Raphaël Le Bidan
email: raphael.lebidan@imt-atlantique.fr

Hugo Meric
email: hugo.meric@cnes.fr

Julien Sommer
email: julien.sommer@cnes.fr

Frame format and DSP receiver design for a 56-GBaud GEO DP-QPSK coherent optical feeder link

Keywords: GEO feeder link, coherent detection, digital signal processing, synchronization, equalization Coherent TX X-POL DAC

This paper addresses the design of a custom digital signal processing receiver and companion frame format for a 56 GBaud coherent optical Dual-Polarization Quadrature Phase-Shift Keying GEO feeder link. The selected algorithms combine standard practice from both coherent optical fiber and RF satellite communications. Robust blind algorithms are used in the first stages, followed by data-aided algorithms for subsequent fine carrier and phase recovery. A simplified blind equalizer architecture is proposed for fast and robust polarization demultiplexing, based on a modified constant-modulus algorithm. Our simulations demonstrate the possibility to establish links at an electrical SNR as low as -6 dB, in the presence of realistic values of system impairments, using less than 5% signaling overhead.

I. INTRODUCTION

Satellite communications systems require to reduce the overall cost per bit in order to remain competitive with terrestrial solutions and to offer affordable high-speed Internet services. Replacing or complementing radio-frequency (RF) feeder links by optical links is regarded as a promising solution to cope with the increasing RF bandwidth scarcity problem. Free-space optical (FSO) communications are expected to deliver substantially higher data rates than RF-based solutions, by an order of magnitude at least, with less interference, and comparable onboard terminal footprint [START_REF] Roy | Optical feeder links for high throughput satellites[END_REF], [START_REF] Mata Calvo | Optical feeder links for very high throughput satellites -System perspectives[END_REF].

In 2018, the French space agency CNES initiated a study to analyze the technical and economic feasibility of optical feeder links for geostationary (GEO) satellites. After evaluating several trade-offs (telescope size, waveform, etc.) from a system point of view, a series of technical recommendations were issued, accompanied by a roadmap to develop the corresponding technological building blocks: optical amplifiers, multiplexers, modems, etc. A coherent 56-GBaud dual-polarization QPSK (DP-QPSK) waveform was selected at the physical layer, as a good compromise between achievable throughput, increased receiver sensitivity, expected robustness against atmospheric turbulence, and long-term evolution capabilities. We note that discussions are currently underway at the CCSDS Optical Communications Working Group to incorporate coherent signaling in a revision of the Blue Books.

In principle, the use of 1550 nm lasers makes it possible to leveraging some of the commercial off-the-shelf (COTS) optical components and 100G/400G coherent transceiver subsystems developed for terrestrial fiber-optic networks. The problem is that they are designed to cope with different impairments than those encountered in space optical links. COTS coherent transceivers are mainly interference-limited. They are intended to operate at high signal-to-noise ratio (SNR), and a large part of the baseband digital-signal-processing (DSP) receiver is dedicated to the compensation of the interference introduced by linear (chromatic dispersion and polarizationmode dispersion) and non-linear fiber impairments. On the other hand, space optical communications are primarily powerlimited. The development of high-power optical amplifiers for long-distance GEO links through the atmosphere proves to be challenging, and the modem has to be able to lock at very low electrical SNR (< 0 dB). This raises the question of the extent to which it is possible to re-use the signal formats and DSP algorithms developed for coherent fiberoptic communications in such conditions. Recently, the authors of [START_REF] Schieler | On-orbit demonstration of 200-Gbps laser communication downlink from the TBIRD CubeSat[END_REF] have successfully demonstrated 100-Gbps and 200-Gbps laser LEO communication downlinks, based on COTS DP-QPSK transceivers supplemented by an automatic repeat request (ARQ) protocol to mitigate the effects of atmospheric turbulence. While very cost effective in the LEO setting, this approach does not come without limitations for high-capacity coherent optical GEO feeder links, which may have much to benefit from a custom receiver architecture [START_REF] Conroy | Demonstration of 40 Gbaud intradyne transmission through worst-case atmospheric turbulence conditions for geostationary satellite uplink[END_REF], [START_REF] Geisler | Coherent architectures for free-space optical communications[END_REF].

This paper reports on a second study supported by the CNES, that addressed the problem of designing a dedicated frame format and DSP receiver for the selected 56-GBaud DP-QPSK GEO feeder scenario. It is organized as follows. Section II provides an overview of the communication link model, performance requirements, and design goals of the project. Section III details the proposed frame format and receiver architecture. The selected DSP algorithms are reviewed in Section IV. Simulation results are presented in Section V.

II. LINK MODEL AND DESIGN REQUIREMENTS A. Transmission scenario

The communication link under consideration is depicted in Fig. 1. Transmission uses DP-QPSK signaling at symbol rate R s = 56 GBaud, with a square-root raised cosine pulseshape having a very small roll-off factor α = 0.1. Prior section 8.2 due to its complexity and importance in space lasercom. Other fundamentals concepts specific of free-space links such as modulation, coding, sensitivity, etc. are introduced in this section, as well as a link-budget calculation as the basic design tool in any lasercom system.

Brief historical overview

While optical-fiber communications experienced a huge growth in the 1970s, the effects of the atmosphere on the optical signals made the development of FSOC slow down for two decades. In 1992, NASA carried out the first laser transmission to space with the GOPEX project, by emitting 532-nm pulses with a power of MW from the Earth and detecting them with a camera onboard Galileo probe up to 6 million km [START_REF] Roy | Optical feeder links for high throughput satellites[END_REF]. In 1994, NICT carried out the first demonstration of a space-to-ground communication downlink by using the Japanese geostationary satellite ETS-VI [START_REF] Mata Calvo | Optical feeder links for very high throughput satellites -System perspectives[END_REF]. After 2000s, there was Alberto Carrasco-Casado, Ramon Mata-Calvo "Free-space optical links for space communication networks" https://doi.org/10.1007/978-3-030-16250-4_34

Free-space optical links for space communication networks

In recent years, wireless communications have witnessed an unprecedented explosion. From cellular networks to satellite links, unguided telecommunications have enabled countless new services, firmly establishing as a basic part of the current information society. In particular, Free-Space Optical Communication (FSOC), despite its recent emergence, provides a number of advantages that allow the materialization of completely-new applications such as quantum communications, as well as the promise to revolutionize traditional applications like satellite communications. FSOC can be applied in a wide variety of scenarios, from crosslinks to up-and-downlinks between satellites, aircraft, ships, and ground standing or mobile terminals. This chapter focuses on communication links where one of the terminals is in space, and it is organized as follows: Section 8.1 gives a quick overview of key concepts of FSOC that help understand how this technology is used to design lasercom links. Section 8.2 describes how the laser-communication signals are affected as they propagate through the atmosphere. Sections 8.3 and 8.4 explain the two applications with more potential, each of them having unique characteristics and advantages. Finally, section 8.5 provides an insight of what optical satellite networks may progress towards in the future.

Principles of free-space optical communication

This section reviews the most fundamental parameters in a space laser communication (lasercom) link.

Basically, any space lasercom system will encode some information on a laser beam, collimate and transmit it by means of a telescope, and, after been propagated through free space, it will be collected in other distant telescope and focused on a small spot in the focal plane, where a photodetector will transform the optical signal into an electrical one, which will be decoded to extract the original information. A typical space-lasercom link (see Fig. 8.1) exhibits three different types of impairments, representative of its free-space nature, i.e. geometrical and pointing losses, degradation of the SNR (Signal-to-Noise Ratio) with the background noise, and losses and perturbation of the received signals due to atmospheric effects. This section reviews the first two types, leaving the atmospheric effect to section 8.2 due to its complexity and importance in space lasercom. Other fundamentals concepts specific of free-space links such as modulation, coding, sensitivity, etc. are introduced in this section, as well as a link-budget calculation as the basic design tool in any lasercom system. system studies conducted by the CNES concluded that, apart from atmospheric turbulence, the main impairments to be considered or addressed in the digital receiver design were: • timing drift in the analog-to-digital converters (ADCs) A discrete-time over-sampled channel model of the link has been developed, capturing all the above-mentioned impairments. Following standard practice, we used a Wiener phase noise model for the laser phase noise, and a white gaussian noise model for ASE noise. Atmospheric turbulence was not explicitly simulated in this work, as it fluctuates at a time scale much larger than the frame duration, and the fading caused by scintillation cannot be compensated by DSP algorithms alone, but rather through adaptive optics, forward-error-correction (FEC) coding and interleaving, or other forms of diversity.

B. Performance requirements

The following performance requirements were specified:

• maximum X-Y skew tolerance of 0.25 ns;

• peak PDL tolerance of 5 dB;

• maximum cumulated -3dB bandwidth ∆ν = 600 kHz for the laser phase noise; • maximum CFO of ±5 GHz;

• tolerance of timing clock drift up to ±30 ppm;

• minimum operating electrical SNR E s /N 0 = -6 dB The large X-Y polarization time skew arising from the use of distinct transmit amplifiers for each polarization, which may span up to 15 symbol periods, is a notable difference with fiber-optic communications. Another one is the minimum electrical SNR of -6 dB, which translates into a received optical power in the range -60 ≤ ROP ≤ -55 dBm for typical optical receiver front-ends. Our discretetime equivalent channel model does not simulate atmospheric turbulence. Yet scintillation is tacitly taken into account in this minimum operating SNR level, which includes sufficient margin to ensure reliable communication at the desired average throughput, with the envisioned FEC code and interleaver, in conditions of turbulence that are typical of GEO links.

