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Jognes Panasiewicz
email: jognes.panasiewicz@isae-supaero.fr

Nisrine Arab
email: nisrine.arab@isae-supaero.fr

Fabien Destic
email: fabien.destic@isae-supaero.fr

Gefeson M Pacheco
email: gpacheco@ita.br

Angélique Rissons
email: angelique.rissons@isae-supaero.fr

All-Digital Optical Phase-Locked Loop for satellite communications under Turbulence Effects

Keywords: atmospheric turbulence, coherent detection, FSO communications, optical digital phase-locked loop

This study proposes a new architecture for the phase detector in an all-digital Optical-Phase-Locked Loop (OPLL) to demodulate a digitally modulated optical signal from a longdistance Free Space Optics (FSO) communications system. The performance of the proposed architecture is evaluated and compared with an OPLL using an analog phase detector. In addition, a non-negligible delay is considered in the system analyses. Finally, the impact of wind speed on communication quality through the obtained Bit Error Rate (BER) from the recovered data is studied under three different atmospheric turbulence scenarios.

I. INTRODUCTION

FSO has recently emerged as a promising communications technology. The need to shift to optical carriers has emerged due to a significant increase in data and multimedia service demands and the congestion of the radiofrequency (RF) spectrum [START_REF] Kaushal | Optical Communication in Space: Challenges and Mitigation Techniques[END_REF]. Furthermore, FSO has numerous advantages over RF communication, including large modulation bandwidth, narrow beam divergence, high security, high directivity and no licensing requirement. However, optical signal propagation is affected by absorption, scattering, and changes in the atmospheric refractive index or turbulence, which causes optical signal attenuation and deteriorates the link quality [START_REF] Andrews | Laser Beam Propagation through Random Media[END_REF].

For long-distance FSO, such as satellite-to-ground links, as shown in Fig. 1, coherent optical communication is advantageous since it allows higher-order modulation formats and provides better signal sensitivity [START_REF] Winzer | Fiber-optic transmission and networking: the previous 20 and the next 20 years[END_REF]. Optical phase-locked loop (OPLL) or digital signal processing (DSP) are the two techniques used to recover the data in a coherent optical system. OPLL is generally used in inter-satellite links, while adaptive optics (AO) coupled with DSP techniques can be employed for satellite-to-ground links [START_REF] Carrasco-Casado | Space Optical Links for Communication Networks[END_REF]. Despite using digital control loops, the OPLL used in inter-satellite links uses analog phase detectors to provide phase error to the control loop [START_REF] Yue | Homodyne coherent optical receiver for intersatellite communication[END_REF], [START_REF] Liu | Study of multistage composite loop control based on optical phase-locked loop technology[END_REF]. However, the analog detector is unsuitable for systems subject to fading as its gain is related to the signal level and could render OPLL unstable. Therefore, taking advantage of OPLL and DSP techniques, such as real-time data recovery and flexibility of parameters reconfiguration, this study proposes a fully digital OPLL to demodulate a digitally modulated optical signal. Additionally, the optical signal propagating an LEO slant path shown in Fig. 1 will be considered. Thus, OPLL will use a digital phase detector to handle a fading signal resulting from the effects of atmospheric turbulence. The system, supported by simulations using VPIphotonics under a co-simulation technique, will be evaluated under three scenarios of atmospheric turbulence and several wind speed values.

II. COHERENT OPTICAL RECEIVER

A. LEO slant path

Fig. 1 shows a general overview of the implemented system and the power level diagram for the satellite downlink. First, the transmitter section performs the quadrature phase-shift keying (QPSK) with a bit rate of 20 Gbps. Next, as the data signal employs nonreturn-to-zero, an optical band-pass filter (OBPF) filters the undesired side lobes, thereby limiting the bandwidth of the transmitted modulated signal. The signal is then amplified by an Erbium-Doped Fiber Amplifier (EDFA) to reach the required optical power. After that, the optical signal propagating an LEO satellite slant path with a minimum elevation of 20 • and an altitude of 700 km is considered. Through these considerations, the received power is approximately -16 dBm, given a margin of 4 dB over the required receiver input power. The required power is based on the coherent detector characteristics as photodetector responsivity of 0.04 A/W. Finally, the transmitted data are demodulated at the Optical Ground Station through the proposed fully digital OPLL. Its performance is evaluated under different scenarios of atmospheric turbulence. The a scintl in the power level diagram corresponds to the attenuation induced by turbulence.

