Cooperative Beam Hopping for Accurate Positioning in Ultra-Dense LEO Satellite Networks

We consider an orthogonal frequency division multiplexing (OFDM) based ultra-dense LEO multibeam satellite network. The satellite system operates at a center frequency $f_{0}$ with total system bandwidth $W$. To mitigate intra-satellite inter-beam interference, a frequency/polarization reuse factor of $\rho$ is applied. For inter-satellite interference reduction, time/frequency/code domain multiplexing can be utilized, and such kind of interference is neglected. For example, the positioning reference signal design with frequency reuse and muting configuration in 3GPP LTE can mitigate inter-cell interference. In the network, there are a set of $\mathcal{I}=\{1,\ldots,I\}$ LEO satellites. A satellite $i\in\mathcal{I}$ is equipped with a set of $\mathcal{B}=\{1,\ldots,B\}$ beams. For beam allocation, we further define a binary variable $\gamma_{i,b}=1$ if beam $i$ is allocated by satellite $i$, and $\gamma_{i,b}=0$ otherwise. A set of $\mathcal{J}=\{1,\ldots,J\}$ user equipments (UEs) are distributed requesting for positioning service. For a user $j\in\mathcal{J}$, the associated set of satellites used for positioning is denoted as $\mathcal{I}_{j}=\{1,\ldots,I_{j}\}$. The TDOA positioning scheme is adopted to calculate UE location results.

As per 3GPP NTN specifications [4], the following satellite antenna pattern is considered

where $J_{1}(\cdot)$ is the Bessel function of the first kind and first order with the argument, $a$ is the antenna aperture radius, $\theta$ is the steering angle, and $c$ is the light propagation speed. Meanwhile, the total path loss consists of following components:

where $SF$ is the shadow fading modeled by a log-normal distribution $N(0,\sigma_{SF}^{2})$, $d$ is the slant path distance, and $PL_{g}$ and $PL_{s}$ represent the atmospheric absorption and scintillation loss, respectively. The power $P_{i,b}^{j}$ received by the $j$-th user from $b$-th beam of $i$-th satellite is

where $\theta_{i,b}^{j}$ and $d_{i,b}^{j}$ denote the angle and distance between user $j$ and beam $b$ in satellite $i$, respectively. Besides, $EIRP_{i,b}=P_{i,b}+G_{T}$ is the transmitted Equivalent Isotropically Radiated Power density allocated to beam $b$ in satellite $i$, $P_{i,b}$ is the beam transmit power, and $G_{T}$ and $G_{R}$ denote the constant transmit and receive antenna gain, respectively. To this end, the overall signal-to-interference plus noise ratio (SINR) is

where $N_{0}$ is the noise power determined by UE noise figure and antenna temperature [4].

As a promising solution, BH is devised to employ only a small subset of transceivers/beams for serving extensive satellite coverage area. More specifically, assuming that a satellite aims to achieve a coverage of $B_{cov}$ beams by using $B$ beams with $B<B_{cov}$. For this purpose, at each time stamp, a maximum number of $B$ beams are assigned to illuminate a portion of the whole satellite coverage area, and time-division multiplexing approach is implemented to manipulate the set of $B$ beams into different portions within the coverage area of $B_{cov}$ beams. Through flexible beam allocation, full satellite coverage can be eventually served.

To support accurate positioning, we exploit BH to adaptively tune the physical beam layout from communication-oriented to positioning-oriented design. With cooperative BH from multiple neighboring satellites, localization performance can be significantly improved for UEs in a target satellite. An example BH for dual functional communication and positioning beams in a 4-satellite scenario is depicted in Fig. 1. On the one hand, communication beams are generally pointed to coverage area centered at the satellite nadir as in Fig. 1(a), such that seamless coverage is formed. On the other hand, to enhance localization performance, a set of neighbor satellites need to direct their beams to the target satellite. The center point of positioning beams normally stays far away from satellite nadir point. The cooperative BH patterns for positioning of Sat-1 and Sat-2 are sketched in Fig. 1(b) and Fig. 1(c), respectively.

An efficient UV plane based BH scheme (UVBHS) is devised for beam operation. As plotted in Fig. 2, UV plane is defined as the perpendicular plane to the satellite-earth line on the orbital plane. In UVBHS, a hexagonal beam layout is defined on the UV plane with UV coordinate of the nadir of the reference satellite setting to (0,0) for communication beams. For positioning beams, there are two different configuration situations. Firstly, to perform localization in its own service area, the satellite can simply reuse the communication configurations for positioning usage. Secondly, to assist localization for a neighboring satellite, the satellite needs to translate the beams centered at (0,0) to the center of the neighbor satellite’s nadir in the UV plane denoted by $(u,v)$. We can derive $u=sin\theta cos\varphi$ and $v=sin\theta sin\varphi$, where $\theta$ and $\varphi$ represent beam bore-sight steering angle and azimuth, respectively.

