n}\geq 0$. Intuitively, rank-1 constraints imply that only one
eigenvalue is larger than zero. Thus, according to the lemma 2 above, we can
replace the rank-1 constraints of (40g) and (40g) with the following equation:

$\displaystyle\text{tr}(\mathbf{W}_{i})-\sum_{i=1}^{K_{1}}\lambda_{1,i,\max}=0,$
(43)

$\displaystyle\text{tr}(\mathbf{V}_{m})-\sum_{m=1}^{K_{2}}\lambda_{2,m,\max}=0.$
(44)

Then, we can rebuild the objective function (40a) by introducing the penalty
function with the constraints (43) and (44).

$\displaystyle\underset{\mathbf{W}_{i},\mathbf{V}_{m},\lambda_{1,i},\lambda_{2,m}}{\mathop{\text{min}}}\,\sum_{i=1}^{K_{1}}\alpha_{1,i}\text{tr}(\mathbf{W}_{i})+\sum_{m=1}^{K_{2}}\alpha_{2,m}\text{tr}(\mathbf{V}_{m})$
(45)
$\displaystyle+\rho_{1}\bigg{(}\sum_{i=1}^{K_{1}}\big{(}\text{tr}(\mathbf{W}_{i})-\lambda_{1,i,\max}\big{)}+\sum_{m=1}^{K_{2}}\big{(}\text{tr}(\mathbf{V}_{m})-\lambda_{2,m,\max}\big{)}\bigg{)}$

where $\rho_{1}>0$ denotes the penalty factor used to penalize the violations
of the equality constraints (43) and (44). It is important to note that as
$\rho_{1}$ approaches infinity, the best result of problem ($\mathcal{Q}3$)
will always satisfy the rank-1 constraints (40g) and (40g), making it the best
result of problem ($\mathcal{Q}3$). Therefore, our focus shifts to solving
problem ($\mathcal{Q}3$) in the subsequent steps.

Unfortunately, the presence of the penalty function in the new objective
function renders it nonconvex. To solve such a nonconvex issue, we adopt a
successive convex approximation method to get the convex one. More precisely,
for the solutions $\mathbf{W}_{i}^{(t)}$ and $\mathbf{V}_{m}^{(t)}$ during the
$t$-th iteration, respectively, we can obtain the inequality as below

$\displaystyle\text{tr}(\mathbf{W}_{i}^{(t+1)})-(\mathbf{e}_{1,i,\max}^{(t)})^{H}\mathbf{W}_{i}^{(t+1)}\mathbf{e}_{1,i,\max}^{(t)}$
(46)
$\displaystyle\geq\text{tr}(\mathbf{W}_{i}^{(t+1)})-\lambda_{1,i,\max}^{(t+1)}\geq
0,$

$\displaystyle\text{tr}(\mathbf{V}_{m}^{(t+1)})-(\mathbf{e}_{2,m,\max}^{(t)})^{H}\mathbf{V}_{m}^{(t+1)}\mathbf{e}_{2,m,\max}^{(t)}$
(47)
$\displaystyle\geq\text{tr}(\mathbf{V}_{m}^{(t+1)})-\lambda_{2,m,\max}^{(t+1)}\geq
0,$

where $\mathbf{e}_{1,i,\max}$ and $\mathbf{e}_{2,m,\max}$ denote the unit
eigenvector corresponding to $\lambda_{1,i,\max}$ and $\lambda_{2,m,\max}$,
respectively. Thus, by replacing (43) and (44) with their upper bound (46) and
(47), the optimization problem ($\mathcal{Q}4$) becomes convex, which is shown
at the bottom of the page. By using some available optimization software
packages, e.g., CVX, it can be effectively solved. The solving process of the
problem ($\mathcal{Q}4$) is summarized in Algorithm 1.

Algorithm 1 Joint Robust Beamforming for Integrated Satellite-Terrestrial
Maritime Communication

0: $K_{1}$, $M_{1}$, $K_{2}$, $M_{2}$, $P_{S}$, $P_{T}$, $\alpha_{1,i}$,
$\alpha_{2,m}$, $\gamma_{1}$, $\gamma_{2}$, $\sigma^{2}_{0}$,
$\varepsilon_{1}$, $\varepsilon_{2}$

0: $\mathbf{w}_{i}$, $\mathbf{v}_{m}$

1: $\mathbf{Initialization:}$ Set positive constant $\tau$, penalty factor
$\rho_{1}$, maximal iteration number $t_{\max}$, accuracy $\epsilon_{1}$,
$\epsilon_{2}$, feasible solution $\mathbf{W}_{i}^{(0)}$,
$\mathbf{V}_{m}^{(0)}$ and iteration index $t=0$