B. Turbulence model

The refractive index fluctuations caused by atmospheric turbulence induce harmful effects on an optical signal propagating through the atmosphere. Due to the non-linear turbulence characteristics, the refractive index structure C 2 n is used to characterize its effects [START_REF] Andrews | Laser Beam Propagation through Random Media[END_REF]. The parameter C 2 n can be considered a constant when dealing with propagation along a horizontal path. However, when studying the vertical links, C 2 n varies according to altitude h. Several profile models of C 2 n (h) exist to describe the turbulence profile for ground-tospace and space-to-ground applications. Among these models, one commonly used in the literature is the Hufnagel-Valley (H-V) model [START_REF] Andrews | Laser Beam Propagation through Random Media[END_REF].

For this model, the nominal value of C 2 n (0) at ground level will be considered with three different values of 1×10 -13 m -2/3 , 5×10 -14 m -2/3 , and 1×10 -15 m -2/3 , corresponding respectively to strong, moderate and weak turbulence.

In order to calculate the corresponding ground wind speed, the International Telecommunication Union ITU-RP.1621-1 recommendation is used through the expression v rms = v 2 g + 33.11v g + 360.31 , where v g is the ground wind speed. The simulated rms wind speeds v rms correspond to ground wind speeds ranging from zero to 43 m/s (zero to 154.8 km/h). For the rest of this article, we use the term "wind speed" to define "rms speed". Choosing this range of rms wind speeds allowed us to simulate the optical link under different weather conditions (zero m/s during a calm summer day, 154 km/h during a storm). Fig. 2 shows the profile of C 2 n (h) for different wind speeds for moderate turbulence. For different wind speeds values, the C 2 n (h) profile is modified only for an altitude between 5000 m and 20000 m according to the Bufton model [START_REF] Andrews | Laser Beam Propagation through Random Media[END_REF]. Note that the same behavior has also been obtained for weak and strong turbulence.

C. All-Digital OPLL description

The block diagram of the implemented coherent optical receiver used in the OGS of Fig. 1 is shown in Fig. 3. The trans-impedance amplifier (TIA) is used in the coherent receiver for loss compensation. The laser, the dual-parallel Mach-Zehnder modulator (DPMZM), and the VCO form an optical voltage-controlled oscillator (OVCO), enabling the the optical signal to be fine-tuned [START_REF] Ferrero | Optical Phase Locking techniques: An overview and a novel method based on Single Side Sub-Carrier modulation[END_REF]. The analog-to-digital converter (ADC) provides the interface of a continuous-time system to a discrete-time, and a digital-to-analog converter (DAC) carries out the opposite conversion. The OPLL loop filter (LF) is based on a first-order active filter transfer function that results in a second-order system when combined with the VCO integrator. The LF transfer function is given by F (z) = K 1 +K 2 /(z-1), where K 1 , and K 2 are the loop filter gains. In addition to the LF, a signal-to-noise (SNR) estimator was also implemented to determine the noise variations on the demodulated signal and the OPLL. The SNR algorithm is based on the second and fourth-order moments estimator, M2M4 [START_REF] Pauluzzi | A comparison of SNR estimation techniques for the AWGN channel[END_REF].

The green dashed block drawn in Fig. 3 shows the phase detector's block diagram. It provides phase error information from a tracked carrier signal. However, a digitally modulated signal does not have a carrier since modulation suppresses it. Therefore, according to the modulation scheme, a different circuit is employed to remove the modulation signal, resulting in an unmodulated carrier to track. In the QPSK format, for example, a circuit multiplied by four is used to generate an error signal proportional to K D sin 4θ [START_REF] Schaefer | Coherent receiver design based on digital signal processing in optical high-speed intersatellite links with M-phase-shift keying[END_REF]. A circuit consisting of logic gates could achieve this multiplication, thus obtaining

K D = R 4
T IA R 4 P 2 s P 2 LO /4. R T IA is the trans-impedance gain, R is the responsivity, P s is the modulated signal power, and P LO is the power of the local oscillator (LO).

