Interference Situational Aware Beam Pointing Optimization for Dense LEO Satellite Communication System

Abstract

Recently, the low earth orbit (LEO) mega-constellation faces serious time-varying interferences due to spectrum sharing, dense deployment, and high mobility. Therefore, it is important to study the interference avoidance techniques for the dense LEO satellite system. In this paper, the interference situational aware beam pointing optimization technique is proposed. Firstly, the angle of departure (AoD) and angle of arrival (AoA) of the interfering links are obtained to represent the time-varying interference. Then, the interference avoidance problem for dense LEO satellite systems is modeled as a non-convex optimization problem, and a particle swarm optimization (PSO) based method is proposed to obtain the optimal beam pointing of the user terminal (UT). Simulations show that the relative error of the mean signal-to-interference plus noise ratio (SINR) obtained by the proposed method is $0.51$%, so the co-channel interference can be effectively mitigated for the dense LEO satellite communication system.

1. Introduction

Recently, the low earth orbit (LEO) satellite constellation has received much attention for its low transmission delay, small path loss, large bandwidth, and global seamless coverage ability [1,2]. However, the available spectrum for the emerging LEO satellite constellation is scarce. The declared frequency bands for the typical mega-constellation satellites, such as Starlink, OneWeb, and Telesat, are mainly located in Ku-band, Ka-band, and Q/V-band [3]. Moreover, the same frequency bands available for LEO satellites will also be used by the existing geostationary earth orbit (GEO) satellite systems and terrestrial mobile systems. Thus, severe intra-system and inter-system co-channel interference will occur when the LEO satellite constellations are deployed, and interference mitigation techniques are essential to guarantee the performance of these systems. The existing interference suppression schemes can be divided into several types, including spatial isolation (SI), cognitive radio (CR), adaptive power control (APC), beam hopping (BH), beam pointing optimization (BPC), etc.

SI refers to setting the exclusion zone for the interfering system. Then, the interfering system operating outside the exclusion zone can avoid harmful interference [4]. The fixed exclusive angle is designed for the GEO satellite earth station to reduce the in-line interference from the LEO system [5,6]. However, the fixed exclusion zone is not suitable for the non-geostationary earth orbit (NGEO) satellite with high mobility. Thus, a dynamic protection area is defined to mitigate the harmful interference between GEO–NGEO [7] or NGEO–NGEO satellite systems [8]. However, the disadvantage of SI is that the protection area for the interfered satellite is the non-communication zone for the interfering satellite.

CR is also an effective technique to mitigate co-sharing spectrum interference [9]. It can sense the available frequency band in the spectrum for use, which can improve spectrum efficiency and reduce the mutual interference between systems [10,11,12]. Zhang et al. have proposed a spectrum strategy to distinguish the GEO signal from the interfering NGEO and noise [13], and the GEO satellite can use the appropriate power to protect its normal communication. In [14], a spectrum-sensing and power allocation-aided spectrum-sharing method is proposed to ensure the LEO system can work simultaneously with GEO systems in the interference region. However, CR cannot guarantee the long-term performance of the secondary systems.

APC has received great attention as a solution to reduce interference between satellite systems. In [15], an adaptive power control technique is proposed to mitigate the in-line interference for the GEO–NGEO coexistence scenario. In [16], a dynamic beam power adjustment strategy is proposed to avoid the in-line interference between the GEO–LEO satellite systems. APC can adapt the transmit power of the interfering system to satisfy the desired quality of service while protecting the interfered system. However, the co-channel interference is severe and dynamic for the LEO mega-constellation, and the power balance among many satellites is difficult.

BH can realize full frequency reuse over a certain beam hopping pattern and suppress the co-channel interference. In [17], a beam hopping strategy is proposed to match uneven traffic demands and avoid interference to GEO ground stations. Moreover, the multi-satellite interference avoidance problem for NGEO satellite communication systems is solved by designing BH patterns with spatial isolation characteristics [18]. Thus, a suitable BH pattern is designed for dynamically allocating resources and resisting interference. However, the complexity of BH pattern design will rapidly increase as the number of LEO satellites increases.

BPO can be used to adjust the antenna beam pointing direction to mitigate interference. In [8], the harmful interference can be mitigated by turning off the beam or adjusting the satellite beam pointing. In [19], the in-line interference between the LEO and GEO system can be mitigated by tilting the normal direction of the phased array antennas of LEO satellites. In [20], the progressive pitch method and the coverage-expanding method are proposed to reduce interference, but the interference can be only partially solved for the high latitudes [21]. Moreover, the adjustment of the satellite antenna beam pointing of the LEO mega-constellation means a huge computational overhead, which is difficult to achieve with the limited computing resources on the satellite. If the calculation of the optimal beam pointing is completed by the ground control centers with stronger computing capabilities, the time-varying antenna adjusting information should be sent back to the satellites frequently, which leads to an additional delay and signaling overhead.

However, the co-channel interference for the dense LEO satellite systems will be more complicated due to the overlapped coverage and time-varying interfering links. The above interference mitigation techniques cannot be directly applied to the dense LEO satellite systems. Thus, a distributed beam pointing optimization method based on interference situational awareness is proposed in this paper. Firstly, the interference situational database composed of the angle of departure (AoD) and the angle of arrival (AoA) of the time-varying interfering links can be collected at each UT. Then, the optimal beam pointing at each UT can be obtained by PSO optimization. The main contributions of this paper are summarized as follows:

• With the time-varying interfering satellite set and minimum elevation constraint, the AoD and AoA of communication and interfering links can be calculated. Then, the AoD and AoA of interfering links are stored to construct the interference situational aware database, which is used for further beam pointing optimization;
• With the interference situational aware database, the optimal beam pointing can be modeled as a non-convex optimization problem, and a distributed method based on particle swarm optimization (PSO) is proposed to maximize the signal-to-interference plus noise ratio (SINR) of each UT;
• The performance of the proposed interference situational aware beam pointing optimization technology is verified by the simulation results.

