Power Adaptation for Suborbital Downlink with Stochastic Satellites Interference

The downlink effective bandwidth can be guaranteed by increasing the transmit power of the SN [3]. Still, it is limited by a power upper limit due to the hardware restriction, etc. Hence the transmit power strategy of the base station is segmented, i.e.

where $P_{t}$ denotes the minimum transmitted power needed to meet the QoS, and $P_{u}$ means the maximum transmitted power allowed by the transmitter in the SN. $g_{th}$ is the equivalent threshold of the small-scale fading coefficient $g$ (i.e. when $g$ is larger than $g_{th}$, $P_{t}$ can be smaller than $P_{u}$, and in this case the downlink transmission is carried out normally).

$P_{u}$ is relatively large to meet most of the conditions, but there are still rare occasions caused by interference that require higher $P_{t}$. Accordingly, there is a probability that $g$ is smaller than $g_{th}$ denoted as $\varepsilon^{t}$. Here suppose the SN still sends the downlink packet with maximum transmitted power $P_{u}$ if $\varepsilon^{c}$ cannot be guaranteed, which is dubbed TISPA. Thus the essential approach to reduce costs and optimize the global power strategy is to obtain the minimum $P_{u}$.

After the derivation in the next section, the optimization for TISPA is formulated as

where we mark

for simplicity. $\eta$ is the packet size in bits per packet. $N_{0}$ represents the unilateral power spectral density (PSD) of Gaussian white noise, $B$ is the bandwidth of the channel used, and $Q^{-1}$ means the inverse of the Gaussian Q-function. Additionally, the path loss, the period of a frame, and the power of stochastic interference caused by satellites are denoted by $G$, $T_{f}$, and $I$ respectively. $\mathbb{E}(I)$ stands for the expectation of total stochastic interference from satellites. Explicitly, (2) is the core problem of the power adaptation at an SN.

The interference of each satellite is jointly determined by the channel gain and spatial gain. The commonly used Ku-band channel of SoI is considered as LoS [8]. Assuming SoI transmitting full-time, $I$ is a cumulative power from satellites distributions of different orbits, i.e.

where $i$ is the index of orbits, $\Phi={x_{1},x_{2},...}\subset\mathbb{R}^{2}$ represents the stochastic point process of interference satellites on the orbital plane. We assume that the launch power of each satellite is subject to uniform distribution within a certain range. By a simple derivation of the compound PP, the expectation of launch power $P_{\rm{SN}}$ is what we only need. Additionally, the path loss from each satellite, the power gain of the antennas at satellites, and the power gain of the antennas at the receiving station are denoted by $f_{\rm{PL}}$, $G_{tx}$, and $G_{rx}$ respectively.

$I$ in (4) is a function of the stochastic point position ${x}$. The distribution on the orbital plane is modeled as a homogeneous Point Poisson Process (PPP) in a spherical surface $\mathbb{R}^{2}$ when observed at the geocentric angle or the length of the satellite orbit.

One theorem used in this part is the maximum coding rate of short packets [3], which is an indicator to measure the downlink of the SN. With approximation, it can be written as

where $\gamma$ stands for the SINR at the receiving station. $\sigma$ is the approximated channel dispersion

according to [3]. $\sigma\approx 1$ when $\gamma$ is high. Thus, for a fixed $\sigma$, the optimal coding rate $R^{*}$ will increase as block-length $n$ tends to be a large value. With $n=T_{f}B$, and $\eta$ denote the packet size, $R^{*}$ can be expressed as

Besides, the minimum service rate of the SN should be larger than its effective bandwidth determined by the queuing process of downlink packets. To establish the relationship between SINR and the transmission rate of short packets at the SN, it is necessary to adopt the effective bandwidth $E_{B}(\theta)\triangleq\Delta(\theta)/\theta$. And can be expressed as $E_{B}(\theta)=\Omega/2\theta T_{f}$ by extending the similar scenario in [2] to PP, where

in which $D_{max}$ represents the maximum tolerable delay. $\theta$ denotes the rate of the exponential tail of the queue length distribution, which is a variable in the function of minimum envelope rate [12]. And then $E_{B}(\theta)$ can be derived as

where $\varepsilon^{u}$ denotes the upper bound of delay violation probability. $\lambda_{q}=N_{u}*T_{f}$ is the density of data generated, and $N_{u}$ represents the packet arrival rate.

