Power Allocation for Reliable and Energy-Efficient Optical LEO-to-Ground Downlinks with Hybrid ARQ Schemes

Abstract

Satellites in low earth orbit (LEO) are currently being deployed for numerous communication, positioning, space and Earth-imaging missions. To provide higher data rates in direct-to-user links and earth observation downlinks, the free-space optics technology can be employed for LEO-to-ground downlinks. Moreover, the hybrid automatic repeat request (HARQ) can be adopted since the propagation latency is low for LEO satellites. In this work, a power allocation methodology is proposed for optical LEO-to-ground downlinks under weak turbulence employing HARQ retransmission schemes. Specifically, the average power consumption is minimized given a maximum transmitted power constraint and a target outage probability threshold to ensure energy efficiency and reliability, respectively. The optimization problem is formulated as a constrained nonlinear programming problem and solved for Type I HARQ, chase combining (CC) and incremental redundancy (IR) schemes. The solutions are derived numerically via iterative algorithms, namely interior-point (IP) and sequential quadratic programming (SQP), and validated through an exhaustive (brute-force) search. The numerical simulations provide insight into the performance of the retransmission schemes regarding average power. More specifically, Type I HARQ has the worst output, CC has a moderate one, and IR exhibits the best performance. Finally, the IP algorithm is a slower but more accurate solver, and SQP is faster but slightly less accurate.

1. Introduction

As early as the 1960s, the development of optical and laser pumping brought free-space optical (FSO) communication to life [1,2]. Since then, FSO has been employed throughout the industry either for space telecommunication applications such as the modern SpaceX Starlink project for satellite interconnection, optical satellite feeder links and even terrestrial commercial and military applications, e.g., inter-building links [1]. The rollout of the fifth generation (5G) and beyond has especially shifted the research paradigm to optical frequency technologies in order to meet the strict broadband, reliability and latency requirements, leading to the massive installation of fiber optics and optical telescopes [2]. Albeit a relatively new field of communication, FSO systems are considered mature enough to be employed as they hold many noteworthy advantages. More specifically, it is easy to establish point-to-point optical links due to the small-size equipment, lack of any digging, huge bandwidth availability and data rates, unlicensed operation, their strong immunity to unwanted interferences, low-power to ensure eye-safety and improved security due to employment of narrow beams [1,2,3]. Therefore, assuming good optical signal propagation conditions (clear sky), FSO links guarantee fast, convenient, economic, secure and reliable deployment as well as the efficient usage of the spectrum [1,2,3].

On the other hand, in the case of atmospheric impairments such as opaque fog and clouds or line-of-sight interruption in general, the irradiance losses can reach hundreds of dBs, leading to an optical link outage [1,2,3]. Moreover, the atmospheric refractive index is not spatially or temporally homogeneous but is varied with wind speed, temperature and wavelengths, which subsequently cause atmospheric turbulence [2,3,4]. The turbulence in turn influences the slant path propagation, causing scintillation of the received irradiance and beam spreading [2,3,4]. There are also many more sources of deterioration such as pointing jitter and background noise, but these are beyond the scope of this work [4]. Typical mitigation techniques regarding the physical layer include aperture averaging, adaptive optics, hybrid radio frequency (RF)/FSO and spatial diversity, but there are also upper-layer solutions such as automatic repeat request (ARQ) retransmission schemes [3,4].

In particular, hybrid ARQ (HARQ) combines error detection with error correction by adding redundancy bits to the transmitted frames, and if the message decoding fails, a series of retransmission rounds are performed [5,6,7,8]. Depending on whether the erroneous message is discarded or stored, the HARQ is categorized as Type I HARQ or soft combining HARQ, respectively [5,6,7,8]. The latter is performed in practice based on the following two methods: chase combining (CC), in which the transmitter sends identical copies of the corrupted frame on each round and the receiver employs maximal ratio combining, whereas in incremental redundancy (IR) the transmitter sends more parity bits on each round, increasing the successful decoding probability [5,6,7,8]. The HARQ is superior when reliability and link adaptation are required due to the joint error detection and correction, while cost-wise it is potentially cheaper. HARQ for optical links has shown good performance and can operate in parallel with the aforementioned physical layer techniques [3].

A low Earth orbit (LEO) satellite operates at an altitude less than 2000 km, has a full period of about two hours and exhibits a latency of a few milliseconds (17 times lower than GEO) [2]. If equipped with an optical transmitter it can provide high-definition data at reduced latency. Several optical LEO-to-ground experiments have been conducted to study the feasibility and obtain measurements for channel characterization and modeling [2,9,10]. In [6] a new HARQ protocol for FSO multi-user systems is proposed, and in [8,11,12] a performance analysis of FSO HARQ systems and estimations of the packet error probability are derived. In [13,14,15] power allocation strategies for RF HARQ links under Rayleigh fading are reported. The LEO satellite-to-ground links are also used for quantum key distribution (QKD). LEO-to-ground QKD links have also very recently been demonstrated to reach distances up to 1200 km and key rates up to kbps [16]. The key rate is the exchange rate of polarized photons (encryption keys) over an optical fiber or FSO link. Moreover, a study has been evaluated for QKD performance on a hypothetical constellation with ten satellites in sun-synchronous LEOs that are assumed to communicate over a period of one year with an optical infrastructure (three optical ground stations) located in Greece [17]. The atmospheric effects of turbulence and the background solar radiance have been considered [17].

