1.

INTRODUCTION

The adoption of satellites as a communication medium for multimedia connectivity systems requires optimization of the cost per bit transmitted in order to remain competitive with terrestrial solutions. For systems dedicated to broadband connectivity, increasing the total capacity of the system is a natural solution to achieve this objective. In this context, Free Space Optical (FSO) Communications on feeder links is a promising solution for the future generation of Very High Throughput Satellites^1–3 in geosynchronous orbit (GEO). Besides increasing the capacity, FSO payloads may offer many advantages in comparison to their RF counterparts like (reduced number of gateways, spectrum availability and regulation, lower size and weight of the satellite’s feeder section…). However, these come at the expense of new challenges such as the fact that optical links through the atmosphere are subject to more adverse propagation effects. These later are notably due to the propagation channel (clouds, aerosols, optical atmospheric turbulence… that are site location-dependant) on one hand and due to the optical front-end design (from the optical head to the specific photonic sub-systems) on the other. Obtaining accurate analytical and developing end-to-end simulators or models relevant to satellite-to-ground optical propagation channel alone is very challenging even if some approximations were proposed in the literature (cf⁴ and therein references).

To the authors knowledge, the most reliable models that are used currently to assess optical feeder link design and analysis are rather based on times series (TS). TS represents the varying received optical power during time and can be obtained either via real ground-onboard transmissions using a simplified architecture⁵⁶ or via proprietary optical channel simulators configured with the right physical parameters.^{4, 7} Relevant and representative experiments are not widely spread as GEO feeder telecommunications through the atmosphere are still at early stages. Accurate simulated data able to cover a wide range of relevant propagation parameters are costly to generate as they often rely on complex wave-optics coupled to end-to-end mitigation systems simulators. They often require an non-negligible amount of simulation time (depending on the configuration) in order to obtain a-few-seconds-long time series.

In this paper, we address the case where we have a relevant but short TS representation of a given ground-to-GEO transmission scenario. Often, system engineers aiming at conducting end-to-end performance analyses and optimizations require longer series with different statistical realizations. This enables Monte Carlo system analyses used for the reliable prediction of fading and inter-fading durations used to evaluate error correction code performance, modem synchronization thresholds and re-synchronization delays, channel interleaver design and their impact on service availability. We propose a framework that allows to generate, from a given TS (either acquired using actual transmissions or generated with advanced optical channel simulators), new TS series of any length on the fly. Our goal is not to generate TS that represent more reliably the aforementioned optical transmission effects, but rather that assist analysing the system performance (synchronization, error correction code, interleaving …) and evaluating equipment behavior for free-space laser propagation through atmospheric turbulence. To this aim, we propose a two-fold framework: first using a distribution fitting procedure to approach the statistical gain of the atmospheric optical channel and second using a discrete Markov state machine to model the time correlation of the TS.

This paper is organized as follows. We introduce our system architecture and the underlying TS. Then, our proposed framework to model and generate such time series is described. And finally, the obtained results are presented and discussed.

2.

SYSTEM DESCRIPTION

The optical feeder link considered in this paper implements a half optical link from the optical ground station (OGS) to the satellite followed by a half radio-frequency (RF) link from the satellite to the end users. This architecture is depicted in Fig. 1.

Figure 1:

Considered optical feeder link scenario

The end-to-end performance of the physical layer in this case is complex to simulate in comparison to a conventional RF transmission. In particular, it is essential to know the impact of the optical half-link on the overall performance in order to reliably dimension the optical link. Optical links through the atmosphere are subject to different propagation effects than RF links. The optical spectrum that is envisioned is located primarily in the C-band [1530-1565 nm] and in the L-band [1565-1625 nm]. At such wavelengths, the optical propagation phenomena in the atmosphere can be divided into two main classes:

These impairments can be very complicated to model analytically because (i) macro-scale phenomena depends on various physical and chemical parameters and (ii) optical turbulence theory coupled to mitigation techniques such as adaptive optics can rarely be captured by simple analytical frameworks. Consequently and in order to assess the impact of the propagation channel during design phase, we generally resort to simulated TS to evaluate the optical communication system (OCS) performance. Such TS can represent, for instance, the coupled optical flux in a single mode component characterizing the input of receiving OCS. The stochastic nature of such variations depends on various parameters such as the characteristic turbulence and wind profiles, adaptive optics (AO) correction, telescope diameter and link elevation. Figure 2 plots an example of a downlink coupled optical flux for a given set of system parameters. *.