According to the effective bandwidth, with $E_{B}T_{f}\leq R^{*}$, we consider the lower bound as the service rate and convert (7) into a function of $P$, i.e.

and $P\geq{P_{t}}$. Moreover, $\nu$ is the same in (3), which is actually the SINR at the receiver when the SN is transmitting. Consistent with (1), ${P_{t}}$ is the minimum required to transmit power to meet the QoS of the downlink. In this way, the SN transmits packets with a rate of effective bandwidth. Since $R^{*}$ depends on the maximum transmission power $P_{u}$ when $g$ is fixed to $g_{th}$, the maximum achievable rate of $R^{*}$ can be denoted as

Since the downlink channel from the SN is different from other satellites and has a lower altitude, the small-scale fading $g$ needs to be considered differently from the satellite-ground transmission. Thanks to [11], the finite blocklength theorem is available in the quasi-static multi-antenna channel within a frame of the downlink transmission. The probability distribution function (pdf) of $g$ is a Nakagami distribution [13], which is

where $N_{t}$ denotes the number of transmitting antennas. The TISPA scheme of (1) considers the probability of $g<g_{th}$ occurring is small if there is a relatively large $P_{u}$. Consistent with (2), the probability of $g\leq g_{th}$ is written as

which can be considered as the error probability introduced by TISPA. According to the expression of $g_{th}$ from (10) and (13), the total transmission error probability $\varepsilon$ can be approximated as

Basically, minimizing $P_{u}$ is equivalent to minimizing $\nu$ under fixed $\varepsilon^{t}$ according to (2), i.e.

By solving the second-order differential of the expression above, it can be proved that the sub-problem (15) is convex w.r.t. $\varepsilon^{c}$ and $\varepsilon^{q}$.

After the specific analysis above, the solution to the optimization problem in (2) can be summarized in Algorithm 1.

The distribution of satellites is homogeneous w.r.t. the geocentric angle. Thus, the density distribution of interference is related to the geometric relationship of the observation points, i.e., azimuth angle $\vartheta$ and elevation angle $\varphi$ from the receiving station. $L$ represents the height of the satellite orbital plane, the radius of the earth is $C_{0}$, and $d$ denotes the transmission distance to a certain spatial position on the orbital plane. The geometric relationship under each observation domain is shown in Fig. 2.

Since the area where the main lobes of SoI and the receiving station overlap are small, it can be approximately considered that the corresponding elevation angle of SoI and the receiving station that generates interference share the same $\varphi$.

The stochastic point process $\Phi=\{x_{1},x_{2},...\}$ is a space on the orbital plane with two dimensions, and the distribution of satellites can be converted to the union process $\Phi_{a}=\{\vartheta_{1},\vartheta_{2},...\}$ and $\tilde{\Phi}=\{\varphi_{1},\varphi_{2},...\}$ w.r.t. the azimuth and elevation angle respectively. For an equivalent array antenna covering the hemisphere, the main lobe of the beam is considered to be fixed in the middle of the azimuth. Assuming that the base of the receiving station can rotate horizontally while tracking the SN. Hence, the distribution of the path loss changes in azimuth does not cause a difference in distance, i.e., $f_{\rm{PL}}$ is only a function of $\varphi$. Then the array antenna degenerates into an equivalent linear array. Thus, the antenna gain at the receiving station is fixed for different azimuth angles, i.e., both $G_{tx}$ and $G_{rx}$ are only functions of $\varphi$. Based on the most widely used low earth orbit (LEO), medium earth orbit (MEO), and geosynchronous earth orbit (GEO), we consider the geometric distribution of satellites as orbital planes at 3 altitudes to generalize satellites in adjacent regions at that altitude. For more accurate results, the division of the orbital plane can be more detailed, which will not be discussed here. Thereupon, (4) can be specifically written as

where $i$ is the index of the 3 orbits mentioned.