The rest of this paper is organized as follows: in Section 2, the system model and problem formulation are described. In Section 3, the scheme of interference situational aware beam pointing optimization is analyzed. In Section 4, simulation results and complexity analysis are presented. In Section 5, conclusions are drawn. Matrices and vectors are denoted by bold letters. $\overrightarrow{AB}$ denotes the vector that goes from A to B. $|a|$ is the magnitude of a. $\hat{a}$ is the estimation of a. The major variables adopted in the paper are listed in

Table 1. Symbol list.

Notation	Definition
$G_t^{\mathcal{S}},G_r^{\mathcal{S}}$	Antenna gain of transmitter and receiver
${\theta }_t^{\mathcal{S}},{\phi }_t^{\mathcal{S}}$	Elevation and azimuth angles of AoD
${\overline{\theta }}_t^{\mathcal{S}},{\overline{\phi }}_t^{\mathcal{S}}$	Beam center pointing of the satellite antenna
${\theta }_r^{\mathcal{S}},{\phi }_r^{\mathcal{S}}$	Elevation and azimuth angles of AoA
${\overline{\mathit{\vartheta }}}_r^{C_U}=[{\overline{\theta }}_r^{C_U},{\overline{\phi }}_r^{C_U}]$	Beam center pointing of UT antenna
${\theta }_r^S,{\phi }_r^S$	Elevation and azimuth angles of AoA of communication link
${\Theta }_{3dB},{\Phi }_{3dB}$	Horizontal and vertical 3 dB beamwidth of the UT antenna
$N_{tx},N_{ty}$	Number of arrays placed along x-axis and y-axis of the satellite antenna
$d_{tx},d_{ty}$	Separation of two array elements on x-axis and y-axis of the satellite antenna
$N_{rx},N_{ry}$	Number of array elements placed along x-axis and y-axis of the UT antenna
$d_{rx},d_{ry}$	Separation of two array elements on x-axis and y-axis of the UT antenna
$[{\theta }_t^S,{\phi }_t^S],[{\theta }_r^S,{\phi }_r^S]$	AoD and AoA of communication link
$[{\theta }_t^{I_{q,d}},{\phi }_t^{I_{q,d}}],[{\theta }_r^{I_{q,d}},{\phi }_r^{I_{q,d}}]$	AoD and AoA of dth interfering link from qth interfering satellite
$[{\overline{\theta }}_t^S,{\overline{\phi }}_t^S]$	Beam center pointing of serving satellite antenna
${\widehat{\mathit{\vartheta }}}_r^{C_U}=[{\widehat{\theta }}_r^{C_U},{\widehat{\phi }}_r^{C_U}]$	Optimized beam center pointing of UT antenna

.

2. System Model and Problem Formulation

2.1. System Configuration

In this paper, we consider the dense LEO satellite communication system composed of many LEO satellites with different orbital planes, which leads to overlapped coverage areas. Both satellites and UTs are equipped with uniform rectangular arrays (URAs). Every satellite can generate several fixed pointing circular beams to provide service for ground users, as shown in Figure 1. These satellite beams move with the motion of the satellite. The UTs are generated randomly, and each UT has one flexible circular beam to realize beam alignment. Thus, the existence time of the overlapping coverage area for fixed-position UTs is limited; moreover, the full frequency reuse is considered among all satellites. Generally, the satellite ephemeris broadcasted periodically can be utilized to obtain the satellite location. The time-varying beam center position can also be obtained via its relative position to the location of the nadir of the satellite, as shown in Figure 1.

2.2. Antenna Model

In this paper, planar URA is employed for the satellite and the UT. According to 3GPP TR 37.840 [22], the LEO satellite and UT antenna pattern can be derived. The composite array radiation pattern of the satellite is expressed as follows:

$$\begin{array}{c}\hfill G_t^{\mathcal{S}}=G_E({\theta }_t^{\mathcal{S}},{\phi }_t^{\mathcal{S}})+10{log}_{10}(|{\tilde{W}}_t({\theta }_t^{\mathcal{S}},{\phi }_t^{\mathcal{S}},{\overline{\theta }}_t^{\mathcal{S}},{\overline{\phi }}_t^{\mathcal{S}}){|}^2),\end{array}$$

(1)

where $G_t^{\mathcal{S}}$ is the satellite antenna gain at the transmitter, which is composed of single element gain $G_E({\theta }_t^{\mathcal{S}},{\phi }_t^{\mathcal{S}})$ and array gain $10{log}_{10}(|{\tilde{W}}_t({\theta }_t^{\mathcal{S}},{\phi }_t^{\mathcal{S}},{\overline{\theta }}_t^{\mathcal{S}},{\overline{\phi }}_t^{\mathcal{S}}){|}^2)$, ${\theta }_t^{\mathcal{S}}$ and ${\phi }_t^{\mathcal{S}}$ are the elevation and azimuth angle of AoD, ${\overline{\theta }}_t^{\mathcal{S}}$ and ${\overline{\phi }}_t^{\mathcal{S}}$ denote the electrical down-tilt steering and the electrical horizontal steering, which represent the beam center pointing of the satellite antenna.