C. Design goals and constraints

The present work had two main objectives: i) to propose a frame format and a comprehensive DSP receiver for the transmission scenario of Fig. 1, and ii) to evaluate its performance against the specified performance requirements. The proposed DSP algorithms should have low latency and be amenable to parallel processing in order to comply with the 56 GBaud symbol rate constraint. Other practical design constraints included a total overhead for the training sequences and pilot symbols limited to 5% of the symbol rate at most, and DSP receiver processing at no more than 2 samples per symbol, to comply with existing 128-Gs/s commercial ADCs.

The driving motivation behind this work was to assess to what extent can one re-use in outer space the coherent DSP algorithms and chips developed either for high-performance, long-haul optical fiber communications [START_REF] Torbatian | Performance oriented DSP for flexible long haul coherent transmission[END_REF] or for low-power, short-reach data center optical interconnects [START_REF] Nagarajan | Low power DSP-based transceivers for data center Ootical fiber communications[END_REF].

III. DSP FRAME FORMAT AND RECEIVER ARCHITECTURE A. Design methodology

We first compared the frame format and baseline receiver architecture for a popular RF satellite communication standard, DVB-S2/S2X [START_REF]Second Generation Framing Structure, Channel Coding and Modulation Systems for Broadcasting, Interactive Services, News Gathering and other Broadband Satellite Applications, ETSI Std[END_REF], and for the only public coherent fiber-optic standard we are aware of, namely the OIF 400ZR agreement for interoperable data center interconnect [START_REF]Implementation Agreement 400ZR, Optical Internetworking Forum (OIF) Std[END_REF]. 400ZR is designed for low-power short-reach coherent communication at a symbol rate of nearly 60 GBaud, in the presence of a limited amount of fiber dispersion, 120 km at worse, which comes close to our scenario in several aspects. The goal of the comparison was to highlight the convergence points as well as the major differences. RF satellite receivers are not designed to cope with the polarization and other impairments, e.g., laser phase noise, specific to coherent optical transmissions (although cross-polarization equalization can be required also on certain RF channels). On the other hand, the 400ZR DSP

Symbol

Det.

Symbol

Det. r X,I (n) r X,Q (n) r Y,I (n) r Y,Q (n) Carrier Phase Recovery b Y (n) b X (n)

Sample-level processing Symbol-level processing

To FEC X-Pol

To FEC Y-Pol

Bit-level processing frame and receiver algorithms does not allow for reliable frame acquisition and carrier/phase recovery at the very low SNRs that can be encountered in space optical links. This led us to define a custom frame format and an entire DSP receiver architecture, FEC excluded, that could meet all the constraints and performance requirements of the envisioned GEO link.

B. Frame format

The proposed DSP frame format is depicted in Fig. 3. It achieves a synthesis between DVB-S2 and 400ZR framing. Each frame starts with a known preamble (frame header) of size H symbols, followed by I consecutive packets. Each packet consists of D payload symbols followed by a block of P pilot symbols. Similar to 400ZR, the X and Y polarizations use distinct frame headers and pilot symbol sequences.

As will be demonstrated shortly after, good synchronization and transmission performance can be achieved by choosing H = 64, D = 248, and P = 8. Such a choice has the additional benefit of providing header and packet sizes that are powers of two, which can substantially facilitate parallel processing at the receiver side, e.g. through block frequencydomain filtering. The pilots block size P = 8 and spacing D = 248 result from a trade-off between estimation accuracy and fine tracking capability of the expected worst-case laser phase noise dynamics. We note that 31 symbols only separate (single) pilots in 400ZR. But the latter has been designed for lasers having worse frequency noise characteristics.

C. DSP receiver architecture

Several candidate DSP solutions have been evaluated and compared at each stage of the digital receiver. The goal was to identify practical and robust DSP algorithms that translate well into high-throughput hardware processing. Accordingly, the choice was made to favor open-loop solutions at each step. The proposed receiver architecture is described in Fig. 2, in which we distinguish between operations performed at the sampling rate (twice the symbol rate in our case), at the symbol rate, and at the bit rate, respectively. Robust blind algorithms are used in the first stages. Data-aided algorithms then follow for the fine carrier and phase recovery steps. Section IV provides details about the purpose of each stage and the selected algorithms.

D. DSP receiver operation

At receiver startup, before regular data demodulation can begin, a fixed period of time is required for blind initialization and training of the coarse frequency correction, timing recovery, and adaptive equalization stages. Our simulations have shown that less than 50µs was sufficient to achieve steadystate convergence, even at the lowest SNR. Once done, the receiver can enter the frame acquisition process, which could take another 50µs to complete. All in all, initialization and lock-in was achieved in (usually much) less than 100µs, even in the worst SNR and PDL conditions.

IV. DSP ALGORITHMS

A. Coarse frequency synchronization

At system startup, the initial CFO may be very large. The function of the coarse frequency correction stage is to bring the offset down to a small fraction of the symbol rate. Not Coarse CFO estimation needs only be done once, during the training period, and is then disabled to save power. The M tentative pre-corrections can be carried out in parallel. Assuming a frequency spacing ∆ = δf i+1 -δf i between tentative corrections and correct coarse CFO estimation, the residual CFO after coarse estimation and correction will be at most ±∆/2. One can trade estimation accuracy and peak CFO tolerance for hardware complexity through the number M of candidate CFO to test. In our simulations, we used M = 6 candidate CFOs {±0.025, ±0.075, ±0.125} × R s , and averaged over N 0 = 4096 samples.

B. Matched filtering

Chromatic dispersion is negligible in FSO links. Accordingly, static equalization reduces to simple matched filtering, and can be efficiently carried out by block-filtering in the frequency domain with the overlap-save trick, see e.g. [START_REF] Torbatian | Performance oriented DSP for flexible long haul coherent transmission[END_REF], [START_REF] Nagarajan | Low power DSP-based transceivers for data center Ootical fiber communications[END_REF]. We chose to implement matched filtering prior to timing recovery in order to improve SNR for the latter.

C. Timing recovery

The function of the timing recovery block is to compensate for the sampling clock offset and reconstruct a discrete-time over-sampled signal at exactly twice the symbol rate used at the transmitter. Symbol rate down-sampling will happen later on, at the adaptive equalizer output. Timing recovery is performed block-wise, in the manner shown in Fig. 5, wherein a sampling time offset estimate is calculated for each block, and then corrected by digital interpolation of the block samples. Due to X-Y skew, the sampling time offset will generally be different on the two polarizations, requiring independent timing recovery functions for each. Blind timing estimation under the constraint of feed-forward processing at 2 samples per symbol leaves little choice but Lee's estimator [START_REF] Lee | A new non-data-aided feedforward symbol timing estimator using two samples per symbol[END_REF], or its asymptotic unbiased variant [START_REF] Wang | An alternative blind feedforward symbol timing estimator using two samples per symbol[END_REF]. When averaging over blocks of 5000 symbols or more, (an equivalent PLL would have a normalized loop bandwidth B L T s ≈ 10 -4 ), this estimator can provide accurate timing correction in the presence of phase noise and CFO, as demonstrated by the mean-square error (MSE) performance in Fig. 6. Other simulations, not reported here, have assessed its robustness to crosspolarization impairments (skew and SOP rotation), as well as its ability to track timing clock drifts of ±30 ppm.

D. Adaptive cross-polarization equalization

As FSO links are not impaired by PMD, we can revisit and simplify the usually power-intensive 2×2 adaptive equalization stage at the core of coherent optical DSP receivers [START_REF] Faruk | Digital signal processing for coherent transceivers employing multilevel formats[END_REF]. In

J(a, b, g X , g Y )= E (|s X (n)| 2 -1) 2 + E (|s Y (n)| 2 -1) 2 + λ (|a| 2 + |b| 2 -1) 2 (1)
where λ is a non-negative real coefficient that balances the relative weight given to the CMA constraint and to the firststage unit norm constraint, respectively, in the tap coefficients update. Adaptation of the scalar demultiplexing coefficients (a, b) and filter taps for (g X , g Y ) uses a standard stochastic gradient descent approximation of the cost function [START_REF] Roy | Optical feeder links for high throughput satellites[END_REF]. Equalizer adaptation is performed block-wise, to enable fast adaptive frequency-domain filtering [START_REF] Faruk | Digital signal processing for coherent transceivers employing multilevel formats[END_REF], [START_REF] Zia-Chahabi | Efficient frequency-domain implementation of block-LMS/CMA fractionally spaced equalization for coherent optical communications[END_REF]. Our simulations used 16-taps filters for (g X , g Y ). Equalizer coefficients were adapted once every 32 symbol periods, with step-size parameter µ = 10 -3 for training, and µ = 5.10 -4 for tracking.

Parameter λ was set to 0.7.

Blind equalization was preferred to decision-directed adaptation as the latter can lead to catastrophic error propagation at very low SNR. In addition the use of tentative decisions would require a feedback path from the carrier and phase recovery stages, thereby increasing latency. In contrast, the CMA has the advantage of being insensitive to CFO and phase noise. Owing to its first stage, the proposed equalizer is also structurally immune to the well-known CMA singularity problem, whereby the CMA may happen to converge to

. If ! !" > ! !# and ! $" > ! $# # %,! , # %,$ = (' !" , ' $# ) . If ! !# > ! !" and ! $" > ! $# Permute ) " and ) # # %,! , # %,$ = (' !# , ' $" )
. Otherwise declare failure # %,! , # %,$ = (? , ? )

Search Header Search Header

Search Header Search Header

s! * (ℓ) )"(') ) # (') s $ * (ℓ) #%,!, #%,$ ! !" , ' !" ! $# , ' $# !!#, '!# ! $" , ' $" ! !" " #!" " * ! !" * #(%) s * (ℓ)
)"(%)

*"(%) Every DSP subframe has the same structure based on a fixed TS with the first symbol processed as a pilot.