As the range of antenna scanning is small when interference occurs, it is approximately considered that the stochastic distribution on the satellite orbits under the $\varphi$ observation domain is quasi-homogeneous. The propagation in the Ku-band is LoS, and the path loss in free space is inversely proportional to the square of $d$, i.e., $f_{PL}\propto(1/d)^{2}$. However, there is still a part of the energy in the atmosphere, and the equivalent path loss on the whole link is slightly higher. Therefore, the path loss of the interfering signal is considered a segmented function of the track height, which is organized as a function of transmission distance according to the geometric relationship, i.e.,

where both path loss factors ${\alpha_{0}}$ and ${\alpha}$ in outer space and inside the atmosphere are considered, and $L_{r}$ denotes the reference height of the boundary of the atmosphere.

In the satellite-ground link, besides high energy efficiency, a low enough fluctuation is needed, i.e., not affecting the gain of stochastic interference in the space domain. An HBF array design with better side lobe suppression and smaller main lobe ripple is adopted. The design process needs to solve an optimization problem with passband flattening and hybrid decomposition with details in [7]. Since the beam of common satellites is fixed or moves within a small range, we assume that their main beams are fixed in the center. A certain scanning elevation angle is selected for the receiving station, and then the corresponding gain can be determined by the HBF.

Subsequently, the function of the stochastic point process in (16) is converted into a function w.r.t. $d$, and the space of $\varphi$ is also mapped to the space of $d$, i.e.,

where $\Phi^{*}=\{d_{1},d_{2},...\}$ is an inhomogeneous point process in the distance domain as $d$ is unevenly distributed. Still, on a relatively small area of the orbital plane, the distribution is homogeneous when viewed from a distance $r$ from the center of the orbital plane (i.e. the radius in the orbital sub-plane, which is introduced as a reference variable for a homogeneous distribution w.r.t. subset $X\in\mathbb{R}$ of $\Phi_{r}=\{r_{1},r_{2},...\}$ and is equivalent to the expression in (16)). Let $Y\in\mathbb{R}$ be a subset of $\Phi^{*}$, and according to the geometric relationship, the mapping function from the original subset of $\Phi_{r}$ is $f(x)=\sqrt{\parallel x{\parallel^{2}}+{L^{2}}}$.

With known geometric relations, the measure of this distribution in the space w.r.t. $d$ is

where $\lambda_{d}$ denotes the density of distribution in the orbit that can be calculated from the number of satellites $n_{L}$ in orbit and $L$. And then, the density measure is ${\lambda^{*}}(r)=2{\lambda_{d}}\pi r=2{\lambda_{d}}\pi\sqrt{{d^{2}}-{L^{2}}}$.

Finally, the measure function under the original distribution can be expressed as

where the partial differential representation is used to prevent confusion of $d$.

According to Campbell’s Theorem [9], the expectation of a function of a random point process can be expressed as the R-S integral w.r.t. the density measure. Thus, combining (18) and (20), the expectation of $I$ is obtained, i.e.

where $F(d)={{f_{{\text{PL}}}}(d){G_{t}}(d){G_{r}}(d)}$. The upper limit $d_{ui}$ and lower limit $d_{li}$ of the integral is determined by the actual overlap of the beams on the i-th orbit. Due to the expression of $f_{\rm{PL}}$ and the numerical relationship of $G_{r}$ and $G_{r}$, the integral can be solved numerically.

Employ a uniform interval of $\cos\varphi$ to get a vector of ${\sqrt{{d^{2}}-{L^{2}}}}/d$, and then perform linear interpolation on the expression to obtain the numerical relationship of $G_{r}$ and $G_{t}$ with $d$. Eventually, the integral can be solved by the complex trapezoidal integral with no more discussion here.