2: $\mathbf{repeat}$

3: Use CVX to deal with problem ($\mathcal{Q}4$), then get
$\mathbf{W}_{i}^{(t)}$ and $\mathbf{V}_{m}^{(t)}$

4: if
$|\text{tr}(\mathbf{W}_{i}^{(t)})-\lambda_{1,i,\max}^{(t)}|>\epsilon_{1}$ and
$|\text{tr}(\mathbf{V}_{m}^{(t)})-\lambda_{2,m,\max}^{(t)}|>\epsilon_{2}$ then

5: Set $\rho_{1}=\rho_{1}+\tau$

6: end if

7: Set $t=t+1$

8: $\mathbf{until}$ $t=t_{\max}$ or problem is solved

9: $\mathbf{Finally}$, apply eigenvalue decomposition (EVD) to
$\mathbf{W}_{i}^{(t)}$, $\mathbf{V}_{m}^{(t)}$ and get $\mathbf{w}_{i}$,
$\mathbf{v}_{m}$.

TABLE I: Main Simulation System Parameters For LEO Satellite IoT precoding scheme Parts |  Complexity in order of ln$\frac{1}{\eta}c_{i}$
---|---
Part 1 with $n_{1}=\mathcal{O}(K_{1}M_{1}^{2})$ |  $\hat{c}_{1}=\sqrt{2K_{1}(M_{1}+1)}\cdot n_{1}\cdot\\{K_{1}[(M_{1}+1)^{3}+M_{1}^{3}+1]+K_{1}n_{1}[(M_{1}+1)^{2}+M_{1}^{2}+1]+n_{1}^{2}$$\\}$
Part 2 with $n_{2}=\mathcal{O}(K_{2}M_{2}^{2})$ |  $\hat{c}_{2}=\sqrt{K_{2}(2M_{2}+3)}\cdot n_{2}\cdot\\{K_{2}[(M_{2}+1)^{3}+M_{2}^{3}+2]+K_{2}n_{2}[(M_{2}+1)^{2}+M_{2}^{2}+2]+n_{2}^{2}$$\\}$

###  III-C Algorithm Analysis

In this subsection, we analyze the performance of the proposed Algorithm 1 in
terms of convergence behavior and computational complexity.

_Convergence Analysis:_ For ($\mathcal{Q}4$), each individual optimization
variable is convex when the rank-1 constraints are disregarded. As a result,
we can obtain the stationary point for each variable and the best solution for
the problem during the iterations. By utilizing Algorithm 1, we can ensure
that the objective values of problem ($\mathcal{Q}4$) with solution
$\mathbf{W}_{i}^{(t+1)}$ and $\mathbf{V}_{m}^{(t+1)}$ are always less than or
equal to the solution $\mathbf{W}_{i}^{(t)}$ and $\mathbf{V}_{m}^{(t)}$. This
guarantees the existence of a lower bound for the objective value, indicating
that Algorithm 1 will ultimately converge to a minimum value after a certain
number of iterations.

_Complexity Analysis:_ The problem ($\mathcal{Q}4$) solely consists of linear
matrix inequality (LMI) and second-order cone (SOC) constraints [33].
Consequently, the computational complexity of the proposed Algorithm 1 can be
analyzed by the standard interior-point method (IPM), where the complexity
consists of two components: the complexity per iteration and the complexity
introduced by iterations. Given a $\eta>0$, the number of iterations needed to
achieve a $\eta$-optimal solution to problem ($\mathcal{Q}4$) is approximately
proportional to $\zeta\cdot\ln(1/\eta)$, where $\zeta$ represents the barrier
parameter to measure the geometric complexity of the conic constraints. During
each iteration, a search direction is determined by dealing with a system of
$n$ linear equations with $n$ unknowns. Besides, the computation cost is
primarily influenced by the construction and factorization of the coefficient
matrix of the linear system. We can divide the computational complexity of
Algorithm 1 into two parts which are summarized in Table I. Consequently, the
overall complexity of Algorithm 1 is given by $\hat{c}_{1}+\hat{c}_{2}$, which
can obtain the sub-optimal solutions within polynomial time complexity. The
details of the computational complexity expressions are presented in Appendix.

Figure 2: Convergence performance of the proposed algorithm with different
required date rates. Figure 3: Total transmit power versus required data rates
with different numbers of near-shore users. Figure 4: Total transmit power
versus required data rates with different numbers of off-shore users.

##  IV Simulation Results

In this section, we provide extensive simulation results to demonstrate the
effectiveness of the proposed robust beamforming design algorithm for
integrated satellite-terrestrial maritime communications. The simulation
parameters are outlined in Table II.