! !(#!") " #!" #!" * ! !(#!") * s * (ℓ)
The TS includes 11 QPSK symbols for each polarization. The TS is different between X and Y polarizations

The PS sequence includes (1+115) QPSK symbols based on PRBS. The first TS symbol is also the first symbol of the PS sequence.

12.1 First DSP sub-Frame

The first DSP sub-frame of the super-frame includes a 22 symbol Frame Alignment Word (FAW) used to align to the 5 SC-FEC Frames. 76 additional symbols are reserved for future use/innovation. The First DSP sub-frame includes: 22 symbols used as the Super Frame Alignment Word (FAW). The FAW is different between X and Y polarizations. 76 symbols are reserved to be used for future proofing and for innovation. These symbols should be randomized to avoid strong tones. These symbols should be selected from 16QAM modulation. Every DSP subframe has the same structure based on a fixed TS with the first symbol processed as a pilot.

The TS includes 11 QPSK symbols for each polarization. The TS is different between X and Y polarizations

The PS sequence includes (1+115) QPSK symbols based on PRBS. The first TS symbol is also the first symbol of the PS sequence.

12.1 First DSP sub-Frame

The first DSP sub-frame of the super-frame includes a 22 symbol Frame Alignment Word (FAW) used to align to the 5 SC-FEC Frames. 76 additional symbols are reserved for future use/innovation. The First DSP sub-frame includes: 22 symbols used as the Super Frame Alignment Word (FAW). The FAW is different between X and Y polarizations. 76 symbols are reserved to be used for future proofing and for innovation. These symbols should be randomized to avoid strong tones. These symbols should be selected from 16QAM modulation. the same input on both outputs, entailing the loss of one polarization. Our equalizer reached steady-state convergence with proper polarization demultiplexing in less than 30µs in most simulations, even at negative SNRs. This came as a surprise for the authors which did not expect a blind algorithm to be so robust at these noise levels.

E. Frame synchronization

Frame synchronization is in charge of resolving the polarization ambiguity at the blind adaptive equalizer output, and locating the frame header on each of the two polarizations. At this stage the FSE should have taken care of any fractional delay (with respect to symbol rate) on each polarization through the two filters (g X , g Y ). However, due to X-Y skew, there may still be a relative integer delay, multiple of the symbol rate, between the two polarizations. It follows that the search for frame headers has to be performed independently on each, to compensate and correct for this integer delay.

Frame header search uses the robust differential correlation metric proposed in [START_REF] Lee | Robust frame synchronization for low signal-to-noise ratio channels using energy-corrected differential correlation[END_REF]. In addition to being virtually immune to CFO and highly robust to laser phase noise, one can easily trade synchronization performance for hardware complexity by increasing or decreasing the maximum correlation order. Note that the correlation calculations at different orders can be carried out in parallel, and that higher-order correlations need less hardware resources than lower-orders.

The whole frame synchronization and polarisation ambiguity resolution proceeds as illustrated in Fig. 8. We search in parallel for the X and Y frame headers, on each of the X and Y signals delivered by the blind equalizer. Once a header has been detected on both polarizations, we verify that the solution is consistent, i.e. that we have either received X on X and Y on Y, or X on Y and Y on X. Any other situation is

MSE( fT)

MCRB M&M order-32 =0 M&M order-32 =600kHz PN limit Fig. 9. Mean-square error performance of Mengali and Morelli CFO estimator with a block size of 64 symbols, a maximum correlation order of 32, with and without laser phase noise; Comparison with the theoretical limits (MCRB for AWGN only, and high-SNR MCRB limit for AWGN + PN) regarded as a synchronization failure for the current position. A header size H = 64 symbols with a maximum correlation order of 32 for the synchronization metric, supplemented by a simple synchronization protocol that checks and confirms the header location within two consecutive frames, was found to be sufficient for reliable frame acquisition down to -6 dB, in less than 50µs, even with 5 dB of PDL.

F. Fine carrier-frequency recovery

Fine carrier-frequency recovery uses the Mengali and Morelli data-aided frequency estimator [START_REF] Mengali | Data-aided frequency estimation for burst digital transmission[END_REF]. The reason is twofold. First it has an estimation range and threshold very close to those of the maximum-likelihood data-aided estimator of Rife and Boorstyn. Second, it makes direct re-use of the correlations calculated at the frame synchronization stage, which greatly simplifies receiver design. From the simulation results in Fig. 9, which assume a block size H = 64 symbols with maximum correlation order H/2 = 32, one observes that this estimator has optimal performance at SNR > -4 dB, and acceptable performance below, with and without phase noise.

CFO estimation is performed once per frame, using the frame header. Estimation is carried out jointly on the two polarizations, for maximum accuracy (better noise averaging). The calculated frequency correction is then applied to the whole frame, and the process repeated frame after frame. Simple first-order exponential smoothing is applied to the CFO estimates from one frame to the other, in order to further improve accuracy over time.

G. Fine carrier-phase recovery

Two feed-forward carrier-phase recovery schemes very popular in the fiber-optics community are the Viterbi and Viterbi (V&V) estimator [START_REF] Viterbi | Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission[END_REF], and the blind phase search (BPS) 𝑟 ! (𝑛) 𝑟 " (𝑛) Fig. 10. Carrier-phase recovery scheme algorithm [START_REF] Zhou | Efficient clock and carrier recovery algorithms for singlecarrier coherent optical systems: A systematic review on challenges and recent progress[END_REF]. Both allow fast tracking of phase noise in a hardware-friendly, non-data-aided manner. Our studies have shown that the performance of both approaches collapse at low SNR (E s /N 0 < 5 dB). Specifically, the V&V is hindered by excessive noise amplification caused by 4-th power elevation. BPS does not have this problem, but its use of tentative decisions gives rise to catastrophic error propagation. We have chosen to work instead with pilot-based ML estimation, which is a much more robust solution, particularly at negative SNR. Simple linear interpolation between unwrapped phase estimates is used to coarsely track the phase noise variations in-between consecutive pilot blocks [START_REF] Casini | DVB-S2 modem algorithms design and performance over typical satellite channels[END_REF], as well as any nonnegligible residual CFO that might have survived thus far. carrier-phase recovery scheme is depicted in Fig. 10. Carrier-phase estimation can be performed either independently on each polarization, or jointly. Joint estimation improves estimation accuracy by averaging over the two polarization. Equivalently, it can reduce by half the number of pilot symbols required to reach a given MSE performance, as compared to separate estimation. However joint estimation requires that both polarizations share the exact same phase offset. One can show that this assumption does not hold in the present case, due in part to X-Y skew, but also to a π/2 phase ambiguity at the output of the adaptive CMA equalizer. Fortunately it turns out that both issues can be taken care of easily, by simple prior estimation and correction of the relative phase shift between polarizations (not shown in Fig. 10), making joint phase possible after all and with little additional calculation.

H. Symbol detection

Symbol detection consists in computing a soft-decision (loglikelihood ratio) on each modulated bit, for subsequent softdecision FEC decoding.

V. RECEIVER PERFORMANCE

The performance of the proposed receiver has been assessed under several transmission scenarios. Several performance metrics have been considered, including the uncoded biterror rate (BER), and also the generalized mutual information (GMI), for FEC performance prediction [START_REF] Alvarado | Replacing the soft-decision FEC limit paradigm in the design of optical communication systems[END_REF]. As an illustration, Fig. 11 Fig. 11. Mutual information at the DSP receiver output, with different impairments and receiver configurations (PN = phase noise, CFO = carrier frequency offset, CPR = carrier-phase recovery, CFR = carrier-frequency recovery, EQ = adaptive equalizer, TM = timing recovery) (4.8% overhead). At every SNR, 400 frames were used for training, followed by a maximum of 300 frames for frame acquisition. BER and GMI were then evaluated over the next 512 frames.

We first verify here that blocks of 8 pilots about 256 symbols apart enable fine estimation and tracking of phase noise over the all SNR range. Switching on carrier-frequency recovery worsens the performance at negative SNR, due to the residual frequency error. The pilot symbols could be used to improve estimation accuracy and reduce this error. It is noteworthy that, in the absence of turbulence and PDL, the two main sources of degradation in communication performance are equalization noise at low-to-medium SNR, and laser phase noise at high SNR, respectively. The later was expected. The former is more surprising, and arises independently of X-Y skew or SOP rotation. It could be due to noise intrinsic to the adaptive tap coefficient update (stochastic gradient descent). One can note also that clock drifts do not hinder performance.

In the absence of PDL, we simulate a single rotation between the transmit and receive lasers SOPs. PDL simulation uses a lumped PDL model with a Jones matrix of the form

R θ2 √ 1 + γ 0 0 √ 1 -γ R θ1
, with Γ = 10 log 10 1+γ 1-γ the PDL factor, in dB. In that case two SOP rotations occur, one before the introduction of PDL, to align with the most-and leastlossy PDL axis, and another one after, to align with the LO polarization axis. DSP alone cannot compensate for the PDL power loss, which impacts performance at all SNRs and to varying degree, depending on the first SOP rotation θ 1 . In the setting in Fig. 11, the power loss distributes evenly on X and Y due to the θ 1 = 45°rotation prior to PDL introduction. This explains that the two polarization show similar BER and GMI performance in that particular case.