Similarly, the antenna model of the UT is as follows:

$$\begin{array}{c}\hfill G_r^{\mathcal{S}}=G_E({\theta }_r^{\mathcal{S}},{\phi }_r^{\mathcal{S}})+10{log}_{10}(|{\tilde{W}}_r({\theta }_r^{\mathcal{S}},{\phi }_r^{\mathcal{S}},{\overline{\theta }}_r^{C_U},{\overline{\phi }}_r^{C_U}){|}^2),\end{array}$$

(2)

where $G_r^{\mathcal{S}}$ is the UT antenna receiving gain, ${\theta }_r^{\mathcal{S}}$ and ${\phi }_r^{\mathcal{S}}$ are the elevation and azimuth angle of AoA, ${\overline{\theta }}_r^{C_U}$ and ${\overline{\phi }}_r^{C_U}$ express the electrical down-tilt steering and the electrical horizontal steering, which denote the beam center pointing of UT antenna.

2.3. Signal Model

As shown in Figure 1, when multiple satellites share the same frequency bands, the UT will receive communication and interfering signals simultaneously. Moreover, satellite-to-ground communication undergoes several stages of propagation and attenuation. Free space path loss, atmospheric absorption, and rain attenuation are mainly considered in path loss (PL). Then, PL can be given as follows:

$$P{L}^{\mathcal{S}}=P{L}_b(d^{\mathcal{S}},f_c)+P{L}_g(e{l}^{\mathcal{S}},f_c)+P{L}_r\left(e{l}^{\mathcal{S}}\right).$$

(3)

where $P{L}_b(d^{\mathcal{S}},f_c)=32.45+20{log}_{10}\left(f_c\right)+20{log}_{10}\left(d^{\mathcal{S}}\right)$ is the free space path loss [23], $P{L}_g(e{l}^{\mathcal{S}},f_c)=\frac{A_{\mathrm{zenith}}\left(f_c\right)}{sin\left(e{l}^{\mathcal{S}}\right)}$ is the atmospheric absorption [23]. Specifically, $d^{\mathcal{S}}$ is the distance between the satellite and the UT, $f_c$ is the carrier frequency, $A_{\mathrm{zenith}}\left(f_c\right)$ is the zenith attenuation, and $e{l}^{\mathcal{S}}$ is the elevation angle. Moreover, rain attenuation is the most dominant cause of signal degradation for the Ka-band, and it can be represented as follows: $P{L}_r\left(e{l}^{\mathcal{S}}\right)=A_{0.01}{\frac{p}{0.01}}^{-(0.655+0.033ln\left(p\right)-0.045ln\left(A_{0.01}\right)-\beta (1-p)sin\left(e{l}^{\mathcal{S}}\right))}$ [24], in which $A_{0.01}$ is the predicted attenuation exceeded for $0.01\%$ of an average year, p is the exceedance probability, $e{l}^{\mathcal{S}}$ is the elevation angle, and $\beta$ is given as follows:

$$\begin{array}{c}\hfill \beta =\left\{\begin{array}{cc}0,& p>1\%\mathrm{or}|la{t}_U|\ge {36}^{\circ }\\ -0.005\left(|la{t}_U|-36\right),& p<1\%\mathrm{and}|la{t}_S|<{36}^{\circ }\mathrm{and}e{l}^{\mathcal{S}}\ge {25}^{\circ }\\ -0.005\left(|la{t}_U|-36\right)+1.8-4.25sin\left(e{l}^{\mathcal{S}}\right),& \mathrm{other}\end{array}\right.\end{array}$$

(4)

and $la{t}_U$ is the latitude of the UT. The above rain attenuation model has been used in the existing research [25,26]. Moreover, the effect of rain attenuation during radio propagation and the new rain attenuation model have been studied in [27,28,29,30,31]; however, this is not the focus of this paper.

The received signal strength is given as follows:

$$\begin{array}{c}\hfill P^{\mathcal{S}}=p_t+G_t^{\mathcal{S}}+G_r^{\mathcal{S}}-{PL}^{\mathcal{S}},\end{array}$$

(5)

where the subscripts t and r denote the transmitter and receiver. $\mathcal{S}$ is the set of visible satellites, which is composed of the serving satellite S and the interfering satellites I. $p_t$ denotes the satellite antenna transmit power. $G_t^{\mathcal{S}}$ and $G_r^{\mathcal{S}}$ are the antenna transmitting and receiving gain according to (1) and (2). Moreover, the received signal strength of serving signal and interfering signal are obtained by (5), and can be represented as $P^S$ and $P^I$.

Thus, the SINR at the receiver is written as follows:

$$\begin{array}{c}\hfill \mathrm{SINR}\left(\mathrm{n}\right)=-10{log}_{10}({10}^{-0.1(P^S\left(n\right)-KT{B}_w)}+{10}^{-0.1(P^S\left(n\right)-P^I\left(n\right))}),\end{array}$$

(6)

where $P^I={\sum }_{q=1}^Q{\sum }_{d=1}^D{P}^{I_{q,d}}$, Q is the total number of interfering satellites at the time index n, D is the total number of interfering links generated by an interfering satellite, which is equal to the number of beams that use the same frequency bands. $I_{q,d}$ denotes the dth interfering link of the qth interfering satellite. K is the Boltzmann’s constant and, given as $-228.6$ dBW/K/Hz, T represents the noise temperature, and $B_w$ means the bandwidth of the LEO satellite communication system.

2.4. Problem Formulation

In Figure 1, the beam, colored black, points directly to its serving satellite without considering the co-channel interference generated by other satellites. Actually, the beam pointing of the UT antenna needs to be shifted to other directions for interference mitigation, represented as the green beam in Figure 1. Moreover, interference avoidance for the time-varying dense LEO networks is challenging.