For optical satellite downlinks with HARQ schemes under weak turbulence, there has not been a power allocation investigation in the literature. In this contribution, a power allocation methodology is proposed for optical LEO-to-ground downlinks under weak scintillation conditions employing HARQ retransmission schemes. The contributions of this work are summarized as follows:

• Three power allocation methodologies based on the Type I HARQ, CC and IR schemes are proposed, and their performances are compared and ranked from best to worst in terms of the average power consumption.
• The energy efficiency and the reliability of the optical links are optimized by formulating the optimization problem as a constrained nonlinear programming problem with an objective function, the average power usage, constraints, the maximum transmitted power and a target outage probability accordingly.
• Only the channel statistics (long-term channel state information) are required to obtain the optimal power allocation strategy and not the instantaneous channel states.
• The proposed solutions are derived numerically via iterative algorithms, namely interior-point and sequential quadratic programming, and validated through an exhaustive or brute-force search [18,19].
• Simulations are executed for various channel conditions and system settings by simulating a LEO passing over various turbulence intensities and ground weather conditions to investigate the sensitivity of the three HARQ schemes to weak scintillation, path loss and target outage probability. Novel numerical results are reported and commented on.

The remainder of the paper is structured as follows: In Section 2 the LEO-to-ground system model is given along with the necessary FSO theory and assumptions regarding the weak fluctuation model. In Section 3 the optimal power allocation problem under maximum power and outage probability constraints is developed by taking into account the three HARQ schemes, and the proposed solutions are reported. In Section 4 simulations of various scintillation, weather conditions and constraints are obtained using the proposed methodology, and numerical results are derived, compared and commented on. Finally, Section 5 concludes this article and future work is proposed.

2. System Model

A single, cloud-free optical LEO-to-ground communication downlink is considered subject to path losses and weak atmospheric turbulence. The optical channel is generally dynamic due to the elevation angle–varying link distance and the LEO satellite’s slew rate, thus leading to temporal signal fluctuations known as scintillation [2,9]. For the transmission, the intensity modulation with on-off-keying (OOK) is assumed, and direct detection is used for the reception. It is also hypothesized that a negative acknowledgment (NACK) or no acknowledgment at all to a particular frame transmission by the receiving terminal will initiate a series of retransmissions via an HARQ protocol [5,6,7,8]. The maximum number of HARQ rounds is predefined and equal to M. During these M rounds, the receiver either successfully decodes the message and responds with a positive acknowledgment (ACK) or fails to decode it, and the re-transmission stops. In order to achieve independent fading states, the minimum retransmission time between rounds must be equal to the coherence time ${\tau }_0(\sec )$ of the optical channel. According to the weak turbulence model the coherence time is given by [9,10]:

$${\tau }_0={\left(118{\lambda }^{-2}\sec \left(90°-e\right){\displaystyle \underset{H_{\mathrm{OGS}}}{\overset{H_{\mathrm{LEO}}}{\int }}{C}_n^2(\mathrm{z})V^{5/3}(\mathrm{z})\mathrm{dz}}\right)}^{-3/5},$$

(1)

where $\lambda \left(m\right)$ is the communication wavelength, $\sec (x)$ is the secant function, $e\left(deg\right)$ is the elevation angle, $H_{\mathrm{OGS}}\left(m\right), H_{\mathrm{LEO}}\left(m\right)$ are the altitudes of the optical ground station (OGS) and LEO satellite, $C_n^2\left(z\right)$ is the refractive index structure parameter usually given by the Hufnagel–Valley model, $z\left(m\right)$ is the altitude, and $V\left(z\right)\left(m/s\right)$ is the wind speed (vertical path) usually described by the Bufton model. Therefore, for transmission periods greater than ${\tau }_0$ the fading states can be considered uncorrelated. In Figure 1, the system model is depicted.

Weak turbulence is often represented by the lognormal (LN) distribution, which fits well and exhibits good agreement with first-order statistics from experimental data [2,20]. The LN model does not fully apply only for elevation angles <20°, and the reason is the saturation of scintillation [2]. The normalized received irradiance $I\left(W/m^2\right)$ is an $LN$ random variable according to the following probability density function (PDF) [4,20]:

$$p(I)=\frac{1}{I\sqrt{2\pi }{\sigma }_I}\exp \left\{-\frac{{\left[\ln \left(I/\langle I\rangle \right)+0.5{\sigma }_I^2\right]}^2}{2{\sigma }_I^2}\right\},$$

(2)

where $\langle I\rangle$ is the average irradiance, and ${\sigma }_I^2$ is the so-called scintillation index (SI). A theoretical expression for SI in the case of weak turbulence (SI < 0.5) and point receivers is derived from Rytov, which is expressed by the formula [9,20]:

$${\mathsf{\sigma }}_I^2=2.25k^{\frac{7}{6}}{\sec }^{\frac{11}{6}}\left(90°-e\right){\displaystyle \underset{H_{\mathrm{OGS}}}{\overset{H_{\mathrm{LEO}}}{\int }}{C}_n^2\left(\mathrm{z}\right){\left(\mathrm{z}-H_{\mathrm{OGS}}\right)}^{\frac{5}{6}}\mathrm{dz}},$$

(3)

where $k\left(rad/m\right)$ is the wavenumber, $e\left(deg\right)$ is the elevation angle, and the rest of the parameters are defined as in (1). The Kolmogorov spectrum is assumed, and it is also assumed that the optical wavefront is approximated by a plane wave far from the source. For moderate (SI~0.5) or strong turbulence (SI~1), other distributions such as gamma–gamma are more suitable [2]. If experimental data are to be employed then SI is simply the normalized variance of I: ${\sigma }_I^2=\langle I^2\rangle /{\langle I\rangle }^2-1$ [9,20].