VI. SUMMARY

We have demonstrated the possibility to establish reliable GEO links at an electrical SNR as low as -6 dB, in the presence of realistic system impairments, using less than 5% signaling overhead for the header and pilots. This was made possible by the design of a custom DSP receiver based on simple, proven feed-forward algorithms that borrow both from fiberoptic communication and from RF satellite communications. The proposed DSP receiver is just a proof-of-concept whose performance can be improved in many ways, e.g. by using the pilots to refine CFO estimation, supplementing feed-forward clock recovery by a lower-frequency feedback loop driving the ADC frequency, or by more accurate tracking of phase noise in-between pilots blocks.

These results underscore the value of a dedicated receiver architecture, both from the point of view of performance and hardware complexity (see e.g. equalizer simplification). However it is also worth recognizing that coherent fiber-optic DSP receivers would need little modification to reach full operational capability on GEO feeder links.

Atmospheric turbulence was not simulated, yet accounted for in the minimum operating SNR. In practice, however, deep fades will ineluctably appear and possibly span many frames. How to modify the proposed receiver so as to stay locked as long as possible in such a situation, and to recover quickly from hang-up, requires further work. Finally, albeit designed for a GEO link, the proposed frame format and DSP receiver should apply to LEO links as well with few changes.

Fig. 8 . 1 .

 81 Fig. 8.1. Basic diagram of a generic lasercom system.

Fig. 8 . 1 .Fig. 1 .

 811 Fig. 8.1. Basic diagram of a generic lasercom system. 8.1.1 Brief historical overview While optical-fiber communications experienced a huge growth in the 1970s, the effects of the atmosphere on the optical signals made the development of FSOC slow down for two decades. In 1992,

•

  polarization time skew (X-Y skew) induced by separate optical pre-amplification of the two polarizations at the transmitter; • polarization-dependent losses (PDL) introduced by the optical transmit and receive subsystems; • accumulated laser phase noise (PN) from the transmit laser and receive local oscillator (LO); • an unknown and possibly time-varying state-ofpolarization (SOP) rotation at the receive LO; • a carrier-frequency offset (CFO) at the receive LO; • optical amplified spontaneous emission (ASE) noise;

Fig. 3 .

 3 Fig. 3. DSP frame format

Fig. 4 .

 4 Fig. 4. Principle of coarse frequency-offset estimation

Fig. 5 .

 5 Fig. 5. Block diagram of timing recovery

Fig. 6 .

 6 Fig.6. Mean-square error performance of Lee's timing estimator with a block size of 5000 symbols, a normalized CFO of 0.025/Ts, and 600 kHz of laser phase noise; Comparison with the modified Cramer-Rao bound (MCRB)

Fig. 7 .

 7 Fig. 7. Structure of the proposed adaptive blind cross-polarization equalizer

)A

  Figure 34: First DSP sub-frame of super-frame

  Figure 34: First DSP sub-frame of super-frame

Fig. 8 .

 8 Figure 34: First DSP sub-frame of super-frame

  Fig.[START_REF] Wang | An alternative blind feedforward symbol timing estimator using two samples per symbol[END_REF]. Uncoded BER at the DSP receiver output, with different impairments and receiver configurations (same settings and legend than in Fig.11)

  Block diagram of the DSP receiver (dashed lines show possible feedback paths, not implemented here for latency constraints)

	1/𝑇 $ e	2/𝑇 !
	Matched
	Filtering
	Fig. 2.

  and 12 shows the GMI and BER measured at the DSP receiver output in the presence of various transmission impairments, and with different receiver configurations, some stages being enabled or disabled depending on the impairments. The frame size was 64 + 16 × 256 = 4160 symbols

		2
		1.8
		1.6
									Marker x : X-Pol
		1.4							Marker + : Y-Pol
	GMI (bits/sym)	0.8 1 1.2
							AWGN (ref)
		0.6					PN=600kHz (with CPR only) PN=600kHz, CFO=5GHz (with CFR+CPR)
							PN=600kHz, CFO=5GHz, XY Skew=0.25ns, XY Rot=45°(
		0.4					with EQ+CFR+CPR) PN=600kHz, CFO=5GHz, XY Skew=0.25ns, XY Rot=45°,
							Clock Drift=20ppm (with TM+EQ+CFR+CPR)
		0.2					PN=600kHz, CFO=5GHz, XY Skew=0.25ns, XY Rot=45° + PDL=5dB + XY Rot=45°,
							Clock Drift=20ppm (with TM+EQ+CFR+CPR)
		0
		-6	-4	-2	0	2	4	6	8	10	12	14	16
							SNR (dB)

ACKNOWLEDGMENT

This research was carried out under contract with the Centre National d'Etudes Spatiales (CNES). The authors would like to thank Yves Jaouen from Telecom Paris and Géraldine Artaud from CNES for many fruitful discussions.


Melanie Ehrler

David C Bellinger

Adam R Cassidy

Jane W Newburger

Johanna Calderon
email: johanna.calderon@inserm.fr

Johanna Calderone

Social cognition and behavioral outcomes in congenital heart disease: profiles and neuropsychiatric comorbidities

Keywords: Autism traits, social cognition, congenital heart disease, executive functions, anxiety symptoms

come

deficit/hyperactivity disorder (ADHD) frequently display co-existing impaired social cognition and executive dysfunction [START_REF] Miranda | Social cognition in children with high-functioning autism spectrum disorder and attention-deficit/hyperactivity disorder. Associations with executive functions[END_REF][START_REF] Oerlemans | Co-segregation of social cognition, executive function and local processing style in children with ASD, their siblings and normal controls[END_REF], which seem to be directly associated with each other [START_REF] Fong | The role of executive functioning in predicting social competence in children with and without autism spectrum disorder[END_REF][START_REF] Pineda-Alhucema | Executive function and theory of mind in children with ADHD: A systematic review[END_REF] and may share underlying neuronal correlates [START_REF] Miranda | Social cognition in children with high-functioning autism spectrum disorder and attention-deficit/hyperactivity disorder. Associations with executive functions[END_REF]. Furthermore, patients with diagnosed autism or ADHD are at increased risk for anxiety symptoms [START_REF] Bishop | An examination of the association between anxiety and social functioning in youth with ADHD: A systematic review[END_REF][START_REF] Van Steensel | Anxiety disorders in children and adolescents with autistic spectrum disorders: A meta-analysis[END_REF]. In ADHD, more social difficulties have been associated with elevated anxiety [START_REF] Bishop | An examination of the association between anxiety and social functioning in youth with ADHD: A systematic review[END_REF].

In individuals with CHD, the prevalence of executive function deficits is high, with more than 70% of the children exhibiting some degree of executive dysfunction (e.g., lower working memory, inhibitory control and cognitive flexibility) (Cassidy et al., 2015), which can interfere with their ability to read social cues. In addition, youth with CHD present with elevated psychiatric symptoms, predominantly anxiety disorders [START_REF] Demaso | Psychiatric disorders in adolescents with single ventricle congenital heart disease[END_REF][START_REF] Freitas | A cohort study on psychosocial adjustment and psychopathology in adolescents and young adults with congenital heart disease[END_REF][START_REF] Pike | Brain abnormalities in cognition, anxiety, and depression regulatory regions in adolescents with single ventricle heart disease[END_REF] that might, in turn, hinder their capacity to process social information. [START_REF] Calderon | Executive function and theory of mind in school-aged children after neonatal corrective cardiac surgery for transposition of the greatarterie[END_REF] provided the first evidence that lower scores on inhibitory control are associated with lower ToM skills (i.e., first and second order false belief understanding) in children with d-TGA [START_REF] Calderon | Executive function and theory of mind in school-aged children after neonatal corrective cardiac surgery for transposition of the greatarterie[END_REF]. Further evidence suggests that a computer-based working memory training (i.e., Cogmed) may improve "autism-like" symptoms in children with CHD. However, the authors only found an improvement at the 3month follow-up but not at post-intervention [START_REF] Calderon | Randomized controlled trial of working memory intervention in congenital heart disease[END_REF]. Moreover, Werninger suggested that there is a substantial prevalence of comorbid ADHD-related symptoms and "autism-like" symptoms in children with CHD (Werninger et al., 2020). However, to date, the co-existence and the potential interplay of executive functions in everyday life, anxiety, social and behavioral difficulties in children CHD have not been examined.

The aim of this study is to provide additional characterization of the profile of social communication difficulties, unusual behaviors, and impaired theory of mind in school-aged children with complex CHD using a comprehensive set of standardized measures and to quantify the cumulative burden of neurodevelopmental and psychiatric sequelae by assessing potential comorbidities. First, we characterized parent-reported social and behavioral symptoms as measured by autism symptom scales, children's ToM abilities, and parentreported empathy skills in children with CHD who underwent cardiac surgery in infancy and did not have a formal autism diagnosis. We hypothesized that, compared to the general population, children with CHD will display worse outcomes in areas such as social communication and higher-order ToM. We also predicted that children with ToM impairments will have social communication difficulties and restricted repetitive behaviors in everyday life, as reported by their parents. Second, we examined potential comorbidities of social and behavioral difficulties, particularly problems in executive function and anxiety symptoms. We predicted that more social and behavioral difficulties will be associated with worse outcomes in executive functions and with greater anxiety symptoms.

Materials and methods

Study sample

This cross-sectional single-center cohort study was conducted at Boston Children's Hospital between 2019 and 2020. Eligibility criteria included a diagnosis of CHD requiring cardiopulmonary bypass (CPB) surgery within the first year, age between 7 years 0 months and 12 years 11 months, fluency in English, no chromosomal or genetic syndrome, no severe physical or sensory impairments (hearing, visual or psycho-motor), and absence of a confirmed autism diagnosis (based on medical records and parent interviews). We also screened for intellectual or developmental impairment in the medical record and during parent interview, and we excluded children who had severe intellectual or developmental impairment (e.g., severe language or motor delay) that would prevent completion of the planned study assessment.