In the mean squared error (MSE) sense, CRLB gives the lowest possible variance that an unbiased linear estimator can achieve. To this end, we use CRLB as the accuracy indicator for TDOA positioning error analysis.

Based on the envisioned cooperative BH framework, i.e., UVBHS, an average CRLB optimization problem is formulated to improve positioning accuracy.

In problem (9), the objective is to minimize average CRLB of UEs by optimizing decision variables $\bm{\delta}=\{\delta_{i,b}^{j}\}$, $\bm{P}=\{P_{i,b}\}$, and $\bm{\gamma}=\{\gamma_{i,b}\}$. Constraints C1 states that each user should be associated to at most one beam at a given satellite; C2 specifies that the beam should be allocated once a user is associated to the beam, where $\Lambda$ is a sufficiently large integer; C3 is the individual beam power constraint; and C4 is imposed to guarantee that the allocated power at a satellite should not exceed its total available power. The NP-hardness of the optimization problem is shown below.

The problem (9) is solved by decomposing it into three sub-problems, namely, user association sub-problem, BH design sub-problem, and power allocation sub-problem.

The computational complexity of the devised algorithm is discussed herein. Recall that there are $J$ users, $B$ beams per satellite, and $N=I_{j}$ satellites used for positioning. The complexity of FBHCA comprises three parts: 1) User association with the complexity of $O(JNB)$. The complexity comes from a total of $JNB$ SINR calculations and comparisons for all UEs; 2) BH design with the complexity of $O(JNB^{2})$. This is because for a $B$ points Voronoi diagram, the complexity is $O(B^{2})$ [16]. Besides, the Voronoi graph is generated for $N$ satellites, and $J$ calculations are required to decide the beam allocation variable $\gamma_{i,b}$. This results in a total complexity of $O(JNB^{2})$; 3) Power allocation with the complexity of $O(NB^{4.5})$. For each satellite, solving the SDP problem incurs $O(B^{4.5})$ worst-case complexity, and a set of $N$ satellites needs to be calculated for power allocation.

The algorithm runs iteratively to obtain the desired solution. There are $M$ iterations in total, with each iteration generating a complexity of $O(JNB)+O(JNB^{2})+O(NB^{4.5})=O(JNB^{2}+NB^{4.5})$. Overall, the computational complexity of FBHCA is derived as $O\left(M(JNB^{2}+NB^{4.5})\right)$.

The set of key parameters used for simulations are summarized in Table I. Particularly, the satellite network comprises a total of 2400 satellites, with an orbit height varying from 800 km to 1500 km and inclination of 87.5 degree. The synchronization signal blocks (SSBs) in 3GPP NR are used for ToA measurements. The value of $\sigma_{SF}^{2}$ for shadow fading is a function of elevation angle following Table 6.6.2-3 in [4]. For dynamic simulation, the orbit period is divided into 100 snapshots of equal time duration. A total of 500 stationary UEs are randomly deployed in the target area with longitude and latitude setting to [-70,-60] and [-5,5], respectively. Note that all simulations are performed by extending the already calibrated platform for 3GPP NTN [17].

For comparison, two benchmark algorithms are used. One is the traditional method using communication beams (TMCB) for multi-satellite positioning as shown in Fig. 1(a). In TMCB, maximum SINR based user association and equal power allocation among all satellite beams are assumed. The other one is termed as UVBHS-EPA, which combines the proposed UVBHS framework with equal power allocation.

Fig. 4 depicts the average CRLB performance for different schemes as orbital height changes. As orbit height increases, the CRLB of all the three algorithms first decreases. The reason is that the constellation of 2400 satellites is not sufficient to provide full coverage for lower orbit height than around 1100 km. Lower orbit can cause worse SINR and positioning performance. With orbit height further rising, the CRLB increases as well due to the larger experienced path loss. Besides, both UVBHS-EPA and FBHCA significantly outperform TMCB in terms of CRLB. This is because in TMCB, although the received signal quality from the serving satellite is favorable, the signal quality from neighboring satellites is very poor due to the long distance between UE and beam center. Nonetheless, in the other two methods, neighboring satellite beams are directed to cover UEs of interest, and thus multiple signals with good quality can be measured to enhance positioning accuracy. To verify the performance gain, we list the SNR values of different satellites in Table II.

The CRLB results for different schemes as time snapshot varies are given in Fig 5. The orbit height is fixed to 1200 km in the simulation. All the algorithms exhibit CRLB fluctuation as time snapshots change. The variation tendency is quite complicated, due to the intertwined influence by time varying inter-beam interference and dynamic geometric dilution of precision. Besides, as the number of positioning satellites increases from 6 to 8, the CRLBs for all the three algorithms improve as well. This phenomena can be expected, because more signals and better network geometry are obtained by exploring satellite diversity.