To solve the time-varying interference problem, we will decompose the continuous time-varying interference situation into multiple discrete time slots. Then, we intend to maximize the SINR of the communication link at the receiver to obtain the optimal beam pointing of the receiving antenna at every moment, and the problem is formulated as follows:

$$\begin{array}{cc}\hfill & \underset{\begin{array}{c}{\overline{\theta }}_r^{C_U},{\overline{\phi }}_r^{C_U}\end{array}}{max}\left\{\mathrm{SINR}\left(\mathrm{n}\right)\right\},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill s.t.& {\theta }_r^S\left(n\right)-{\Theta }_{3\mathrm{dB}}\le {\overline{\theta }}_r^{C_U}\left(n\right)\le {\theta }_r^S\left(n\right)+{\Theta }_{3\mathrm{dB}},\hfill \end{array}$$

(7a)

$$\begin{array}{cc}\hfill & {\phi }_r^S\left(n\right)-{\Phi }_{3\mathrm{dB}}\le {\overline{\phi }}_r^{C_U}\left(n\right)\le {\phi }_r^S\left(n\right)+{\Phi }_{3\mathrm{dB}},\hfill \end{array}$$

(7b)

where ${\Theta }_{3\mathrm{dB}}$ and ${\Phi }_{3\mathrm{dB}}$ are the horizontal and vertical 3 dB beamwidth of the main beam of the UT antenna, ${\theta }_r^S$ and ${\phi }_r^S$ are the elevation and azimuth angles of AoA of the communication link. (7a) and (7b) represent the range of the UT antenna beam center adjustment. The above constraints can help to reduce the beam search space.

3. Interference Situational Aware Beam Pointing Optimization

The beam pointing optimization problem in (7) is a non-convex optimization problem, and the non-convexity is introduced by SINR in (6) [32]. Generally, the non-convex problem is challenging to be solved directly. However, as a heuristic optimization method, PSO can solve convex and non-convex optimization problems by finding the optimal solution in the search space through a collaborative particle swarm optimization and a fitness function [33]. In this section, the scheme of an interference situational aware beam pointing optimization is introduced, as shown in Figure 2. Firstly, the satellite position data are obtained. Based on the coordinate system conversion, the database of AoD and AoA is constructed. Then, the database can be utilized for PSO to optimize the beam center pointing of the UT antenna by maximizing the SINR.

3.1. Interference Situational Awareness

For a given UT, the visible satellites change dynamically, as shown in Figure 3. Thus, the visible satellite set is as follows:

$$\mathcal{V}=\{\mathcal{S}|e{l}^{\mathcal{S}}\ge e{l}_{th}\}.$$

(8)

where the elevation angle of satellite $e{l}^{\mathcal{S}}$ observed by UT can be calculated using the location of satellite and UT, $e{l}_{th}$ is the minimum visibility elevation angle.

According to the visible satellite set $\mathcal{V}$, the serving satellite S is selected based on the maximum remaining service time, while the other satellites are interfering satellites I. Due to satellite movement, the interfering satellites observed by the UT are time-varying. The time-varying interfering satellite set at the time index n is expressed as $I\left(n\right)=\{I_1\left(n\right),I_2\left(n\right),\dots ,I_Q\left(n\right)\}$, where Q is the total number of interfering satellites and varies with time.

To obtain the AoA and AoD of communication and interfering links, coordinate conversion is implemented. Firstly, local coordinate systems (LCSs) of the satellite and UT are defined. Then, the location information of the satellite and UTs in the global coordinate system (GCS) represented by the Earth-centered Earth-fixed (ECEF) coordinate system should be converted to the LCS for each satellite and UT. The detailed conversion steps are provided in 3GPP TR 38.901 [34].

As shown in Figure 4, S, I, and U denote the serving satellite, interfering satellite, and UT. $O\text{-}XYZ$ is the GCS, $O\text{-}{X}_S{Y}_S{Z}_S$ is the local coordinate system of serving satellite (LCS−S), $O\text{-}{X}_I{Y}_I{Z}_I$ is the local coordinate system of interfering satellite (LCS−I), and $O\text{-}{X}_U{Y}_U{Z}_U$ denotes the local coordinate system of the UT (LCS−U). Additionally, $C_S$ and $C_I$ denote the beam center of the serving and interfering satellite, respectively. $C_U$ is the beam center of the UT antenna.

Figure 5 and Figure 6 show the serving satellite and UT antenna placed in LCS−S and LCS−U, respectively. The total number of antenna elements of the satellite and UT are $N_t=N_{tx}\times N_{ty}$ and $N_r=N_{rx}\times N_{ry}$. $N_{tx}$ and $N_{ty}$ are the number of antenna elements along x-axis and y-axis at the transmitter, while $N_{rx}$ and $N_{ry}$ are the numbers of antenna elements along x-axis and y-axis at the receiver. $d_{tx}$, $d_{ty}$, $d_{rx}$, and $d_{ry}$ are the distance between adjacent antenna elements along x-axis and y-axis at the transmitting and receiving antenna. ${\phi }_t^S$ and ${\theta }_t^S$ are defined as the azimuth and elevation angle of AoD of the communication link, while ${\phi }_r^S$ and ${\theta }_r^S$ are the azimuth and elevation angle of AoA of the communication link.