The clinical database for all patients treated at Boston Children's Hospital was reviewed to identify eligible children. In total, 377 children met the inclusion criteria and were contacted by e-mail or postal mail. Of those, 294 families did not respond and could not be reached by phone and 6 families refused participation. Of the 77 families who were contacted, a screening interview was completed by telephone. Seven children did not meet the inclusion criteria due to confirmed autism diagnosis or other severe neurodevelopmental impairment, and 14 families did not consent to participate. A total of 56 families (15% of all initially contacted patients; 80% of all patients contacted by telephone and proved to be eligible) agreed to participate. Neuropsychological assessment was conducted remotely by a pediatric research neuropsychologist (JC), and questionnaires were filled out online by the child's primary caregiver. Remote assessment was conducted due to the ongoing COVID-19 pandemic. While research on the validity of remote neuropsychological evaluations remains sparse, it has been suggested that remote assessment of several neuropsychological tests is feasible (e.g., subtests of the WISC-V, [START_REF] Cox | Integrating telehealth into neurodevelopmental assessment: A model from the cardiac neurodevelopmental outcome collaborative[END_REF]. All assessments were conducted using a laptop, computer, or tablet (no cellphone screen was used). Families who did not have a proper device were sent a tablet to complete the assessment remotely. The examiner conducted all assessment in an office and the families were at home. Verbal responses were recorded where pointing was not possible. Children completed the assessments without parental assistance.

The study was approved by the hospital's Institutional Review Board. Primary caregivers of all participants gave written informed consent prior to participation.

Outcome measures

Primary outcomes

Primary outcomes for this study were the Autism Spectrum Rating Scale (ASRS) [START_REF] Goldstein | Autism spectrum rating scales (ASRS): Manual[END_REF] and the Theory of Mind Task Battery (ToM-TB) [START_REF] Hutchins | Test-retest reliability of a theory of mind task battery for children with autism spectrum disorders[END_REF][START_REF] Hutchins | Test-retest reliability of a theory of mind task battery for children with autism spectrum disorders[END_REF].

Parent-reported autism symptoms were assessed with the full-length ASRS (6-18 years) [START_REF] Goldstein | Autism spectrum rating scales (ASRS): Manual[END_REF]. The ASRS is a validated 71-item questionnaire to measure autism characteristics according to DSM-5 diagnostic criteria. The total score and the three ASRS subscales were used for the current study. The social/communication subscale assesses the use of verbal and non-verbal social communication skills in social interactions. The unusual behavior subscale assesses the level of tolerance for changes in routine, engagement in apparently purposeless or stereotypical behaviors, and overreaction to certain sensory inputs. The self-regulation subscale assesses how well the child can control behavior and thoughts, maintain focus, and resist distraction. Raw scores were transformed into T-scores (Mean = 50, SD = 10) based on representative US norms including children with and without clinical conditions. Cronbach's alpha ranges from 0.92 to 0.97 indicating excellent internal consistency. Higher ASRS scores indicate more "autism-like" symptoms.

ToM was assessed using the ToM-TB [START_REF] Hutchins | Test-retest reliability of a theory of mind task battery for children with autism spectrum disorders[END_REF] as part of the virtual neuropsychological assessment. The ToM-TB consists of 15 test questions within 9 tasks. Tasks are presented in a storybook format via short visual vignettes depicting social situations of increasing complexity. The ToM-TB provides a total score (0-15) with higher scores indicating better ToM abilities and three subscales corresponding to the three developmental stages of ToM (i.e., early, basic, and advanced). The early ToM subscale evaluates basic emotion recognition, which emerges in typically developing children between 1 and 3 years of age. Patients were classified as having "fully developed" early ToM if they correctly passed all subtests. The basic ToM subscale measures social cognition typically emerging between 3.5 and 5 years of age and include visual perceptive-taking and first-order false belief tasks (i.e., the ability to infer somebody's false belief). Patients were classified as "fully developed" basic ToM if they correctly passed all subtests. The advanced ToM subscale evaluates secondorder theory of mind including emotional and false belief tasks typically mastered between 5.5 and 8 years of age. Second-order emotion and false belief tasks assess the ability to infer what one person thinks about another person's mental or emotional states. For example, saying that Paul thinks Mary is upset, when in reality, she is not. Patients were classified as "fully developed" advanced ToM if they correctly passed at least one subtest of each task (in accordance with the technical manual [START_REF] Hutchins | Test-retest reliability of a theory of mind task battery for children with autism spectrum disorders[END_REF] the ToM-TB subtest Second-Order False Belief task was only administered to children > 8 years of age). Cronbach's alpha ranges from 0.91 to 0.94 indicating excellent internal consistency.

Secondary outcomes

The Social Responsiveness Scale (School-age form), 2nd Edition (SRS-2) (Constantino, 2012) and the Empathy/Systemizing Quotient Child Version (ESQ-C) [START_REF] Auyeung | The children's empathy quotient and systemizing quotient: Sex differences in typical development and in autism spectrum conditions[END_REF] were used to investigate social impairments beyond the two primary outcomes (i.e., ASRS and ToM-TB).

The SRS-2 (Constantino, 2012) measures social skills and autistic mannerisms. T-scores (Mean = 50, SD = 10) were calculated based on representative US norms for the total score and the DSM-5 subscales (Social Communication Impairments, Restricted Repetitive Behavior). Higher SRS-2 scores indicate more "autism-like" symptoms. The ESQ-C [START_REF] Auyeung | The children's empathy quotient and systemizing quotient: Sex differences in typical development and in autism spectrum conditions[END_REF] measures the child's ability to feel empathy and relate to other people's feelings in everyday situations. Sex-specific T-scores (Mean = 50, SD = 10) were calculated for the Child Empathy Quotient using sex-specific means and standard deviations of typically developing children reported by [START_REF] Auyeung | The children's empathy quotient and systemizing quotient: Sex differences in typical development and in autism spectrum conditions[END_REF]. Higher Child Empathy Quotient scores imply better empathy skills.

The Behavior Rating Inventory of Executive Function, 2nd Edition (BRIEF-2) [START_REF] Gioia | BRIEF2: Behavior rating inventory of executive function[END_REF] and the Multidimensional Anxiety Scale for Children, 2nd Edition (MASC-2) (March, 2012) were used to investigate comorbidities of social impairments. The BRIEF-2 [START_REF] Gioia | BRIEF2: Behavior rating inventory of executive function[END_REF] The Conners-3 (Conners, 1997), subtests of the Wechsler Intelligence Scale for Children, 5th edition (WISC-V) [START_REF] Wechsler | Wechsler intelligence scale for children-fifth edition[END_REF] and primary caregiver's highest education were considered as descriptive variables and/or covariates. The Conners-3 (Conners, 1997) assesses ADHD-related behaviors in children. T-scores (Mean = 50, SD = 10) were calculated based on representative US norms for the total score. Higher Conners-3 scores imply more ADHD-related symptoms. The following WISC-V subtests were administered Matrix Reasoning, Digit Span, Similarities, and Vocabulary. Age-normed scaled scores were calculated for all subtests (Mean = 10, SD = 3). An age-normed Verbal Comprehension Index score (Mean = 100, SD = 15) was calculated from the scores on two subtests, Similarities and Vocabulary. Higher scores indicate better verbal comprehension. For all of the questionnaires used, Cronbach's alpha scores indicate good to excellent internal consistency [START_REF] Auyeung | The children's empathy quotient and systemizing quotient: Sex differences in typical development and in autism spectrum conditions[END_REF][START_REF] Conners | Manual for the Conners' rating scales -revised[END_REF]Constantino, 2012;[START_REF] Gioia | BRIEF2: Behavior rating inventory of executive function[END_REF]March, 2012). Clinical and sociodemographic factors were collected from patient charts and by means of a standardized questionnaire filled out by the primary caregiver.

Statistical analysis

Clinical and sociodemographic characteristics were reported. Number and probabilities were reported for nominal data, median and interquartile range (IQR) for ordinal-scaled data, and mean and standard deviation (SD) for continuous data. Total scores and subscales of the parent-reported social functioning questionnaires (i.e., ASRS, SRS-2, ESQ-C) were compared to normative data with one-sampled t-tests (Reference Mean = 50, SD = 10). Numbers and proportions of children scoring outside the normal range (ASRS, SRS-2: T ≥ 60, ESQ-C: T ≤ 40) were reported and compared to normative data with Chi-square tests (Reference proportion = 16%). To describe ToM abilities, number and proportion of children with "not fully developed" ToM stages (i.e., subscales early, basic, and advanced ToM of the ToM-TB) were reported. In order to explore the profile of social and behavioral difficulties, the proportions of children with significant parent-reported outcomes and/or impaired ToM were displayed in a Venn Diagram. For this purpose, numbers and proportions of children who scored ≥ 1SD above the mean indicating worse outcomes in the ASRS subscales (T ≥ 60) and who have "not fully developed" advanced ToM (ToM-TB) were considered.