TABLE II: Main Simulation Parameters For Integrated Satellite-Terrestrial Communication Systems Parameter | Value
---|---
Satellite orbit | LEO
Carrier frequency | 160 MHz
Bandwidth | 20 MHz
Number of TBS antennas | 8
Number of satellite antennas | 8
Number of near-shore users | 4
Number of off-shore users | 6
Satellite antenna gain | 55 dBi
TBS antenna gain | 30 dBi
Receive antenna gain | 20 dBi
Noise temperature | 300 K
Altitude of orbit | 550 km
Variance of AWGN | -110dBm
Boltzmann’s constant |  1.38$\times 10^{-23}$ J/m
Rain fading mean | -2.6dB
Rain fading variance | 1.63dB
3dB angle | $0.4^{\circ}$
Rician factor | 10
Required data rates of near-shore users | 0.1 bps/Hz
Required data rates of off-shore users | 0.1 bps/Hz
TBS total transmit power | 47dBm
Effective antennas height of TBS | 50m
Effective antennas height of users | 10m

To begin with, we examine the convergence behavior of the proposed Algorithm
1. To support the entertainment and communication needs for maritime
passengers, the minimum data rate required is from 1 to 3 Mbps with an ideal
data rate of 4 to 8 Mbps [26]. Therefore, our simulation results setting up
0.1 to 0.3 bps/Hz with the 20 MHz bandwidth can match the practical scenarios.
As shown in Fig. 2, the total transmit power of the TBS and the LEO satellite
gradually decreases and converges to a stable point within 15 iterations,
demonstrating the rapid convergence of the proposed Algorithm 1. In addition,
with the increment of the data rates required by users, the total transmit
power increases accordingly.

Figure 5: Total transmit power versus required data rates with different
numbers of TBS antennas. Figure 6: Total transmit power versus required data
rates with different numbers of LEO satellite antennas. Figure 7: Total
transmit power versus tidal factor $\delta_{1}$ with different required data
rates.

The number of users is also an important factor in the integrated satellite-
terrestrial maritime communications. It is shown in Fig. 3 and Fig. 4 that the
overall transmit power versus the required data rates with different numbers
of near-shore users and off-shore users. It can be seen that the total
transmit power rises as the number of users and required data rate increase.
This is because when the number of users and the required data rates increase,
there may be more interference between users. To satisfy the requirement of
data rate, it is necessary to increase the transmit power. In addition, it is
found that under the same required data rates, when the number of near-shore
users increases, the total transmit power is greater than that of off-shore
users. This is because near-shore users not only have interference between
neighboring users, but also suffer interference from satellite. Therefore,
when the number of users increases, near-shore users need more transmit power
to deal with interference.

Furthermore, we investigate the performance of total transmit power on the
number of antennas with different required data rates. In Fig. 5 and Fig. 6,
it is seen that the total transmit power decreases as the number of antennas
increase for all the required data rates, where we set $K_{1}=K_{2}=8$. This
is because more spatial degrees of freedom can be utilized when the number of
antennas increases. Besides, by optimizing the transmit beamforming of the TBS
and LEO satellite, it is possible to achieve greater array gains and make more
efficient use of the transmit power. In other words, the proposed Algorithm 1
leads to improve efficiency in utilizing the available transmit power.

Figure 8: Total transmit power versus tidal factor $\delta_{2}$ with different
required data rates. Figure 9: Outage probabilities of proposed and non-robust
algorithms versus required data rates with different values of tidal factor
$\delta_{1}$ and $\delta_{2}$. Figure 10: Comparison of total transmit power
with different values of required data rates between proposed algorithm and
other algorithms.

Fig. 7 and Fig. 8 show the total transmit power versus channel phase error for
different required data rate, where the required data rates are set as
$r=\gamma_{1}=\gamma_{2}$ and numbers of users are set as $K_{1}=K_{2}=8$. To
describe the CSI error resulting from wave fluctuations, we define a tidal
factor related to wave fluctuation severity, which is
$\delta_{1}=\varepsilon_{1}/\bar{\theta}_{1}$ and
$\delta_{2}=\varepsilon_{2}/\bar{\theta}_{2}$. We can find that the proposed
algorithm has high robustness, where the total transmit power performance is
good in different tidal factors. When the sea surface environment is extremely
harsh, wave fluctuations not only bring about changes in antenna angle, but
also cause changes in antenna height. In general, we can well resist the
influence of the antenna angle and ignore the influence of the antenna height
change. However, when the environment is extremely harsh, that is, when the
$\delta_{1}$ is high, the antenna height change cannot be ignored resulting in
the turn-up of simulation curve. Besides, the transmit power exhibits an
upward trend as the level of phase uncertainty and required data rates
increase, which is also consistent with theoretical analysis.