In a digital phase detector, the imaginary part of the fourth-power (QPSK) of the sample I/Q signals have the same gain and error signal of a circuit multiplication, where

Im[X M k ] = Im[(I +Q) M ]/M = K D sin(M θ).
M represents the modulation order. For QPSK M = 2 b = 4, where b is the number of bits/symbol. Previously, an all-digital OPLL was demonstrated using a lookup table to implement the fourth power [START_REF] Lu | Digital-analog hybrid optical phase-lock loop for optical quadrature phase-shift keying[END_REF]. However, the gain K D is related to the input signal level P s that changes the projected control loop behavior rendering the OPLL unstable. Additionally, it uses the approximation of sin(θ) ≈ θ, for small θ, as a phase error signal. Therefore, a Moving Averaging Filter (MAF) followed by the four quadrant arc-tangent function is applied to the power moment of the sampled signals. Then, the estimated phase error is the following:

Θk = 1 M arg { k+L n=k+1 [X k ] M }
, thereby obtaining the phase error without approximation.

This technique is based on the Viterbi-Viterbi algorithm used in DSPs for coherent optical systems. However, it was modified to correct the phase sample-by-sample according to the OPLL sampling rate instead of data blocks as DSPs. The MAF is a particular example of a finite impulse response filter, where all coefficients are equal to 1/L, where L is the average number of samples. Employing it is essential since the drawback of the power moment is the presence of harmonics at high frequencies at its output, increasing the OPLL phase error variance.

III. SYSTEM IMPLEMENTATION AND ANALYSIS

A. OPLL

The OPLL was developed and evaluated through a cosimulation technique. The photonics circuit was implemented in the VPIphotonics Design Suite, while the OPLL algorithm and SNR estimation were implemented in Python. The coherent detector parameters are based on a commercial component whose responsivity R = 0.04 A/W and the R T IA gain was set to 9 kΩ. The P LO optical power was 1 dBm, while P s changed according to the turbulence effects. For all analyses, the Tx and Rx lasers have a linewidth of 1 kHz, and the VCO has a central frequency of 20 GHz with a gain of 500 MHz/V.

The propagation delay was also considered since it can affect the OPLL, rendering it unstable. The MAF represents a delay in the control loop, where L taps introduce a delay of L samples. Additionally, for the ADCs and DAC, a delay of 12 ns was considered based on the commercial components [START_REF] Chen | Z-domain modeling methodology for homodyne digital optical phase-locked loop[END_REF]. While for the atan2, a delay of 15 ns is based on techniques for hardware implementation [START_REF] De Dinechin | Hardware Implementations of Fixed-Point Atan2[END_REF]. The OPLL sampling rate is 10 Gsamples/s (sps), which is motivated by the availability of ADCs, DACs, and even DSPs that can operate at 10 GHz, allowing the OPLL to collaborate with the DSP in a correction system. As the MAF will have 10 taps, it will introduce a delay of 1 ns.

The LF parameters were determined by optimizing the B n parameter (OPLL noise bandwidth) for a total loop delay of 30 ns due to the previously discussed system characteristics. Thus, several values of the B n parameter started based on Norimatisu's relation for OPLLs with delay [START_REF] Norimatsu | PLL propagation delay-time influence on linewidth requirements of optical PSK homodyne detection[END_REF], a power penalty of 0.42 dB, and a standard deviation of the phase error of 2.6 • was found for a B n = 2 MHz. By using the bilinear transformation, the LF parameters were determined through a B n T s = 2×10 -4 for 10 Gsps, where T s is the sample time.

B. Results

The OPLL was evaluated through variations of the turbulence channel. The proposed digital discriminator was compared with the imaginary part of the fourth-power moment in Fig. 5 for a strong turbulence regime and a wind speed of 60 m/s. Each color corresponds to the measured SNR, as shown in Fig. 4 for strong turbulence, representing the fade values where the system is analyzed for thirty different values or thirty iterations.