To investigate potential comorbidities (i.e., lower executive function and more anxiety) of social and behavioral difficulties, two separate regression models were calculated. For parentreported outcomes, a linear regression model was calculated with the ASRS total score as the dependent variable. For ToM impairments, an ordinal regression model was calculated with the ToM-TB total score as the dependent variable. MASC-2 total score, BRIEF-2 total score, and Digit Span scaled score were included as independent variables and age, primary caregiver's education, having an ADHD diagnosis, and WISC-V Verbal Comprehension Index were included as covariates. Conners-3 total score was not considered as a covariate due to its high collinearity with the BRIEF-2 total score (Pearson's r = 0.85). Instead, the presence of an ADHD diagnosis (derived from patient charts and parent reports) was included as covariate. For the linear regression model, standardized ß coefficients with 95% confidence intervals (95-CI) and adjusted R2 were reported. For the ordinal regression model, odds ratio (OR) and Mc Fadden R2 [START_REF] Mcfadden | Conditional logit analysis of qualitative choice behavior[END_REF] were reported. Values between 0.2 and 0.4 in McFadden R2 indicate a very good model fit [START_REF] Cameron | An R-squared measure of goodness of fit for some common nonlinear regression models[END_REF]. Missing values of independent variables and covariates were imputed with the R package MICE (imputations: N = 1, iterations: N = 5) (van Buuren & Groothuis-Oudshoorn, 2011). All variables assessed within the study were considered as potential predictors for missing data estimation. With a sample size of 56 patients, using two-sided Type I error rate of α = 5%, we had 80% power to detect a correlation of r = 0.36 (small to moderate effect size) between a continuous measure of parent-reported outcomes (ASRS total score) and anxiety/executive function outcomes (MASC-2 total score, BRIEF-2 total score, and Digit Span scaled score). p-Values < .05 were considered statistically significant. Analyses were hypothesis-driven and were not corrected for multiple comparisons. Statistical analyses were performed with R (R Core Team, 2020). The STROBE guidelines for reporting observational studies were followed.

Results

Fifty-six children (n = 56, 46% female) with different types of complex CHD (16% univentricular CHD) participated in the study. Children were between 7 and 12 years of age (Mean age: 10.8 ± 1.8 years). Seventy-five percent of the children's primary caregivers had at least a college degree. Patient characteristics are summarized in Table 1. A full list of CHD diagnoses is displayed in Table S1. Participating patients did not significantly differ from nonparticipating patients (eligible and initially contacted, n = 321) regarding sex (p = .6) and cardiac physiology (univentricular vs. biventricular CHD, p = .8). There was a higher proportion of Caucasian patients in the group of participants (82%) compared to non-participants (61%, p = .004). Patients who refused to participate at any time point (n = 20) did not differ from participating patients regarding sex (p = .7), cardiac physiology (p = .9), or race (p = .7).

Social cognition and behavioral profiles in children with CHD

In comparison to the normative population, children with CHD displayed significantly more unusual behavior (p = .002) and self-regulation difficulties (p = .02) as measured by the ASRS parent-report (Table 2). A higher-than-expected proportion of children with CHD displayed clinically elevated scores (T ≥ 60) in those subscales (unusual behavior: p = .03; selfregulation difficulties: p = .003) compared to the norm. ASRS total score did not differ from the norm (p = .07). Social communication abilities were significantly better than the norm (p = .01). See Figure 1 and Table 2 for details. Similar findings were found on the SRS-2: children with CHD displayed more Restricted Repetitive Behaviors (p = .01) in comparison to the normative population, while Social Communication and Interaction abilities (p = .33) and SRS-2 total score (p = .17) did not significantly differ from the norm. Parent-reported empathy skills measured with the ESQ-C did not differ from the norm (p = .17) (Table 2).

On the ToM-TB, all children had "fully developed" early ToM (i.e., emotion recognition). Basic ToM was "not-fully developed" in 49%. Advanced ToM was "not fully developed" in 73% of the children. One-third failed a task that mainly requires visual perception abilities to understand what others see depending on their position (i.e., Line of Sight task of the subscale basic ToM) and 13% failed the first order false belief task of the subscale basic ToM (see list of all subtests in Table S2). One child did not complete the ToM-TB due to fatigue and was therefore excluded from further analyses investigating ToM outcomes.

Figure 2 displays the profile of social and behavioral difficulties in children with CHD. Twelve children (22%) had no social and behavioral difficulties (i.e., T < 60 in all ASRS subscales and "fully developed" advanced ToM). Three children (5%) showed parent-reported difficulties (T ≥ 60 in at least one ASRS subscale) but had "fully developed" ToM. Twentyone children (38%) had "not fully developed" advanced ToM but no parent-reported difficulties. Nineteen children (35%) had both parent-reported difficulties (T ≥ 60 in at least one ASRS subscale) and "not fully developed" advanced ToM. Based on this test's norm expectations, typically-developing children between 5.5 and 8 years of age master advanced ToM abilities [START_REF] Hutchins | Test-retest reliability of a theory of mind task battery for children with autism spectrum disorders[END_REF].

Comorbidities of social and behavioral difficulties in children with CHD

Regression analyses were conducted to identify comorbidities of autism spectrum traits (Table3). Results showed that higher scores for anxiety (p = 0.01) and lower working memory (p = .003) were significantly associated with more parent-reported social and behavioral difficulties on the ASRS. Executive function measured with the BRIEF-2 total score was not significantly associated with parent-reported social and behavioral difficulties (p = .182). However, when the variable Digit Span was excluded from the model, higher BRIEF-2 total scores were significantly associated with more parent-reported difficulties. The Digit Span measures attention and working memory, a core executive function, which was moderately correlated with the BRIEF-2 total score (Pearson's r = -0.515, p = < .001) and may have resulted in collinearity in the model. There was no association between ToM impairments and anxiety (p = .17) or parent-reported executive function (p = .18). This remained the case when the Digit Span was excluded from the model. Lower Digit Span scores were significantly associated with more difficulties in ToM (p = .02). Summary measures are presented in Table S3.

Post hoc, separate regression models were calculated with the subscales of the MASC-2 (i.e., Separation Anxiety/Phobia, Generalized Anxiety Disorder Index, Social Anxiety, Obsessions and Compulsions, Physical Symptoms, and Harm Avoidance) and the BRIEF-2 (i.e., Emotion Regulation, Behavioral Regulation, Cognitive Regulation) as independent variables to investigate whether results were driven by a specific subscale. We further investigated associations with the BRIEF-2 working memory subscale as a parent-reported measure for working memory (supplementing the performance-based measure digit span). All subscales of the BRIEF-2 and the MASC-2, except for the BRIEF-2 working memory subscale (FDR corrected p = .08) and the MASC-2 subscales Social Anxiety (FDR corrected associated with the ASRS total score (all others p < .04). No subscale was significantly associated with ToM-TB total score (all p > .1). Model estimates are displayed in Table S4.

Figure 3 displays comorbidities of social and behavioral difficulties in children with CHD. Of 56 children, 22 (39%) had clinically elevated scores (T ≥ 60) in at least one ASRS subscale (Social Communication, Unusual Behaviors, Self-Regulation). Of those 22 children, 21 (95%) also had clinically elevated scores in the BRIEF-2 total score and/or the MASC-2 total score indicating clinically relevant comorbidity.

Discussion

Children with complex CHD are at increased risk for social and behavioral difficulties even in the absence of a formal autism diagnosis. This study aimed to characterize the profile of social and behavioral difficulties in school-aged children with CHD and to examine the coexistence of potential cognitive and mental health impairments. We showed that despite not having an autism diagnosis, children did have more unusual behaviors and relative deficits in self-regulation and ToM (e.g., first-and second-order false belief) in comparison to the normative population. Social communication and empathy skills were normal. One in three children had combined clinically relevant parent-reported difficulties and compromised ToM. More parent-reported difficulties were associated with more anxiety symptoms, worse working memory and to some extent also with parent-reported executive functions in everyday life. Of particular importance is that among those children with clinically relevant social and behavioral difficulties, the vast majority (i.e., 21 of 22 children) also presented with clinically relevant executive function impairments and/or anxiety symptoms.

Profile of social and behavioral difficulties in children with CHD

Findings from the current study are in line with previous studies in children with CHD [START_REF] Bellinger | Adolescents with d-transposition of the great arteries corrected with the arterial switch procedure[END_REF][START_REF] Calderon | Executive function and theory of mind in school-aged children after neonatal corrective cardiac surgery for transposition of the greatarterie[END_REF][START_REF] Calderon | Impact of prenatal diagnosis on neurocognitive outcomes in children with transposition of the great arteries[END_REF][START_REF] Calderon | Facial expression recognition and emotion understanding in children after neonatal open-heart surgery for transposition of the great arteries[END_REF][START_REF] Sarrechia | Neurodevelopmental outcome after surgery for acyanotic congenital heart disease[END_REF]Werninger et al., 2020). However, an unexpected finding was that the children showed significantly better social communication compared to the norm when measured with the ASRS questionnaire and normal functioning when measured with the SRS-2 questionnaire. Consequently, also the ASRS and SRS-2 total scores did not significantly differ from the norm. Similar to our findings, a previous study in preschoolers with CHD did also not find a significant difference in the SRS between test norms and patients with CHD without a genetic comorbidity (i.e., similar to the inclusion criteria in this study) [START_REF] Brosig | Preschool neurodevelopmental outcomes in children with congenital heart disease[END_REF]. In contrast, several other studies suggested compromised social communication and interaction skills in children with CHD. A systematic review by [START_REF] Clancy | Early emotional, behavioural and social development of infants and young children with congenital heart disease: A systematic review[END_REF] showed that impaired social functioning is already present in early life. Infants with CHD are at increased risk for social withdrawal, socio-emotional delay and social interaction difficulties [START_REF] Clancy | Early emotional, behavioural and social development of infants and young children with congenital heart disease: A systematic review[END_REF]. A study in school-aged children with different types of CHD found more social communication difficulties measured with the parent-reported SRS in patients compared to normative data (Werninger et al., 2020). It is possible that differences in the type of measurement or the developmental timing across studies but also differences in access to neurodevelopmental follow-up programs and early intervention services add to the variability of findings. Further, we excluded children with a formal autism diagnosis and therefore group differences may become smaller. In addition, this study was conducted during the COVID-19 pandemic which may have influenced the parental rating of their child's social functioning due to limited social interactions. Nevertheless, studies are needed to further investigate the risk for social communication and interaction impairments in children with CHD using measurement tools other than parent reports, such as teacher reports, clinician rating, and standardized autism diagnostic assessments (e.g., ADOS).