The elevation and azimuth angles of AoD and AoA of the communication link can be estimated by the following:

$$\begin{array}{cc}\hfill & {\theta }_t^S=arccos\left(\frac{\overrightarrow{S^{\text{LCS-S}}{U}^{\text{LCS-s}}}·\overrightarrow{S^{\text{LCS-S}}{X}_S^{\prime }}}{\sqrt{|\overrightarrow{S^{\text{LCS-S}}{U}^{\text{LCS-S}}}|·|\overrightarrow{S^{\text{LCS-S}}{X}_S^{\prime }}|}}\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\phi }_t^S=arccos\left(\frac{\overrightarrow{S^{\text{LCS-S}}{U}_{Y_{S}O{Z}_S}^{\text{LCS-S}}}·(-\overrightarrow{S^{\text{LCS-S}}{Z}_S})}{\sqrt{|\overrightarrow{S^{\text{LCS-S}}{U}_{Y_{S}O{Z}_S}^{\text{LCS-S}}}|·|(-\overrightarrow{S^{\text{LCS-S}}{Z}_S})|}}\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & {\theta }_r^S=arccos\left(\frac{\overrightarrow{U^{\text{LCS-U}}{S}^{\text{LCS-U}}}·\overrightarrow{U^{\text{LCS-U}}{Y}_U^{\prime }}}{\sqrt{|\overrightarrow{U^{\text{LCS-U}}{S}^{\text{LCS-U}}}|·|\overrightarrow{U^{\text{LCS-U}}{Y}_U^{\prime }}|}}\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & {\phi }_r^S=arccos\left(\frac{\overrightarrow{U^{\text{LCS-U}}{S}_{X_{U}O{Z}_U}^{\text{LCS-U}}}·\overrightarrow{U^{\text{LCS-U}}{Z}_U}}{\sqrt{|\overrightarrow{U^{\text{LCS-U}}{S}_{X_{U}O{Z}_U}^{\text{LCS-U}}}|·|\overrightarrow{U^{\text{LCS-U}}{Z}_U}|}}\right),\hfill \end{array}$$

(12)

where $\overrightarrow{S^{\text{LCS-S}}{X}_S^{\prime }}$ and $\overrightarrow{S^{\text{LCS-S}}{Z}_S}$ are parallel to $\overrightarrow{O{X}_S}$ and $\overrightarrow{O{Z}_S}$ in the LCS-S. $\overrightarrow{U^{\text{LCS-U}}{Y}_U^{\prime }}$ and $\overrightarrow{U^{\text{LCS-U}}{Z}_U}$ are parallel to $\overrightarrow{O{Y}_U}$ and $\overrightarrow{O{Z}_U}$ in the LCS-U. x-axis, y-axis, and z-axis can be represented by the three unit vectors $\overrightarrow{{\mathit{e}}_x}=(1,0,0)$, $\overrightarrow{{\mathit{e}}_y}=(0,1,0)$, and $\overrightarrow{{\mathit{e}}_z}=(0,0,1)$. Moreover, $U_{Y_{S}O{Z}_S}^{\text{LCS-S}}$ is the projection of the UT U onto the $Y_{S}O{Z}_S$-plane in the LCS-S. $S_{X_{U}O{Z}_U}^{\text{LCS-U}}$ is the projection of satellite S onto the $X_{U}O{Z}_U$-plane in the LCS-U.

Moreover, the beam center pointing of the satellite antenna is given by the following:

$$\begin{array}{cc}\hfill & {\overline{\theta }}_t^S=arccos\left(\frac{\overrightarrow{S^{\text{LCS-S}}{C_S}^{\text{LCS-S}}}·\overrightarrow{S^{\text{LCS-S}}{X}_S^{\prime }}}{\sqrt{|\overrightarrow{S^{\text{LCS-S}}{C_S}^{\text{LCS-S}}}|·|\overrightarrow{S^{\text{LCS-S}}{X}_S^{\prime }}|}}\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\overline{\phi }}_t^S=arccos\left(\frac{\overrightarrow{S^{\text{LCS-S}}{C}_{Y_{S}O{Z}_S}^{\text{LCS-S}}}·(-\overrightarrow{S^{\text{LCS-S}}{Z}_S})}{\sqrt{|\overrightarrow{S^{\text{LCS-S}}{C}_{Y_{S}O{Z}_S}^{\text{LCS-S}}}|·|(-\overrightarrow{S^{\text{LCS-S}}{Z}_S})|}}\right),\hfill \end{array}$$

(14)

Similarly, we can replace the serving satellite in the above derivation with interfering satellites. Thus, the AoD and AoA of the interfering links can also be estimated. Then, the interference situational aware database can be constructed, and the time-varying interference can be obtained as follows:

$$\begin{array}{cc}\hfill & {\mathcal{D}}_t^I\left(n\right)=\{[{\theta }_t^{I_{q,d}}\left(n\right),{\phi }_t^{I_{q,d}}\left(n\right)]|q\in \mathcal{Q},d\in \mathcal{N}\},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\mathcal{D}}_r^I\left(n\right)=\{[{\theta }_r^{I_{q,d}}\left(n\right),{\phi }_r^{I_{q,d}}\left(n\right)]|q\in \mathcal{Q},d\in \mathcal{N}\},\hfill \end{array}$$

(16)

where $\mathcal{Q}=\{1,\cdots ,Q\}$,$\mathcal{N}=\{1,\cdots ,N_b\}$.

3.2. Beam Pointing Optimization

Traditionally, the beam center of the UT antenna should point to the communication link AoA. As shown in Figure 7, the beam colored as black points to its serving satellite with ${\mathit{\vartheta }}_r^{C_U}\left(n\right)=[{\overline{\theta }}_r^{C_U}\left(n\right),{\overline{\phi }}_r^{C_U}\left(n\right)]=[{\theta }_r^S\left(n\right),{\phi }_r^S\left(n\right)]$). However, co-channel interference occurs inevitably due to the same frequency reuse among satellites. Therefore, the beam center pointing of the UT needs to be optimized for interference mitigation. In Figure 7, the green beam direction denotes the optimized beam center pointing of the UT antenna (i.e., ${\widehat{\mathit{\vartheta }}}_r^{C_U}\left(n\right))=[{\widehat{\theta }}_r^{C_U}\left(n\right),{\widehat{\phi }}_r^{C_U}\left(n\right)]$).