The optical channel for the downlink follows the LN distribution considering weak scintillation conditions. By incorporating the quantum efficiencies of the transmitter ${\eta }_{\mathrm{LEO}}$ and receiver ${\eta }_{\mathrm{OGS}}$, the atmospheric transmittance $T_{Atm}$, the gains of transmitter $g_{\mathrm{LEO}}$ and receiver $g_{\mathrm{OGS}}$, the large-scale path loss $PL$, the small-scale loss due to scintillation $e^{2X_s}$ where $X_s$ is the log-amplitude of the optical wave and hence Gaussian (normally) distributed, the optical channel is expressed by [4,21]:

$$h_{opt}={\eta }_{\mathrm{LEO}}{\eta }_{\mathrm{OGS}}{T}_{{}_{Atm}}{g}_{\mathrm{LEO}}{g}_{\mathrm{OGS}}PL e^{2X_s}$$

(4)

Additionally, therefore, the signal-to-noise ratio (SNR) is given by [4,21]:

$$SNR=\frac{P_T {\eta }_{\mathrm{LEO}}{\eta }_{\mathrm{OGS}}{T}_{Atm}{g}_{\mathrm{LEO}}{g}_{\mathrm{OGS}}PL\hspace{0.17em}\hspace{0.17em}{e}^{2X_s}}{{\mathsf{\sigma }}_{n_{opt}}^2}=\frac{P_T\hspace{0.17em}{h}_{opt}}{{\mathsf{\sigma }}_{n_{opt}}^2}=P_T\hspace{0.17em}h$$

(5)

where $P_T$ is the transmitted power, ${\mathsf{\sigma }}_{n_{opt}}^2$ is the signal-independent optical noise variance, and $h\left[Watt{s}^{-\hspace{0.17em}1}\right]$ represents the ratio $h_{opt}/{\mathsf{\sigma }}_{n_{opt}}^2$.

In particular, the optical noise is a zero mean, a constant variance random variable that describes the environmental optical interference. It incorporates the background radiation from the various celestial bodies, the amplified spontaneous emission (ASE) from optical preamplifiers and the electronic detection noise [4,21]. By using narrow passband optical filters and small field-of-view telescopes the receiver can eliminate the ambient background radiation and ASE, while the thermal and shot noises can be considered as additive white Gaussian noise [2]. Now, for a fixed ${\mathsf{\sigma }}_{n_{opt}}^2$ it is easy to see that h is also a LN variable, and its PDF is derived from (2) with parameters $\langle h\rangle$, ${\mathsf{\sigma }}_h^2={\mathsf{\sigma }}_I^2$.

3. Power Allocation Problem and Methodology

3.1. Power Allocation Problem Formulation

In this subsection, the power allocation problem for an HARQ optical LEO-to-OGS downlink is formulated under a maximum transmitted power constraint and a guaranteed outage probability constraint. Specifically, assuming M rounds of retransmissions via Type I, CC, IR HARQ schemes the optimization problem considers the minimization of the average total transmitted power:

$$\underset{P_1,P_2,\dots ,P_M}{\arg \min }\hspace{0.17em}\hspace{0.17em}{P}_{Avg}$$

(6)

$$\begin{array}{l}s.t.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\le P_m\le \overline{P}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}1\le m\le M\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\le P_{out,M}\left(C_{out}\right)\le {\epsilon }_{\mathrm{target}}\hspace{0.17em}\hspace{0.17em},\end{array}$$

(7)

where $P_i\hspace{0.17em}(i=1,\dots ,M)$ is the transmitted power at each round, $\overline{P}$ is the peak transmitted power, $C_{out}$ is the defined outage capacity, $P_{out,M}$ is the decoding failure probability after all the M re-transmissions, and ${\epsilon }_{\mathrm{target}}$ is the target outage probability. The $P_{out,m}$ is analyzed below for the three HARQ schemes individually.

The $P_{Avg}$ for all three investigated HARQ schemes is reported as follows [13,14,15]:

$$P_{Avg}=P_1+P_2\cdot P_{out,1}+\dots +P_M\cdot P_{out,M-1}={\displaystyle \sum _{m\hspace{0.17em}=\hspace{0.17em}1}^M{P}_m}\cdot P_{out,m-1}$$

(8)

Note that $P_{out,0}=1$ because no transmission is achieved at round m = 0. Successful decoding occurs when the channel capacity on the mth round $C_m>C_{out}$; otherwise, a re-transmission is requested [13,14]. After M failures, the buffer empties, and the source proceeds to the next packet. In our analysis, the bit error rate (BER) is not included in (7) because it is complex and requires specific knowledge of coding and modulation. The BER performance for a variety of binary modulations can be found in [12].