Basic ToM (e.g., visual perspective taking and first-order false belief task) and advanced ToM (e.g., second-order emotion and false belief tasks) were compromised in a high proportion of patients participating in this study. Previous studies in younger children with d-TGA also found delayed development of ToM measured with first-and second-order false belief tasks [START_REF] Calderon | Executive function and theory of mind in school-aged children after neonatal corrective cardiac surgery for transposition of the greatarterie[END_REF][START_REF] Calderon | Impact of prenatal diagnosis on neurocognitive outcomes in children with transposition of the great arteries[END_REF][START_REF] Calderon | Facial expression recognition and emotion understanding in children after neonatal open-heart surgery for transposition of the great arteries[END_REF]. The current study shows that these impairments seem to persist throughout elementary school. In typically developing children, ToM usually develops sequentially, starting in early childhood and advanced ToM is expected to be fully developed between 5 and 8 years of age [START_REF] Hutchins | Test-retest reliability of a theory of mind task battery for children with autism spectrum disorders[END_REF]. In our study, 3 out of 4 children failed one or more of the advanced ToM tasks at a mean age of 11 years. Thus, our findings suggest substantial ToM impairments in this cohort. Advanced ToM is crucial for inferring other people's intentions, beliefs, and emotions in order to successfully participate in social interactions. This significant delay in children with CHD may put them at a high risk for compromised socio-emotional development in the long term. Further, the extensive ToM test battery used in this study allowed for more in-depth investigation of ToM difficulties in these children: Half of the children also failed one or more of the basic ToM tasks. Basic ToM is expected to consolidate between 4 and 5 years of age in typically developing children. This includes visual perception taking and first-order false belief tasks. Interestingly, many 8-12year-old children with CHD failed a basic ToM task which required visual perception taking. It has been reported that visual-spatial abilities are also vulnerable in individuals with complex CHD [START_REF] Bean | Visuospatial processing in adolescents with critical congenital heart disease: Organization, integration, and implications for academic achievement[END_REF][START_REF] Bellinger | Visual-spatial skills in children after open-heart surgery[END_REF][START_REF] Bellinger | Adolescents with d-transposition of the great arteries corrected with the arterial switch procedure[END_REF]. Potentially, visuospatial impairments in these children may have contributed to children's failure at some of the basic ToM tasks. However, the majority of ToM-TB tasks did not require any visuo-spatial abilities and thus, the observed ToM difficulties are not solely driven by visuo-spatial problems. In addition, ToM performance was correlated with working memory. This association is in line with previous studies in typically developing children showing that working memory capacity is associated with ToM [START_REF] Lecce | Relations between theory of mind and executive function in middle childhood: A short-term longitudinal study[END_REF][START_REF] Mutter | Theory of mind and executive function: Working-memory capacity and inhibition control as predictors of false-belief task performance[END_REF] and may even predict the success of ToM trainings [START_REF] Lecce | Working memory predicts changes in children's theory of mind during middle childhood: A training study[END_REF]. Based on these findings, we suggest that working memory impairments should be considered a red flag for potential theory of mind difficulties. Similar findings have been described previously in adults with CHD. A study in adults with repaired septal defect had more self-and proxy-reported social and behavioral difficulties and lower ToM abilities compared to healthy controls [START_REF] Asschenfeldt | Neuropsychological status and structural brain imaging in adults with simple congenital heart defects closed in childhood[END_REF]. Another study by [START_REF] Chiavarino | Theory of mind deficit in adult patients with congenital heart disease[END_REF] found more ToM impairments in a small sample of patients with different types of CHD compared to healthy controls when assessed with a semi-structured interview [START_REF] Chiavarino | Theory of mind deficit in adult patients with congenital heart disease[END_REF]. These findings provide first evidence that social and behavioral difficulties may persist across life and thus indicate substantial long-term sequelae which may lead to mental health difficulties and compromised well-being.

The underlying etiological pathways leading to the described profile of social and behavioral difficulties in patients with CHD remain unclear. Previous studies found genetic variants that contribute to both CHD and autism (e.g., [START_REF] Jin | Contribution of rare inherited and de novo variants in 2,871 congenital heart disease probands[END_REF][START_REF] Rosenthal | A convergent molecular network underlying autism and congenital heart disease[END_REF]. Indeed, [START_REF] Rosenthal | A convergent molecular network underlying autism and congenital heart disease[END_REF] showed that these shared risk genes are linked to a molecular network of disrupted ion channels, which may play a key role in both heart and brain development [START_REF] Rosenthal | A convergent molecular network underlying autism and congenital heart disease[END_REF]. However, as these studies only investigated patients with a formal diagnosis, it is unclear if genetic variants may also contribute to "autism-like" symptoms without a formal autism diagnosis. Furthermore, the underlying neuronal correlates of social and behavioral difficulties in CHD remain unknown. Importantly, neuroimaging studies in patients with CHD have elucidated macro-and microscopic alterations, which become evident during the fetal and neonatal period (e.g., [START_REF] Dimitropoulos | Brain injury and development in newborns with critical congenital heart disease[END_REF][START_REF] Khalil | Prevalence of prenatal brain abnormalities in fetuses with congenital heart disease: A systematic review[END_REF][START_REF] Mebius | Brain injury and neurodevelopmental outcome in congenital heart disease: A systematic review[END_REF]. Persisting brain alterations in adolescents and young adults with CHD have further been linked to neurodevelopmental impairments including worse executive functions [START_REF] Ehrler | Altered white matter microstructure is related to cognition in adults with congenital heart disease[END_REF][START_REF] Naef | Brain volumes in adults with congenital heart disease correlate with executive function abilities[END_REF][START_REF] Rollins | White matter microstructure and cognition in adolescents with congenital heart disease[END_REF][START_REF] Semmel | Posterior cerebellar volume and executive function in young adults with congenital heart disease[END_REF]. However, future studies are needed to determine the link between social and behavioral difficulties and brain structural and functional alterations in youth with CHD and to elaborate potential interactions between different neurodevelopmental functions. Further, in order to gain additional insights into the underlying etiology of social and behavioral impairments in CHD and the degree of symptom severity, future studies should consider comparing patients with and without a formal autism diagnosis.

Neurodevelopmental and psychiatric comorbidities of social and behavioral difficulties in children with CHD

Our study found that more parent-reported social and behavioral difficulties in children with CHD were associated with worse executive function skills and more anxiety symptoms. Post hoc analyses of the different BRIEF-2 and MASC-2 subscales showed that the associations with social and behavioral difficulties were not driven by any specific subscale. Indeed, worse scores on all three indexes of the BRIEF-2 (i.e., cognitive, behavior, and emotion regulation) and most subscales of the MASC-2 (i.e., Separation Anxiety/Phobia, Generalized Anxiety, Obsessions and Compulsions, Physical Symptoms) were associated with more difficulties. Moreover, when considering those patients with clinically relevant difficulties (≥1 SD outside the normal range), we found a substantial number of patients with co-occurring clinically relevant executive dysfunction and/or anxiety symptoms.

Importantly, co-existing social and behavioral difficulties, anxiety symptoms and executive dysfunction are not unique to individuals with CHD. Executive dysfunction and anxiety are frequently observed in children with an autism diagnosis without CHD [START_REF] Fong | The role of executive functioning in predicting social competence in children with and without autism spectrum disorder[END_REF][START_REF] Miranda | Social cognition in children with high-functioning autism spectrum disorder and attention-deficit/hyperactivity disorder. Associations with executive functions[END_REF][START_REF] Oerlemans | Co-segregation of social cognition, executive function and local processing style in children with ASD, their siblings and normal controls[END_REF][START_REF] Van Steensel | Anxiety disorders in children and adolescents with autistic spectrum disorders: A meta-analysis[END_REF]. Furthermore, social and behavioral difficulties often co-occur with executive dysfunction and anxiety in other clinical populations: in children with ADHD, impaired social cognition and executive dysfunction frequently co-exist and more social difficulties have been associated with elevated anxiety [START_REF] Bishop | An examination of the association between anxiety and social functioning in youth with ADHD: A systematic review[END_REF][START_REF] Miranda | Social cognition in children with high-functioning autism spectrum disorder and attention-deficit/hyperactivity disorder. Associations with executive functions[END_REF]. Children born very preterm, a population with a developmental profile at least somewhat similar to that of children with CHD [START_REF] Easson | A comparison of developmental outcomes of adolescent neonatal intensive care unit survivors born with a congenital heart defect or born preterm[END_REF], are also at increased risk for social and behavioral difficulties [START_REF] Laverty | The prevalence and profile of autism in individuals born preterm: A systematic review and metaanalysis[END_REF], executive function difficulties [START_REF] Van Houdt | Executive function deficits in children born preterm or at low birthweight: A meta-analysis[END_REF], and anxiety [START_REF] Fitzallen | Anxiety and depressive disorders in children born preterm: A meta-analysis[END_REF]. A study by [START_REF] Korzeniewski | Social responsiveness scale assessment of the preterm behavioral phenotype in 10-year-olds born extremely preterm[END_REF] in 10-year-old preterm-born children found that those children who displayed clinically relevant social and behavioral difficulties were also more likely to report anxiety and executive function deficits [START_REF] Korzeniewski | Social responsiveness scale assessment of the preterm behavioral phenotype in 10-year-olds born extremely preterm[END_REF]. Regarding the frequent co-existence of social and behavioral difficulties, anxiety, and executive dysfunction across patient populations, it seems reasonable to hypothesize that these impairments may be developmentally linked to one another irrespective of the underlying disorder.