Next, we compare the outage performance of the non-robust algorithm and
proposed algorithm versus the required data rates with different values of
tidal factor $\delta_{1}$ and $\delta_{2}$. In the case of perfect CSI, the
beamforming vector obtained through the non-robust algorithm may fail to meet
the minimum data rate requirements of the users due to the CSI errors. The
probability of outage is defined as the probability that the minimum data
rates required by the users does not meet. The simulation result shows that
the outage probability of the proposed algorithm is zero in any case, which
proves the effectiveness and robustness of the proposed algorithm.

Finally, we compare the proposed algorithm and some baseline algorithms under
the same conditions. The LD algorithm [34] reformulates the worst-case
beamforming problem into a simplified optimization problem by using Lagrange
duality. Another WCPO algorithm [35] uses the worst-case performance
optimization to solve the power minimization problem. The algorithm of [31] is
designed for a small number of near-shore users. When it is applied to the
scenario considered in this paper, the number of users and the coverage area
increase, thereby more transmit power is required than our proposed algorithm.
It can be observed from the Fig. 10 that the transmit power of all algorithms
increases with the increment of the required data rates, which is also
consistent with the previous analysis. Furthermore, the proposed algorithm
demonstrates reduced transmit power in comparison to the other algorithms,
further substantiating the effectiveness of the proposed algorithm.

##  V Conclusion

In this paper, we provided an integrated satellite-terrestrial framework for
maritime communications with wide coverage. According to the electromagnetic
propagation characteristics in maritime communications, we modeled the
relationship between actual CSI and obtained CSI. With the obtained CSI, we
proposed a joint robust beamforming design algorithm. Both theoretical
analysis and simulation results confirmed that the proposed algorithm can
minimize the total transmit power compared to other existing algorithms, while
guaranteeing the minimum data rate requirements for the maritime users. In
addition, simulation results also demonstrate the robustness of the proposed
algorithm in the different tidal factors.

##  Appendix A Derivation of the computational complexity

The computational complexity of the Algorithm 1 can be divided into two parts,
the first part $\hat{c}_{1}$ can be expressed as

$\displaystyle\hat{c}_{1}=\sqrt{\beta_{1}}\cdot(C_{1}^{\text{form}}+C_{1}^{\text{fact}})\cdot\ln(1/\eta),$
(48)

where $\beta_{1}$ is the so-called barrier parameter, which can be computed as

$\displaystyle\beta_{1}=\underbrace{K_{1}\times(M_{1}+1)}_{\text{due to
(\ref{OP5st1})}}+\underbrace{K_{1}\times M_{1}}_{\text{due to
(\ref{OP5st5})}}+\underbrace{K_{1}\times 1}_{\text{due to
(\ref{OP5st1})}}+\underbrace{1}_{\text{due to (\ref{OP5st3})}},$ (49)

where $C_{1}^{\text{form}}$ is the cost of forming matrix, which can be
computed as

$\displaystyle C_{1}^{\text{form}}=$ $\displaystyle\underbrace{n_{1}\cdot
K_{1}(M_{1}+1)^{3}+n_{1}^{2}\cdot K_{1}(M_{1}+1)^{2}}_{\text{due to
(\ref{OP5st1})}}$ (50) $\displaystyle+\underbrace{n_{1}\cdot
K_{1}M_{1}^{3}+n_{1}^{2}\cdot K_{1}M_{1}^{2}}_{\text{due to
(\ref{OP5st5})}}+\underbrace{n_{1}\cdot K_{1}+n_{1}^{2}\cdot K_{1}}_{\text{due
to (\ref{OP5st3})}},$

where $C_{1}^{\text{fact}}$ is the cost of factorizing matrix, which can be
computed as

$\displaystyle C_{1}^{\text{fact}}=n_{1}^{3}.$ (51)

Thus, the first part of complexity $\hat{c}_{1}$ with
$n_{1}=\mathcal{O}(K_{1}M_{1}^{2})$ can be computed as

$\displaystyle\hat{c}_{1}=$ $\displaystyle\sqrt{2K_{1}(M_{1}+1)}\cdot
n_{1}\cdot\\{K_{1}[(M_{1}+1)^{3}+M_{1}^{3}+1]$ (52)
$\displaystyle+K_{1}n_{1}[(M_{1}+1)^{2}+M_{1}^{2}+1]+n_{1}^{2}\\},$

similarly, the second part of complexity $\hat{c}_{2}$ with
$n_{2}=\mathcal{O}(K_{2}M_{2}^{2})$ can be computed as

$\displaystyle\hat{c}_{2}=$ $\displaystyle\sqrt{K_{2}(2M_{2}+3)}\cdot
n_{2}\cdot\\{K_{2}[(M_{2}+1)^{3}+M_{2}^{3}+2]$ (53)
$\displaystyle+K_{2}n_{2}[(M_{2}+1)^{2}+M_{2}^{2}+2]+n_{2}^{2}\\}.$

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