3.1.1. Type I HARQ

In a Type I HARQ scheme, the received packets are not buffered but discarded on each round. Thus, after the mth round the channel capacity and outage probability are [5,15]:

$$C_m=W{\log }_2\left(1+h_m{P}_m\right)\hspace{0.17em}\hspace{0.17em}\left(bps\right)$$

(9)

$$P_r\left(C_m\le C_{out}\right)=P_r\left(h_m\le \frac{2^{\frac{C_{out}}{W}}-1}{P_m}\right)=\frac{1}{2}\mathrm{erfc}\left(-\frac{\ln \left(\frac{2^{\frac{C_{out}}{W}}-1}{\langle h_m\rangle P_m}\right)+0.5{\sigma }_h^2}{{\sigma }_h\sqrt{2}}\right)$$

(10)

$$P_{out,m}=P_r\left(C_1\le C_{out},\dots ,\hspace{0.17em}{C}_m\le C_{out}\right)={\displaystyle \prod _{i=1}^m{P}_r\left(C_i\le C_{out}\right)}\hspace{0.17em}\hspace{0.17em},$$

(11)

where W is the allocated bandwidth, and erfc(x) is the complementary error function.

For example, in the case of M = 3 rounds then the optimization problem is the following:

$$\underset{P_{1,}{P}_{2,}{P}_3}{\arg \min }\hspace{0.17em}\hspace{0.17em}{P}_1+P_2\cdot \frac{1}{2}\mathrm{erfc}\left(-\frac{\ln \left(\frac{2^{\frac{C_{out}}{W}}-1}{\langle h_1\rangle P_1}\right)+0.5{\sigma }_{h_1}^2}{{\sigma }_{h_1}\sqrt{2}}\right)+P_3\cdot \frac{1}{4}\mathrm{erfc}\left(-\frac{\ln \left(\frac{2^{\frac{C_{out}}{W}}-1}{\langle h_1\rangle P_1}\right)+0.5{\sigma }_{h_1}^2}{{\sigma }_{h_1}\sqrt{2}}\right)\cdot \mathrm{erfc}\left(-\frac{\ln \left(\frac{2^{\frac{C_{out}}{W}}-1}{\langle h_2\rangle P_2}\right)+0.5{\sigma }_{h_2}^2}{{\sigma }_{h_2}\sqrt{2}}\right)$$

(12)

$$P_{out,3}=\frac{1}{8}\mathrm{erfc}\left(-\frac{\ln \left(\frac{2^{\frac{C_{out}}{W}}-1}{\langle h_1\rangle P_1}\right)+0.5{\sigma }_{h_1}^2}{{\sigma }_{h_1}^{\hspace{0.17em}}\sqrt{2}}\right)\cdot \mathrm{erfc}\left(-\frac{\ln \left(\frac{2^{\frac{C_{out}}{W}}-1}{\langle h_2\rangle P_2}\right)+0.5{\sigma }_{h_2}^2}{{\sigma }_{h_2}^{\hspace{0.17em}}\sqrt{2}}\right)\cdot \mathrm{erfc}\left(-\frac{\ln \left(\frac{2^{\frac{C_{out}}{W}}-1}{\langle h_3\rangle P_3}\right)+0.5{\sigma }_{h_3}^2}{{\sigma }_{h_3}^{\hspace{0.17em}}\sqrt{2}}\right)$$

(13)

3.1.2. Chase Combining HARQ

In a CC scheme, the received packets are buffered and MRC-combined on each round. Thus, after the mth round the channel capacity and outage probability are [5,14]:

$$C_m=W{\log }_2\left(1+{\displaystyle \sum _{i=1}^m{h}_i{P}_i}\right)\hspace{0.17em}\hspace{0.17em}\left(bps\right)$$

(14)

$$P_r\left(C_m\le C_{out}\right)=P_r\left(W{\log }_2\left(1+{\displaystyle \sum _{i=1}^m{h}_i{P}_i}\right)\le C_{out}\right)=P_r\left(h_1{P}_1+\dots +h_m{P}_m\le 2^{\frac{C_{out}}{W}}-1\right)$$

(15)

$$P_{out,\hspace{0.17em}m}=P_r\left(C_1\le C_{out},\dots ,\hspace{0.17em}{C}_m\le C_{out}\right)=P_r\left(C_m\le C_{out}\right)$$

(16)

For example, in the case of M = 3 rounds then the optimization problem is the following:

$$\underset{P_{1,}{P}_{2,}{P}_3}{\arg \min }\hspace{0.17em}\hspace{0.17em}{P}_1+P_2\cdot \frac{1}{2}\mathrm{erfc}\left(-\frac{\ln \left(\frac{2^{\frac{C_{out}}{W}}-1}{\langle h_1\rangle P_1}\right)+0.5{\sigma }_{h_1}^2}{{\sigma }_{h_1}^{\hspace{0.17em}}\sqrt{2}}\right)+P_3\cdot P_r\left(h_1{P}_1+h_2{P}_2\le 2^{\frac{C_{out}}{W}}-1\right)$$

(17)

$$P_{out,3}=\underset{h_1=0}{\overset{\frac{2^{\frac{C_{out}}{W}}-1}{P_1}}{\int }}{\displaystyle \underset{h_2=0}{\overset{\frac{2^{\frac{C_{out}}{W}}-1-h_1{P}_1}{P_2}}{\int }}{\displaystyle \underset{h_3=0}{\overset{\frac{2^{\frac{C_{out}}{W}}-1-h_2{P}_2-h_1{P}_1}{P_3}}{\int }}\frac{1}{h_1{h}_2{h}_3{\sigma }_{h_1}^{\hspace{0.17em}}{\sigma }_{h_2}^{\hspace{0.17em}}{\sigma }_{h_3}^{\hspace{0.17em}}\sqrt[3]{2\pi }}{\mathrm{e}}^{\left(\hspace{0.17em}-\mathrm{K}\left(h\right)\hspace{0.17em}\right)}}{\mathrm{dh}}_1}{\mathrm{dh}}_2{\mathrm{dh}}_3$$