During the service time of the serving satellite, the interfering satellites vary dynamically. To characterize the time-varying interference and communication links, the AoA and AoD of interfering and communication links have been divided into discrete status for each time slot in Section 3.1. Thus, the interference is fixed for each time slot. We intend to use the PSO algorithm to maximize the SINR at each time slot to obtain the corresponding optimal beam center pointing.

We take the beam pointing optimization process at time index n as an example. The interference situation at the single time slot can be obtained via (15) and (16). Then, the PSO optimization method can be adopted to achieve the optimal beam pointing of each UT. The advantage and detailed description of the PSO algorithm can be found in [33,35,36]. In this paper, we explain how to implement the PSO algorithm to solve the problem defined in (7). Let the number of the particle swarm of PSO optimization be J, and the maximum iteration is M. The upper and lower bounds of the particle swarm two-dimensional search space are defined as follows: $L_{\theta ,max}={\theta }_r^S\left(n\right)+{\Theta }_{3\mathrm{dB}}$, $L_{\phi ,max}={\phi }_r^S\left(n\right)+{\Phi }_{3\mathrm{dB}}$, $L_{\theta ,min}={\theta }_r^S\left(n\right)-{\Theta }_{3\mathrm{dB}}$, and $L_{\phi ,min}={\phi }_r^S\left(n\right)-{\Phi }_{3\mathrm{dB}}$, where ${\theta }_r^S\left(n\right)$ and ${\phi }_r^S\left(n\right)$ are the communication link AoA estimated by (11) and (12), ${\Theta }_{3\mathrm{dB}}$ and ${\Phi }_{3\mathrm{dB}}$ are the horizontal and vertical 3 dB beamwidth of the UT antenna.

Specifically, the position and velocity of the jth particle are represented as follows:

$$\begin{array}{cc}\hfill & l_j(n,m)=({\overline{\theta }}_{r_j}^{C_U}(n,m),{\overline{\phi }}_{r_j}^{C_U}(n,m)),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & v_j(n,m)=(v_{{\overline{\theta }}_{r_j}}^{C_U}(n,m),v_{{\overline{\phi }}_{r_j}}^{C_U}(n,m)),\hfill \end{array}$$

(18)

where $(n,m)$ denotes the mth iteration at the time index n, $j=1,2,\dots ,J$, and $m=1,2,\dots ,M$. Particle position $l_j$ represents the elevation and azimuth angles of beam center pointing of the UT, ${\overline{\theta }}_{r_j}^{C_U}$ and ${\overline{\phi }}_{r_j}^{C_U}$ are restricted in $[L_{\theta ,min},L_{\theta ,max}]$ and $[L_{\phi ,min},L_{\phi ,max}]$, $v_j$ denotes the particle velocity and is limited between $v_{min}$ and $v_{max}$.

The fitness function is as follows:

$$f_j(n,m)={\mathrm{SINR}}_j(n,m).$$

(19)

Based on the fitness function, the individual and group position optimal value of the particle swarm at the mth iteration are expressed as follows: $p_{\mathrm{best}}(n,m)$ and $g_{\mathrm{best}}(n,m)$. Then, with the guidance of $p_{\mathrm{best}}(n,m)$ and $g_{\mathrm{best}}(n,m)$, the velocity and position of the jth particle at $(m+1)$th iteration can be updated by the following:

$$\begin{array}{cc}\hfill v_j(n,m+1)& =w(n,m)v_j(n,m)+c_1{r}_1(p_{\mathrm{best}}(n,m)-l_j(n,m))\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +c_2{r}_2(g_{\mathrm{best}}(n,m)-l_j(n,m)),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill l_j(n,m+1)& =l_j(n,m)+v_j(n,m+1),\hfill \end{array}$$

(21)

where $c_1$ and $c_2$ are learning factors, and $r_1$ and $r_2$ are random numbers in $[0,1]$ to increase search randomness. The inertia weight will affect the performance and convergence speed of the algorithm [36]. In this paper, the linear inertia factor $w(n,m)$ is adopted as follows [33]:

$$w(n,m)=w_{min}+(w_{max}-w_{min})\frac{M-m}{M}$$

(22)

where $w_{max}$ and $w_{min}$ are the upper and lower bounds of the inertia weight.

The above process will be repeated at the time index n until the maximum iteration M is reached. The corresponding optimized result ${\widehat{\mathit{\vartheta }}}_r^{C_U}\left(n\right)=[{\widehat{\theta }}_r^{C_U}\left(n\right),{\widehat{\phi }}_r^{C_U}\left(n\right)]=p_{\mathrm{best}}(n,M)$. Thus, the optimal result of the whole observation time N can be obtained by applying the above optimization process at each time slot, and it can be expressed as follows: ${\mathsf{\Psi }}_r^{C_U}=\{{\widehat{\mathit{\vartheta }}}_r^{C_U}\left(1\right),{\widehat{\mathit{\vartheta }}}_r^{C_U}\left(2\right),\dots ,{\widehat{\mathit{\vartheta }}}_r^{C_U}\left(N\right)\}$. The detailed process for the PSO optimization process can be found in Algorithm 1.