Fig. 5a and5b show the VCO output frequency behavior using the digital discriminator (atan2) and the imaginary part (sine), respectively. The OPLL using the atan2 discriminator was analyzed under a delay of 30 ns while the sine phase detector was under a total delay of 6 ns since it became unstable for delays longer than 6 ns. Compared to the atan2, the sine discriminator exhibits a higher oscillatory characteristic and a higher standard deviation (STD) of the controlled frequency. Additionally, the OPLL loses the lock during higher fading, unable to recover the track signal, as shown by the deviation of the central frequency. The STD variation is related to the sine discriminator gain since, as shown previously, it depends on the system parameters. As the input power level changes due to the turbulence, the gain also changes, reflecting the OPLL phase error variance and the controlled frequency STD. The proposed digital discriminator used the same gain of 1 for all turbulence scenarios. The atan2 has the advantage of being independent of the system parameters since the computation of the phase error relates to the I/Q signals. Consequently, it was possible to analyze the system under different turbulence scenarios without changing the OPLL project. Additionally, it shows a low STD of the controlled frequency as seen in Fig. 5a. One of the most notable features of the atan2 discriminator is its ability to keep tracking the signal under longer feedback loop delays than the sinusoidal discriminator. Therefore, the atan2 performance under noise and its tracking ability makes it an alternative for digital OPLL. Both discriminators and wind speed values were also compared in other turbulence scenarios. The sinusoidal discriminator always loses lock during fading regime while the proposed discriminator keeps the system locked. The only condition where the sinusoidal discriminator maintained locking was when the system delay was equal to zero, i.e., an unrealistic condition.

Due to the tracking capability of the proposed phase detector under fading regimes, the impact of wind speed on system performance was also studied. Fig. 6 shows the BER variation according to wind speed in three different turbulent scenarios (weak, moderate, and strong). Each point in Fig. 6 corresponds to the mean BER calculated for thirty different events. It was verified that the results are representative by simulating only thirty events. Therefore, the error due to the number of simulated events can be negligible. The thirty events were selected as a compromise between results and simulation time. We notice that for wind speeds lower than 20 m/s, turbulence intensity contributes to the degradation of the transmitted data. However, the influence of turbulence decreases for wind speeds higher than 20 m/s. As seen in Fig. 6, when the wind speed is at 60 m/s, the BER is mainly impacted by wind speed for all the turbulent media. In addition to the quality of the transmitted data, we notice that the quality's stability degrades according to wind speed.

IV. CONCLUSIONS

An all-digital OPLL was presented using a new phase discriminator method, which can track an optical received signal subject to the turbulence effects. The phase discriminator is a digital type that proved capable of tracking a signal under longer feedback loop delays and fading signals. Additionally, it showed a low standard deviation of the controlled frequency and low phase noise, making the digital phase discriminator an alternative for digital OPLLs. Thanks to its characteristics, it was possible to analyze OPLL performance in several scenarios of atmospheric turbulence and wind speed values. It has been seen that the system deteriorates with higher wind speed and strong turbulence. For wind speeds higher than 30 m/s, wind is the leading cause of system degradation. During the simulations, it was observed that the scintillation index behavior is comparable to the obtained BER profiles. Hence, the observed behavior is not caused by system limitations. For better comprehension, further studies will need to be conducted. Future work is underway to implement the hardware of the proposed OPLL.

Fig. 1 .

 1 Fig. 1. Power level diagram for the FSO downlink. G T and G R are the transmitter and receiver telescope gains, respectively, and EIRP stands for Effective Isotropic Radiated Power.

Fig. 2 .

 2 Fig. 2. C 2 n profile according to altitude.

Fig. 3 .

 3 Fig. 3. Block diagram of the implemented all-digital OPLL.

Fig. 4 .

 4 Fig. 4. SNR level variations for wind speeds of 5 m/s and 60 m/s under strong turbulence.

Fig. 5 .

 5 Fig. 5. VCO output frequency behavior under strong turbulence and wind speed of 60 m/s. a) OPLL using the proposed digital phase detector, b) OPLL using the sinusoidal discriminator.

Fig. 6 .

 6 Fig.6. BER variation (mean of 30 events) as a function of wind speed.

ACKNOWLEDGMENT

This work was supported by the National Institute for Space Research (INPE/Brazil), Conselho Nacional de Desenvolvimento Clientfico e Technolgico (CNPq/Brazil), and the French Innovation Defense Agency (AID) through the EPLO ("Emulateur de Propagation Libre Optique").