Figure 2:

An example of a TS representing the optical flux received at the optical ground station from a GEO satellite

In order to evaluate the performance of an end-to-end transmission scheme, one would like to have sufficiently long (and thus statistically representative) time series. This makes the use of these TS generators on the fly not convenient since they must run during a long time in order to generate a-few-seconds-long TS.

A framework which allows to generate, from a parent TS and on the fly, similar TS of independent realizations is detailed in the following section.

3.

PROPOSED FRAMEWORK

As mentioned above, our goal is not to reproduce a time series as finely as possible, but rather to generate time series with different statistical realizations than the original one. To this aim, we mainly focus on the attenuation variations which directly impact the levels of the signal-to-noise ratio at the reception and the digital signal processing algorithms performance (clock and carrier synchronization threshold, forward-error correction (FEC) decoding performance…). The impact of these correlated attenuation can be reduced to two main characteristics:

Consequenlty, in order to generate a new TS from a parent TS, we implement the following two steps:

3.1

Distribution fitting

Optical power attenuation induced by optical atmospheric turbulence are widely modeled using various distributions such as the log-normal, the exponential or the gamma-gamma distribution.^9–11

Without loss of generality, we focus hereafter on the TS illustrated in Fig. 2. The corresponding PDF is shown in Fig. 3a and was found to be very well fit using a log-normal PDF with mean μ = 1.255 and variance σ = 0.399. Also, to estimate the quality of service of the system in terms of availability, we may also consider the cumulative density function (CDF). The original CDF and the obtained one are also plotted in Fig. 3b.

Figure 3:

PDF and CDF of the parent TS (blue histogram) and the corresponding log-normal distribution fitting (red line). The values of the TS points were converted to corresponding positive values for a natural distribution fitting

It is worth noting that this distribution fitting can be further improved considering other more specific distributions or mixture of distributions (as for instance¹²). However, in this case, log-normal distribution was found to present satisfying end-to-end performance analysis results.

3.2

Time correlation

The TS depicted in Fig. 2 presents samples that are correlated in time. This phenomenon is of high interest for the system analysis as it defines the fading and inter-fading statistics. The fading, defined as the received power threshold below which the receiver is not able to retrieve data, can last from a fraction of millisecond to longer than 10ms for GEO-to-ground scenarios. Consequently, repeated and long fading will cause frequent transmission interruptions.

To model this behavior, we consider discrete Markov chains. A discrete Markov chain is a stochastic process that consists of a finite number of states and transition probabilities to go from one state to the other. In this paper, we consider only first-order Markov chain processes in which the conditional probability for getting to a given state depends only on the current state of the system. One of the most simple and well studied Markov chain models in digital communications is the Gilbert-Elliot (GE) model.¹³ It is a time varying channel defined by two states: each state is defined by a binary symmetric channel (BSC) with a given crossover probability. The two states are commonly designated 𝓖 for a good channel (i.e. low crossover probability) and 𝓑 for a bad channel (i.e. a higher crossover probability).