(18)

where

$$\mathrm{K}\left(h\right)\hspace{0.17em}=\frac{{\left(\ln h_1/\langle h_1\rangle +0.5{\sigma }_{h_1}^2\right)}^2}{2{\sigma }_{h_1}^2}\hspace{0.17em}+\frac{{\left(\ln h_2/\langle h_2\rangle +0.5{\sigma }_{h_2}^2\right)}^2}{2{\sigma }_{h_2}^2}+\frac{{\left(\ln h_3/\langle h_3\rangle +0.5{\sigma }_{h_3}^2\right)}^2}{2{\sigma }_{h_3}^2}$$

(19)

3.1.3. Incremental Redundancy HARQ

In an IR scheme, the received packets are buffered, and the information is added on each round because the packets contain new parity bits. Thus, after the mth round the channel capacity and outage probability are [5,13]:

$$C_m={\displaystyle \sum _{i=1}^{m}W{\log }_2\left(1+h_i{P}_i\right)}\hspace{0.17em}\hspace{0.17em}\left(bps\right)$$

(20)

$$P_r\left(C_m\le C_{out}\right)\Rightarrow P_r\left({\displaystyle \sum _{i=1}^{m}W{\log }_2\left(1+h_i{P}_i\right)}\le C_{out}\right)\Rightarrow P_r\left(\left(1+h_1{P}_1\right)\cdot \cdot \cdot \left(1+h_m{P}_m\right)\le 2^{\frac{C_{out}}{W}}\right)$$

(21)

$$P_{out,\hspace{0.17em}m}=P_r\left(C_1\le C_{out},\dots ,\hspace{0.17em}{C}_m\le C_{out}\right)=P_r\left(C_m\le C_{out}\right)$$

(22)

For example, in the case of M = 3 rounds then the optimization problem is the following:

$$\underset{P_{1,}{P}_{2,}{P}_3}{\arg \min }\hspace{0.17em}\hspace{0.17em}{P}_1+P_2\cdot \frac{1}{2}\mathrm{erfc}\left(-\frac{\ln \left(\frac{2^{\frac{C_{out}}{W}}-1}{\langle h_1\rangle P_1}\right)+0.5{\sigma }_{h_1}^2}{{\sigma }_{h_1}^{\hspace{0.17em}}\sqrt{2}}\right)+P_3\cdot P_r\left(\left(1+h_1{P}_1\right)\left(1+h_2{P}_2\right)\le 2^{\frac{C_{out}}{W}}\right)$$

(23)

$$P_{out,3}={\displaystyle \underset{h_1=0}{\overset{\frac{2^{\frac{C_{out}}{W}}-1}{P_1}}{\int }}{\displaystyle \underset{h_2=0}{\overset{\hspace{0.17em}\hspace{0.17em}\frac{2^{\frac{C_{out}}{W}}}{P_2\left(1+h_1{P}_1\right)}-\frac{1}{P_2}}{\int }}{\displaystyle \underset{h_3=0}{\overset{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{2^{\raisebox{1ex}{$C_{out}$}\!\left/ \!\raisebox{-1ex}{$W$}\right.}}{P_3\left(1+h_2{P}_2+h_1{P}_1+h_1{P}_1{h}_2{P}_2\right)}-\frac{1}{P_3}}{\int }}\frac{1}{h_1{h}_2{h}_3{\sigma }_{h_1}^{\hspace{0.17em}}{\sigma }_{h_2}^{\hspace{0.17em}}{\sigma }_{h_3}^{\hspace{0.17em}}\sqrt[3]{2\pi }}{\mathrm{e}}^{\left(\hspace{0.17em}-\mathrm{K}\left(h\right)\hspace{0.17em}\right)}}{\mathrm{dh}}_1}}{\mathrm{dh}}_2{\mathrm{dh}}_3$$

(24)

where K(h) is defined in (19).

The (16) and (22) are based on the fact that in chase combining and incremental redundancy, the channel capacity $C_m$ at the end of mth round is non-decreasing for all fading-sequence realizations because the packets are soft-combined [13,14]. That is: $C_1\le C_2\le \dots \le C_m$. Therefore, if $C_m\le C_{out}$ it means that all the previous rounds’ capacities are also less than $C_{out}.$

The outage probabilities in (18) and (24) are intractable and cannot be solved in closed-form, but even numerical computations are challenging for a large M. In this work, for the sake of simplicity, we will investigate the scenario with M = 3 rounds of re-transmission.

3.2. Power Allocation Methodology

From the aforementioned analysis in Section 3.1, it was observed that both the objective function and the constraints are non-linear and non-convex. The finding of a global minimum is then NP-hard because it may exist in many feasible regions and many local minima, so a global solution is very difficult to obtain. However, according to the Weierstrass theorem, if the objective function is continuous and the feasible region is closed and bounded, then there exists a global optimum [18,19]. All three HARQ power allocation subproblems satisfy these requirements, and the constraints can be plotted to determine if the feasible region is closed and bounded. The global minimum can always be found then with an exhaustive/brute-force search [13].