Algorithm 1 Beam pointing optimization based on PSO

Input:  Maximum iteration M, number of the particle swarm J, particle position $l_j$, particle velocity $v_j$, AoD and AoA of communication link $[{\theta }_t^S,{\phi }_t^S]$ and $[{\theta }_r^S,{\phi }_r^S]$, data set of AoD and AoA of interfering links ${\mathcal{D}}_t^I\left(n\right)$ and ${\mathcal{D}}_r^I\left(n\right)$, search space boundary $L_{\theta ,min}$, $L_{\theta ,max}$, $L_{\phi ,min}$ and $L_{\phi ,max}$, velocity limitation $v_{min}$ and $v_{max}$. Output:  Optimal beam center pointing ${\widehat{\mathit{\vartheta }}}_r^{C_U}\left(n\right)$; 1: Initialize $m=1$; 2: Randomly generate particle position $l_j(n,m)$ and velocity $v_j(n,m)$; 3: Calculate the particle fitness $f_j(n,m)$; 4: Find $f_a(n,m)=max\{f_1(n,m),\cdots ,$$f_J(n,m)\}$, $a\in \{1,2,\cdots ,J\}$; 5: Set $p_{\mathrm{best}}(n,m)=l_a(n,m)$; 6: Set $g_{\mathrm{best}}(n,m)=\{l_1(n,m),\cdots ,l_J(n,m)\}$; 7: repeat   $m=m+1$ 8:     Update the particle position and velocity $v_j(n,m+1)$ and $l_j(n,m+1)$ via (20) and (22); 9:     Calculate the particle fitness $f_j(n,m+1)$; 10:     Find $f_a(n,m+1)=max\{f_1(n,m+1),\cdots ,$$f_J(n,m+1)\}$, $a\in \{1,2,\cdots ,J\}$; 11:     if $f_a(n,m+1)$ > $f_a(n,m)$ then 12:         $p_{\mathrm{best}}(n,m+1)=l_a(n,m+1)$; 13:     else 14:         $p_{\mathrm{best}}(n,m+1)=p_{\mathrm{best}}(n,m)$; 15:     end if 16:     if $f_j(n,m+1)$ > $f_a(n,m)$ then 17:         $g_{{\mathrm{best}}_j}(n,m+1)=l_j(n,m+1)$; 18:     else 19:         $g_{{\mathrm{best}}_j}(n,m+1)=l_j(n,m)$; 20:     end if 21: until (m > Maximum iteration M) 22: Return ${\widehat{\mathit{\vartheta }}}_r^{C_U}\left(n\right)=p_{\mathrm{best}}(n,M)$;

4. Results

In this section, we consider the dense LEO satellite communication scenario consisting of 6372 satellites in the Ka-band and multiple UTs. The orbital parameters of the LEO constellation are listed in

Table 2. Orbital parameters [37].

Altitude	Inclination	Planes	Satellites per Plane
1200 km	87.9°	36	49
1200 km	55°	32	72
1200 km	40°	32	72

[37]. The positions of multiple users are randomly generated in a selected ground area, and the number of users is 16. Moreover, the detailed simulation parameters are shown in

Table 3. Simulation parameters.

Scenario Parameters	Symbol	Value
Earth radius	$R_e$	6378 km
Satellite orbital radius	R	7578 km
Satellite orbital altitude	h	1200 km
Carrier frequency	$f_c$	20 GHz [39]
Bandwidth	$B_w$	400 MHz [39]
Minimum visibility elevation angle of the UT	$e{l}_{th}$	${55}^{\circ }$ [3]
Number of satellite beams	$N_b$	16 [3]
Satellite EIRP density	$E_d$	10 dBW/MHz [39]
Satellite beam radius	r	20 km [39]
Peak gain of satellite antenna	$G_t$	$38.5$ dBi [39]
Peak gain of UT antenna	$G_r$	$39.7$ dBi [39]
Vertical 3 dB beamwidth of UT antenna	${\Theta }_{3\mathrm{dB}}$	1.48°
Horizontal 3 dB beamwidth of UT antenna	${\Phi }_{3\mathrm{dB}}$	1.48°
System temperature	T	290 K
PSO Parameters	Symbol	Value
Number of particles	J	10
Maximum iterations	M	30
Minimum velocity	$v_{min}$	$-0.3$
Maximum velocity	$v_{max}$	$0.3$
Minimum inertia weight	$w_{min}$	$0.4$ [33]
Maximum inertia weight	$w_{max}$	$0.9$ [33]

. Note that the peak gain and 3 dB beamwidth of the satellite antenna in Table 3 are applied for a single beam. The simulation environment is built on a PC with an Intel Core i7-10700F, 16 GB of RAM, and a Windows 10 (64 bit) system. In this paper, the data of satellite position are exported from Satellite Tool Kit (STK) 11.2.0, which provides a physics-based modeling environment for analyzing platforms and payloads in a realistic mission context [38]. Due to the movement of the satellites, the overlapping area appears first and then disappears. Thus, the entire time of users covered by the overlapping coverage area is limited, and the maximum duration time is 21 s. During the entire simulation time, the maximum number of interfering satellites is $Q=28$, and the minimum time slot is 1 s [19].

We take the exhaustive search (ES) as the reference to verify the effectiveness of beam pointing optimization based on PSO. The ES traverses all directions with $grid=0.1°$ and finds the maximum SINR. The PSO algorithm is implemented on each UT and runs independently. To compare the performance of difference schemes clearly, we select a user located at (39.56° N, 116.20° E) to show the simulations in Figure 8, Figure 9 and Figure 10. For clarity, the scheme without a beam pointing optimization for interference avoidance is defined as BPO-WIA, while the schemes with a beam pointing optimization using ES and PSO are denoted as BPO-ES and BPO-PSO, respectively.

In Figure 8, the SINR using BPO-WIA initially decreases and then increases. This is because the motion of the satellites causes the overlapping area to first increase and then decrease. Thus, the interference also increases first and then decreases. The most severe interference occurs when the user is covered by the center of the maximum beam overlapping area, so the SINR is low and even close to 0 dB at time index $n=7$ and $n=8$. The lowest and average SINR is $0.02$ dB (time index $n=8$) and $7.59$ dB, respectively. The SINR with BPO-ES has the optimal SINR performance. However, the computation complexity of BPO-ES is highest, i.e., $\mathcal{O}$$(\frac{\left(2{\Theta }_3\mathrm{dB}\right)}{grid}\times \frac{\left(2{\Phi }_3\mathrm{dB}\right)}{grid})$, where $grid$ is 0.1°. The time consumption for BPO-ES at each time slot is $1.11$ s, as shown in

Table 4. Complexity comparison.