The etiological mechanisms of social and behavioral difficulties and anxiety has been discussed previously: In autism, anxiety has been associated with unusual behaviors (i.e., restricted and repetitive behaviors) in a number of studies (e.g., [START_REF] Joyce | Anxiety, intolerance of uncertainty and restricted and repetitive behaviour: Insights directly from young people with ASD[END_REF][START_REF] Lidstone | Relations among restricted and repetitive behaviors, anxiety and sensory features in children with autism spectrum disorders[END_REF]. [START_REF] Boulter | Intolerance of uncertainty as a framework for understanding anxiety in children and adolescents with autism spectrum disorders[END_REF] suggest that this association is mediated by intolerance of uncertaintiesa finding that has also been reported in children without an autism diagnosis [START_REF] Boulter | Intolerance of uncertainty as a framework for understanding anxiety in children and adolescents with autism spectrum disorders[END_REF]. It has further been discussed that restricted and repetitive behaviors may serve as a coping response to reduce anxiety (e.g., [START_REF] Joyce | Anxiety, intolerance of uncertainty and restricted and repetitive behaviour: Insights directly from young people with ASD[END_REF][START_REF] Rodgers | The relationship between anxiety and repetitive behaviours in autism spectrum disorder[END_REF]. A better understanding of the developmental cascade linking CHD with autism and anxiety is needed to guide therapeutic interventions and improve counseling and care of these patients and their families.

Likewise, previous research has elaborated the causal relationship between ToM and executive function. Several longitudinal studies have suggested that early executive functions including self-monitoring and inhibition are required to develop ToM [START_REF] Hughes | Executive function and theory of mind: Predictive relations from ages 2 to 4[END_REF][START_REF] Marcovitch | A longitudinal assessment of the relation between executive function and theory of mind at 3, 4, and 5 years[END_REF][START_REF] Müller | Knowing minds, controlling actions: The developmental relations between theory of mind and executive function from 2 to 4 years of age[END_REF]. Further, neuroimaging studies suggest both shared and distinct underlying neuronal correlates of executive functions and ToM [START_REF] Wade | On the relation between theory of mind and executive functioning: A developmental cognitive neuroscience perspective[END_REF]. Research investigating developmental associations between unusual behaviors (i.e., restricted and repetitive behaviors) and executive function is sparse. Longitudinal studies are lacking, but a large-scale meta-analysis found a robust correlation between unusual behaviors and executive functions both in children with an autism diagnosis and in typically developing children [START_REF] Iversen | Executive function skills are linked to restricted and repetitive behaviors: Three correlational meta analyses[END_REF]. For those most vulnerable, a better understanding of the social and behavioral profile, associated comorbidities, and underlying mechanisms can inform clinical practice and will be important in developing targeted therapeutic interventions by means of identifying meaningful endpoints to measure efficacy, by targeting comorbidities to mitigate some of these deficits, and by helping to guide the selection of profile-specific therapeutic tools.

Limitations

The following limitations should be considered when interpreting the results. This singlecenter study did not include a control group of typically developing children. Instead, the scores of children with CHD who participated in this study were compared to representative US norms. In the case of ToM abilities, there were no age-adjusted normative scores provided by the manual, and thus ToM abilities were only presented descriptively (e.g., probabilities of fully developed and not fully developed ToM abilities). In addition, even though ToM tasks are informative in understanding cognitive and social development, there are limited standardized measures for clinical practice applications. As a result, ToM tasks are not always considered for diagnostic purposes. Future larger studies are needed to investigate potential medical and demographic predictors of autism-like behaviors in children with CHD.

Patients with a formal autism diagnosis were excluded as the aim of this study was to investigate social and behavioral impairments in the absence of a formal diagnosis. However, autism diagnoses were obtained from medical records and parent interviews, but no formal autism evaluation has been conducted as part of the study. Thus, the sample may have included a few individuals with later childhood undiagnosed autism. Due to a clinical focus on the patients' heart condition or potential atypical/more subtitle presentation of psychiatric symptoms, some individuals may be underdiagnosed. Future studies should therefore include formal evaluations of psychiatric disorders including autism when investigating neurodevelopmental comorbidities in patients with CHD.

Patients with a genetic syndrome and severe intellectual or developmental disabilities were excluded from this study, which limits the generalizability to forms of CHD with important associated anomalies. Further, most patients participating in this study were Caucasian and had a primary caregiver with a college degree. This sample is therefore not representative for the general US population, which limits the generalizability of the results.

Executive functions were assessed with the parent-reported BRIEF-2 and the digit span, a neuropsychological test for working memory. The role of inhibition and flexibility was not examined in this study due to the lack of instruments adapted for remote administration. However, two recent meta-analyses have shown that patients with CHD perform worse in those domains compared to children without CHD [START_REF] Feldmann | Cognitive and executive function in congenital heart disease: A metaanalysis[END_REF][START_REF] Jackson | Executive functions in children with heart disease: A systematic review and metaanalysis[END_REF]. A study in children with d-TGA showed an association between poorer inhibitory control and poor ToM outcomes [START_REF] Calderon | Executive function and theory of mind in school-aged children after neonatal corrective cardiac surgery for transposition of the greatarterie[END_REF]. Further, it has previously been suggested that cognitive flexibility plays an important role in ToM abilities [START_REF] Bock | Specifying links between executive functioning and theory of mind during middle childhood: Cognitive flexibility predicts social understanding[END_REF] and in outcomes in patients with CHD more broadly (see [START_REF] Cassidy | Cognitive flexibility in critical CHD: A target for intervention[END_REF]. Future studies should examine the role of flexibility and other executive functions in social and behavioral difficulties in children with CHD. In addition, "hot" executive functions, which refer to emotional regulatory skills, should be considered in future studies.

Neuropsychological tests were administered remotely due to the ongoing COVID-19 pandemic. The used tests were originally validated for in-person testing and were adapted for remote assessment for the current study. The normative reference data used for this study were obtained in-person and thus, comparability may be limited [START_REF] Van Patten | Introduction to the special issue -neuropsychology from a distance: psychometric properties and clinical utility of remote neurocognitive tests[END_REF]. However, recent research suggests that remote WISC-V assessments are feasible [START_REF] Cox | Integrating telehealth into neurodevelopmental assessment: A model from the cardiac neurodevelopmental outcome collaborative[END_REF] and comparable to in-person assessments [START_REF] Hamner | Equivalency of in-person versus remote assessment: WISC-V and KTEA-3 performance in clinically referred children and adolescents[END_REF][START_REF] Harder | Home-based pediatric teleneuropsychology: A validation study[END_REF]. Nevertheless, these studies only investigated subtests of the WISC-V and overall research validating remote neuropsychological assessments remains sparse. To date, no study has investigated the validity of remotely assessed ToM-TB. Although the developers of the ToM-TB validated the remote application of this test in their own practice and the graphic and picture-based nature of the tasks render the virtual administration possible via screen-share, it remains unclear if the remote setting may impact the child's attention and use of social cues and if potential visuospatial difficulties in patients with CHD may influence the performance in remote ToM tasks.

Conclusion

Children with CHD are at an increased risk for more unusual behavior, less self-regulation and delayed ToM and co-occurring executive dysfunction and anxiety symptoms. Co-existing neurodevelopmental and psychiatric impairment leads to an increased disease burden and should be addressed with therapeutic interventions. Holistic therapeutic interventions for children with CHD, which target social and behavioral difficulties, executive dysfunction and anxiety, are needed to support and improve the long-term outcomes of patients with CHD. Moreover, future studies are needed to investigate the extent of unusual behaviors, less selfregulation and ToM impairments in adults with CHD to more fully appreciate the chronic burden experienced by these patients and to improve guidance and care in the long term.

  measures executive functions in everyday situations. T-scores (Mean = 50, SD = 10) were calculated based on representative US norms for the total score and the three index scales: Behavioral Regulation, Emotional Regulation, and Cognitive Regulation. Higher BRIEF-2 scores imply more executive dysfunction. The MASC-2 (March, 2012) is a comprehensive assessment of anxiety dimensions in children and adolescents. T-scores (Mean = 50, SD = 10) were calculated based on representative US norms for the total score and the six subscales: Separation Anxiety/Phobia, Generalized Anxiety Disorder Index, Social Anxiety, Obsessions and Compulsions, Physical Symptoms, and Harm Avoidance. Higher MASC-2 scores imply more anxiety symptoms.

Acknowledgments

We thank all participating children and their families.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Funding

This work was supported by 2019 OFD/BTREC/CTREC Faculty Career Development Fellowship to Dr. Calderon to conduct all aspects of the study. The Swiss National Science Foundation (Grant Number: 32003B_172914) awarded a grant to Ms. Melanie Ehrler for her research fellowship and participation in the study. The sponsors had no influence on study design, the collection, analysis, and interpretation of data, the writing of the report, or the decision to submit the paper for publication.