In spite of the lack of convexity, the Karush–Kuhn–Tucker conditions are necessary (but not sufficient) for $P^{*}$ to be an optimum solution to the problem. Specifically, for M = 3 rounds and the Lagrangian $\mathrm{L}(P_1^{*},P_2^{*},P_3^{*},{\lambda }^{*})$ [19]:

Stationarity:

$$\frac{\partial \mathrm{L}(P_1^{*},P_2^{*},P_3^{*},{\lambda }^{*})}{\partial P_1^{*}}=\frac{\partial \mathrm{L}(P_1^{*},P_2^{*},P_3^{*},{\lambda }^{*})}{\partial P_2^{*}}=\frac{\partial \mathrm{L}(P_1^{*},P_2^{*},P_3^{*},{\lambda }^{*})}{\partial P_3^{*}}=0$$

(25)

Primal feasibility:

$$0\le P_1^{*},P_2^{*},P_3^{*}\le \overline{P}\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}0\le P_r\left(C_3\le C_{out}\right)\le {\epsilon }_{\mathrm{target}}$$

(26)

Complementary slackness:

$${\lambda }^{*}\cdot \left(P_r\left(C_3\le C_{out}\right)-{\epsilon }_{\mathrm{target}}\hspace{0.17em}\right)=0$$

(27)

Dual feasibility:

$${\lambda }^{*}\ge 0$$

(28)

where ${\epsilon }_{\mathrm{target}}$ is the target outage probability from (7).

Besides the brute-force or exhaustive search, the standard methods for solving constrained non-linear optimization problems include interior-point (IP) methods, sequential quadratic programming (SQP) or even projected gradient descent (PGD) [18]. IP and SQP require the objective and constraint functions to be twice differentiable and exhibit polynomial time complexity for linear and non-linear problems, but the strong advantage of SQP lies in its property that the initial guess and the iteration steps do not need to be feasible points [18]. SQP is an active-set method that works in two stages: Firstly, the objective function is neglected, and a feasible point is obtained that satisfies the constraints. Secondly, the objective function is optimized while keeping the feasibility. In contrast, the IP iterations must stay inside the feasible region and avoid the infeasible region, but it yields better approximations, and it is scalable [18].

4. Numerical Results and Discussion

In this section, the optimal power allocation problem for an optical LEO-to-ground downlink with HARQ schemes is solved, and a variety of simulations are carried out from which conclusive results are drawn. In our work, the MATLAB software is employed to numerically obtain a very good approximation of the global minimum of the constrained non-linear optimization problems. Specifically, the IP and SQP methods are employed, and the outcomes are validated through a brute-force search. Moreover, the average power is minimized for Type I, CC, IR HARQ schemes, and the sensitivity to the scintillation index, average path loss and target outage probability is examined. It must be clarified that we are more focused on the proposed allocation methodologies and less on the transmission characteristics.

A M = 3 rounds HARQ protocol is considered, and an optical LEO-to-ground link is assumed under atmospheric path loss and weak scintillation conditions. The choice of M = 3 is supported by the fact that LEO satellites have a short contact time (~5 min); therefore it is realistic to assume a few HARQ rounds. The wavelength is set to λ = 1550 nm, the slew rate w_s = 0.001, the altitudes H_OGS = 2000 m and H_LEO = H_turb = 20 km and elevation angle e = 50° while the ground weather conditions, i.e., wind speed, $C_n^2\left(z\right)$, were varied to obtain the scintillation indices and the corresponding coherence times using the (1). The Bufton and Hufnagel–Valley models were employed for the wind and $C_n^2\left(z\right)$. From

Table 1. Scintillation index versus the channel coherence time for various weather conditions.

SI	τ0 (ms)
0.1	5.2
0.2	1.8
0.3	1.2
0.4	0.9
0.5	0.8

it is implied that the worse the scintillation effects the less coherence time is needed because the channel fluctuations are greater and rapid. Finally, the maximum transmitted power is set to $\overline{P}=1\hspace{0.17em}\mathrm{W}$ for all simulations.

In the first simulated scenario, the impact of SI on the average total power is evaluated for the three HARQ protocols. The target outage probability, the outage capacity-bandwidth ratio, and the optical channel statistics over the three retransmission rounds are given in

Table 2. First simulation’s input parameters for the optical channels.

${\mathit{\epsilon }}_{\mathbf{target}}$	${\mathit{C}}_{\mathit{o}\mathit{u}\mathit{t}}/\mathit{W}$	$\langle {\mathit{h}}_1\rangle$	$\langle {\mathit{h}}_2\rangle$	$\langle {\mathit{h}}_3\rangle$	${\mathit{\sigma }}_{{\mathit{h}}_1}^2$	${\mathit{\sigma }}_{{\mathit{h}}_2}^2$	${\mathit{\sigma }}_{{\mathit{h}}_3}^2$
0.01	0.5	1	1	1	0.01–0.5	0.01–0.5	0.01–0.5

, while it is assumed that SI₁ = SI₂ = SI₃. The proposed methodology is simulated using the IP and SQP algorithms and validated with a brute-force search. In Figure 2 the numerical results are given.