Computation Complexity	Setting	Time Consumption
BPO-ES = $\mathcal{O}$$(\frac{\left(2{\Theta }_3\mathrm{dB}\right)}{grid}·\frac{\left(2{\Phi }_3\mathrm{dB}\right)}{grid})$	grid = 0.1°	$1.11$ s
BPO-PSO = $\mathcal{O}\left(MJ\right)$	$M=30,J=10$	$0.39$ s
	$M=40,J=10$	$0.51$ s
	$M=30,J=5$	$0.20$ s
	$M=30,J=15$	$0.56$ s

.

In Figure 8, the optimized SINR of different particles and maximum iterations are shown. Moreover, the corresponding convergence curve at the moment of the strongest interference (time index $n=8$) is depicted in Figure 9. In Figure 8 and Figure 9, when the number of particles is set as $J=10$, the obtained SINR using $M=30$ or $M=40$ outperforms that using $M=20$. In addition, when the maximum iterations are set to $M=10$, the optimized SINR using $J=10$ or $J=15$ is better than that using $J=5$. It can be concluded that the optimal value can be found with proper iterative times or particles. Furthermore, the time consumption is related to the number of particles and iterations. As shown in Table 4, the computation complexity of BPO-PSO is $\mathcal{O}$$\left(MJ\right)$. To balance optimization performance and time consumption, we choose $M=30$ and $J=10$ in this paper. Its runtime for the one-shot optimization is approximately $0.39$ s. According to Figure 8, BPO-PSO $(M=30,J=10)$ and BPO-ES can achieve almost the same performance and are better than BPO-WIA during the simulation time. The SINR achieved by BP-ES and BP-PSO at the strongest interference moment is $8.23$ dB (time index $n=8$), and the average SINR is $10.05$ dB. Thus, BPO-PSO is more suitable for the high dynamic LEO satellite scenario than BPO-ES due to its lower time consumption.

The SINR with different numbers of beams is shown in Figure 10. The original SINR with BPO-WIA using $N_b=36$ is lower than $N_b=16$ due to the influence of co-channel interference. This is because the total beam coverage area and interfering links are increased due to the increase in the number of beams, which leads to a higher interference level. Moreover, at the time index $n=2$, the improvement of SINR obtained by BPO-PSO with $N_b=36$ and $N_b=16$ is $3.5$ dB and $3.0$ dB, respectively. At time index $n=8$, the improvement of SINR obtained by BPO-PSO with $N_b=36$ and $N_b=16$ is $10.0$ dB and $8.2$ dB, respectively. Thus, the BPO-PSO proposed in this paper can improve SINR with more gains for stronger interference scenarios, and it can also be applied to scenarios with various numbers of beams.

Furthermore, the simulation results of the multi-user scenario are provided in Figure 11, Figure 12 and Figure 13. The receiving antenna pointing of each user can be optimized separately and achieve the maximum SINR. Based on the above optimized SINR, the complementary cumulative distribution function (CCDF) of the SINR is depicted in Figure 11. The minimum SINR of BPO-WIA is $-7.89$ dB, while the minimum SINR of BPO-ES and BPO-PSO is $-0.63$ dB and $-1.1$ dB. Compared to BPO-WIA, the probability of BPO-PSO can achieve $17.27$% and $8.93$% improvement at SINR $=5$ dB and 10 dB; thus, the SINR is effectively improved for stronger interference. The multi-user average SINR of BPO-WIA, BPO-ES, and BPO-PSO is depicted in Figure 12. The maximum, minimum, and mean SINR of multi-user with BPO-WIA are $10.94$ dB, $2.84$ dB, and $7.85$ dB. With BPO-ES, the maximum, minimum, and mean SINR of multi-user are $11.04$ dB, $8.06$ dB, and $10.03$ dB. With BPO-PSO, the maximum, minimum, and mean SINR of multi-user are $10.99$ dB, $8.06$ dB, and $9.98$ dB. The average SINR of BPO-PSO is enhanced by $27.1$% compared to BPO-WIA. Moreover, compared to BPO-ES, the relative error of mean SINR of multi-user obtained by BPO-PSO is $0.51$%. Furthermore, the equivalent power flux density (EPFD) proposed in ITU Radio Regulations is used to evaluate the interference mitigation performance [40]; the higher EPFD means a higher interference. The cumulative distribution function (CDF) curves of EPFD are given in Figure 13. It can be seen that the CDF curve of EPFD of BPO-PSO basically overlaps with that of BPO-ES. Furthermore, the EPFD performance of BPO-ES and BPO-PSO is better than that of BPO-WIA. For example, the probability of BPO-PSO is improved by $12.86$% compared to BPO-WIA for EPFD = 103 dBW/m².

5. Conclusions

We mainly study the interference avoidance techniques for dense LEO satellite systems by optimizing the user antenna beam pointing. In this paper, the beam pointing optimization is modeled as a non-convex optimization problem. To solve the problem, the interference situational aware database composed of AoD and AoA of interfering satellites can be constructed and used for a distributed PSO-based beam pointing optimization. Then, the optimal beam pointing of the UT antenna can be obtained by maximizing the received SINR of each UT. Simulation results show that the relative error of the mean SINR obtained by BPO-PSO is $0.51$%. So, the proposed method can effectively mitigate the co-channel interference by adjusting the beam pointing of the UT antennas.