Unlike the GE model which is on the bit level, we would like to have a Markov chain (MC) on the analog signal domain and having more than 2 states: each state is defined by fading within a given range of amplitudes. Let us suppose that our TS presents receiver optical powers (or equivalently channel attenuation) ranging between a maximum value 𝓣_max and an minimum value 𝓣_min. Let us define the MC states following a partition of the interval [𝓣_min, 𝓣_max] whose boundaries are given by the real sequence 𝓣_min = 𝓣₀ ≤ 𝓣₁ ≤ … ≤ 𝓣_i ≤ … ≤ 𝓣_n = 𝓣_max. In other terms, the state σ_i is defined as the set of amplitudes laying within the interval]𝓣_i, 𝓣_i+1].

Therefore, as for the GE channel and depending on the state in which our TS generator is, it will output a random received powers comprised between the boundaries of the interval that defines the current state. In this paper, we consider only uniformly distributed sampling. Thus, during calibration phase using the parent TS, it is worth mentioning that there are two main features to optimize in this MC-based generator: the boundaries spacing {𝓣_i}_i and the total number of states n.

The boundaries {𝓣_i}_i can be chosen to be distributed linearly, logarithmically or following any other spacing strategy. Figure 4 illustrates the linear spacing strategy where the boundaries are equidistributed in the interval [𝓣_min, 𝓣_max]. The amplitudes in the bottom-most interval, here [–23.28, –21.30] in Fig. 4a is defined as the worst channel state or σ₀.

Figure 4:

Linear spacing strategy showing the boundaries values {𝓣_i}_i

Figure 5:

Transition matrix learnt from the parent TS

Regarding the transition matrix, if the number of states is too low, for instance 2 as in the GE model, the generated TS will exhibit few bins (2 for GE) in which the amplitudes are going to be generated uniformly. If the number of states is too high, the probabilities in the transition matrix will be too close to each other, at least in the high amplitudes region, and the original TS may be not sufficiently long to allow a good estimation of the transition matrix. Figures 5a and 5b illustrates the transition matrices corresponding to the TS and the state partitions described in Fig. 4.

We observe that due to the time correlation of the TS, the highest probabilities occur on the diagonal of the matrix. This means that, if the current channel state is known, it is most probable that the next attenuation will be within the same or an adjacent amplitude range. Moreover, the matrix become sparse in the bad channel states region (the up region of the transition matrix), which means that once we visit a bad state, we tend to jump to a better state with high probability. A behavior that is explained by the fact that the deep fading regions are steep and sparse in the time domain.

Figure 6:

An example of a generated TS

Figure 7:

Fading and interfading probability of occurrence of the parent and the generated TS for a system communication threshold of –8dB. Parent TS in blue and generated TS in red

4.

RESULTS

Several simulations show that the above estimated log-normal distribution coupled with a MC of 50 states and a linearly spaced boundaries give satisfactory results to approximate the TS depicted in Fig. 2. Figure 6 shows a realization of a random TS generated from our proposed framework.

From a system performance point of view, we would like to compare fading statistics between the two series. Or more precisely, the fading and the inter-fading durations. As an example, suppose that we designed a communication chain that operates when the received coupled optical power is above –8dB. This thus defines (i) the fading period as the durations where the received power drops bellow this threshold and (ii) the interfading period where the received power is above this threshold. In Figures 7a and 7b we plot the probability mass function of different fading and interfading durations as obtained from the parent (blue) and the generated (red) TS. As one can observe, the probability mass functions are very close are are actually shown to give same performance at the system level.

5.

CONCLUSIONS

In this work, a simple framework was proposed to generate on the fly FSO channel time series similar to a parent TS obtained either from specialized simulation tools or real FSO transmissions. An approach combining a distribution fitting and a Markov chain state machine was proposed. First, the distribution fitting and the optimisation of the MC parameters (number of states and the transition matrix) is preformed using the parent TS. Then, once calibrated, the proposed model can be used to generate new samples according to the current state of the channel and the overall distribution. A comparison between the parent and the generated TS were compared in terms of average fading and inter-fading durations. Future work will consider other comparison criteria such as higher order moments or other types of statistical/temporal distances/divergences. Further improvements can be investigated to generalize our proposed framework to various FSO channel models such as considering higher order Markov chains.