In Figure 2, it can be observed that the higher the SI the more power is consumed by all HARQ schemes with a rate of 12% (Type I), 6% (CC) and 5.2% (IR). Type I is the worst and results in the largest average total power and CC is moderate, while IR is the most energy-efficient HARQ protocol. At SI = 0.3, the average total powers of Type I and CC are 47.2% and 6.8% larger than IR, respectively.

In the second scenario of simulations, the impact of $\langle h\rangle$ on the average total power is evaluated for the three HARQ protocols. The target outage probability, the outage capacity-bandwidth ratio, and the optical channel statistics over the three retransmission rounds are given in

Table 3. Second simulation’s input parameters for the optical channel.

${\mathit{\epsilon }}_{\mathbf{target}}$	${\mathit{C}}_{\mathit{o}\mathit{u}\mathit{t}}/\mathit{W}$	$\langle {\mathit{h}}_1\rangle$	$\langle {\mathit{h}}_2\rangle$	$\langle {\mathit{h}}_3\rangle$	${\mathit{\sigma }}_{{\mathit{h}}_1}^2$	${\mathit{\sigma }}_{{\mathit{h}}_2}^2$	${\mathit{\sigma }}_{{\mathit{h}}_3}^2$
0.01	0.5	0.5–1	0.5–1	0.5–1	0.1	0.1	0.1

, while it is assumed that $\langle h_1\rangle =\langle h_2\rangle =\langle h_3\rangle$. The proposed methodology is simulated using the IP and SQP algorithms and validated with a brute-force search. In Figure 3 the numerical results are presented.

In Figure 3, it is shown that the higher the channel gain the less power is consumed by all HARQ schemes with a rate of 12.5% (Type I), 14.5% (CC) and 13.1% (IR). Type I is the worst and results in the largest average total power and CC is moderate, while IR is the most energy-efficient HARQ protocol. At $\langle h\rangle =0.8$, the average total powers of Type I and CC are 28.4% and 5.4% larger than IR, respectively.

In the third hypothetical simulation, the impact of ${\epsilon }_{\mathrm{target}}$ on the average total power is evaluated for the three HARQ protocols. The outage-capacity-to-bandwidth ratio, and the optical channel statistics over the three retransmission rounds, are given in

Table 4. Third simulation’s input parameters for the optical channel.

${\mathit{\epsilon }}_{\mathbf{target}}$	${\mathit{C}}_{\mathit{o}\mathit{u}\mathit{t}}/\mathit{W}$	$\langle {\mathit{h}}_1\rangle$	$\langle {\mathit{h}}_2\rangle$	$\langle {\mathit{h}}_3\rangle$	${\mathit{\sigma }}_{{\mathit{h}}_1}^2$	${\mathit{\sigma }}_{{\mathit{h}}_2}^2$	${\mathit{\sigma }}_{{\mathit{h}}_3}^2$
${10}^{-7}–{10}^{-1}$	0.5	1	1	1	0.2	0.2	0.2

. The proposed methodology is simulated using the IP and SQP algorithms and validated with a brute-force search. In Figure 4 the numerical results are given.

In Figure 4, it is observed that the lesser the ${\epsilon }_{\mathrm{target}}$, the more power is consumed by all HARQ schemes with a rate of 6.7% (Type I), 13.4% (CC) and 14.8% (IR). Type I is the worst and cannot achieve an $\hspace{0.17em}{\epsilon }_{\mathrm{target}}<1\times {10}^{-5}$ and CC is moderate and cannot achieve an ${\epsilon }_{\mathrm{target}}<1\times {10}^{-6}$, while IR is the most energy-efficient HARQ protocol, reaching a threshold of ${\epsilon }_{\mathrm{target}}=1\times {10}^{-7}$. At ${\epsilon }_{\mathrm{target}}=1\times {10}^{-5}$, the average total powers of Type I and CC are 27% and 6.7% larger than IR, respectively.

In the fourth simulation, the channel power distribution is evaluated among the three retransmission rounds. The outage-capacity-to-bandwidth ratio, and the optical channel statistics over the three HARQ rounds, are given in

Table 5. Fourth simulation’s input parameters for the optical channel.

Sim	${\mathit{\epsilon }}_{\mathbf{target}}$	${\mathit{C}}_{\mathit{o}\mathit{u}\mathit{t}}/\mathit{W}$	$\langle {\mathit{h}}_1\rangle$	$\langle {\mathit{h}}_2\rangle$	$\langle {\mathit{h}}_3\rangle$	${\mathit{\sigma }}_{{\mathit{h}}_1}^2$	${\mathit{\sigma }}_{{\mathit{h}}_2}^2$	${\mathit{\sigma }}_{{\mathit{h}}_3}^2$
(a)	${10}^{-7}–{10}^{-1}$	0.5	1	1	1	0.2	0.2	0.2
(b)	${10}^{-4}$	0.44	0.92	0.97	0.88	0.24	0.30	0.22

. The proposed methodology is simulated using the IP and SQP algorithms and validated with a brute-force search. In Figure 5a the numerical results are given only for the IR HARQ protocol, which shows the best performance. In Figure 5b the proposed methodologies are simulated for arbitrary input parameters given in Table 5. The average and allocated power are exhibited.

In Figure 5a, as the ${\epsilon }_{\mathrm{target}}$ gets smaller, it can be observed that the average total power is minimized by allocating the most power on the third round and the least power during the first round until 1 × 10⁻⁷ where maximum power is allocated over all three rounds. This is reasonable because from (23) the $P_3\cdot P_r\left(C_2\le C_{out}\right)$ yields a much smaller term than $P_2\cdot P_r\left(C_1\le C_{out}\right)$ or $P_1$; therefore, $P_3$ is allocated with maximum power first, followed by less power in $P_2$ and finally by the least power in $P_1$. In Figure 5b it can be observed that CC and IR perform much more efficiently than Type I, and in a similar way, by allocating more power on the first and third rounds. It must be mentioned that not all M re-transmissions are required; the proposed methodologies simply indicate the gradual increment of allocated power till the successful decoding or depletion of HARQ rounds [15].

Lastly, a comparison between IP and SQP algorithms has been implemented and shown for the previous case of IR HARQ. The stopping criteria, the termination tolerances of the constraints, the step sizes and the initial power vector $P^{\left(0\right)}$ are given in

Table 6. Optimization parameters of IP and SQP algorithms.

	Stopping Criterion	Termination Tolerance	Step Size	Initial Vector
IP	1000 iterations	1 × 10−6	1 × 10−10	$P^{\left(0\right)}={\left[0.5\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.5\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.5\right]}^T\mathrm{W}$
SQP	400 iterations	1 × 10−6	1 × 10−6	$P^{\left(0\right)}={\left[0.5\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.5\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.5\right]}^T\mathrm{W}$

. In

Table 7. Performance Parameters of Interior-Point Algorithm.

${\mathit{\epsilon }}_{\mathrm{target}}$	Iterations	Function Evaluations	1st Order Optimality	Constraint Violations
1 × 10−1	13	58	3.8 × 10−8	0
1 × 10−2	12	53	4 × 10−7	0
1 × 10−3	20	91	9.3 × 10−7	0
1 × 10−4	22	101	1.6 × 10−8	0
1 × 10−5	29	131	3.2 × 10−7	0
1 × 10−6	29	133	8 × 10−8	0
1 × 10−7	30	139	5.7 × 10−7	5.7 × 10−7

and

Table 8. Performance Parameters of Sequential Quadratic Programming Algorithm.

${\mathit{\epsilon }}_{\mathrm{target}}$	Iterations	Function Evaluations	1st Order Optimality	Constraint Violations
1 × 10−1	10	45	3.5 × 10−7	2.7 × 10−17
1 × 10−2	9	43	6.5 × 10−7	3.1 × 10−11
1 × 10−3	8	37	7.3 × 10−7	2.4 × 10−11
1 × 10−4	8	36	1.9 × 10−8	8.2 × 10−15
1 × 10−5	10	44	1.6 × 10−7	1.4 × 10−17
1 × 10−6	12	52	6.7 × 10−7	3 × 10−15
1 × 10−7	9	68	8.6 × 10−7	8.6 × 10−7

the algorithms’ performance parameters are reported.

Although the two algorithms (IP and SQP) yield the same numerical results in all the examined cases, as exhibited previously in Figure 1, Figure 2, Figure 3 and Figure 4, they require a different number of iterations and function evaluations. Additionally, they converge to a local minimum with different first-order optimality and constraint violations. The first-order optimality translates to the maximum absolute value (infinity norm) of the gradient of the Lagrangian, and ideally it should be zero at the minimum. The constraint violations refer to the nonlinear target outage probability constraints and ideally, they should be zero.

From Table 1 and Table 2 it is observed that IP algorithm requires approximately 3 times more iterations and function evaluations than SQP, especially at ${\mathsf{\epsilon }}_{\mathrm{target}}=1\times {10}^{-7}$, which means that IP is slower. This outcome is reasonable since IP by definition is required to stay inside the feasible region and bounds at all iterations, while SQP allows some constraint and bound violations. On the other hand, IP achieves better first-order optimality has almost no constraint violations, which makes IP a better approximation of the local minimum than SQP.

5. Conclusions

In this paper, optical HARQ-based LEO-to-ground downlinks under weak turbulence conditions are studied, and a power allocation methodology is proposed to ensure reliable and energy-efficient transmission. Specifically, to optimize the system’s energy efficiency and reliability, the average power consumption is minimized given a maximum transmitted power constraint and a target outage probability constraint. Assuming a finite number of rounds with temporal spacing between retransmissions of at least a channel’s coherence time, the optimization problem is constructed as a constrained nonlinear programming problem for the cases of Type I, CC and IR HARQ schemes. The optical channel’s statistics are required to obtain the optimal power allocation rather than the instantaneous channel gains. The solutions are derived numerically via iterative algorithms, namely IP and SQP, and validated through a brute-force search, i.e., a search of all points in the feasible region. Simulations are then executed to evaluate the proposed methodology and provide performance parameters of IP and SQP while also exhibiting the impact of SI, average path loss and target outage probability. The numerical results show that Type I HARQ yields the highest average total power, CC is moderate and IR yields the lowest average total power. The IP algorithm was found to be more accurate in finding the global solution because of its nearly zero first-order optimality and constraint violations, but SQP is faster due to the smaller number of iterations and function evaluations. For future work, the impact of correlation between the channels of the retransmission rounds can be investigated. The case of strong turbulence (SI > 1) for smaller elevation angles may also be studied. Finally, real experimental data from optical LEO-to-ground can be employed to evaluate the feasibility of HARQ schemes.