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OHL-Assisted All-Optical Regenerative Relaying for Pointing-Impaired M-PAM Inter-Satellite Links

Meysam Ghanbari, Mohammad Taghi Dabiri, Zain Ali, Rula Ammuri, Mazen Hasna, Iman Tavakkolnia, Khalid Qaraqe

arXiv:2608.28864v1eess.SP

TL;DR

Pointing-impaired multi-hop optical inter-satellite links need relaying that preserves low latency without forwarding noise or requiring DF’s O/E/O processing. The paper proposes variable-gain EDFA stabilization with parallel OHL-based M-PAM regeneration, and models per-hop and end-to-end SER. Simulations and numerical integration closely agree with the analysis, while results quantify modulation-order, pointing, power, and hop-count tradeoffs.

  • Problem

    Pointing errors degrade narrow-beam inter-satellite links, while AF forwards accumulated noise and DF commonly requires O/E/O processing.

  • Method

    The paper combines variable-gain EDFA stabilization, parallel OHL symbol regeneration, closed-form per-hop SER analysis, and a Markov transition-matrix end-to-end model.

  • Results

    At H = 10, end-to-end SER reaches about 4 × 10^-6, 1 × 10^-5, and 9 × 10^-4 for M = 4, 8, and 16, respectively.

  • Takeaways & Limitations

    The framework provides a design reference for low-delay, spectrally efficient, pointing-aware all-optical regenerative relaying.

Abstract

from arXiv · show

Rapid inter-satellite traffic growth in LEO constellations demands spectrally efficient, low-latency optical relaying. While amplify and forward (AF) relays are latency-efficient, they propagate noise; conversely, decode and forward (DF) relays suppress noise but incur significant complexity via O/E/O conversion. This paper proposes an all-optical regenerative relay for M-ary pulse amplitude modulation (M-PAM) multi-hop links under pointing errors. A parallel optical hard-limiter (OHL) bank performs symbol-level discrimination, regenerating signal levels directly in the optical domain. A variable gain EDFA is employed to stabilize power and define the threshold-stable region. By incorporating pointing-induced fading, various noise sources including ASE-ASE and signal-ASE beat noise and implementation-dependent decision noise, we derive closed-form per-hop symbol error rate (SER) expressions. These are extended to end-to-end performance using a Markov transition-matrix model for arbitrary modulation order and hop count. Analysis of beamwidth, pointing accuracy, and threshold scaling demonstrates reliable multi-hop operation, avoiding both AF noise accumulation and DF O/E/O processing overhead. Verified by Monte Carlo simulations and numerical integration, this framework provides a design benchmark for low-latency, pointing-aware all-optical regenerative relaying.

I. INTRODUCTION

LEO optical relaying must combine low latency, spectral efficiency, and robustness to pointing errors. The paper proposes an all-optical regenerative M-PAM relay that uses OHL-based symbol regeneration, variable-gain EDFA stabilization, closed-form SER analysis, and Markov end-to-end modeling.

  • Motivation: Pointing errors in narrow-beam FSO inter-satellite links cause power fluctuations and higher symbol-error probability, motivating multi-hop relaying.Multi-hop routes divide long paths into shorter hops and add routing flexibility.
  • Motivation: DF suppresses forwarded analog noise through fresh decisions but commonly requires optical-to-electrical-to-optical processing, adding implementation complexity.
  • Motivation: AF relays provide low-latency optical forwarding but repeatedly amplify ASE, background noise, pointing fluctuations, and waveform distortion.
  • Proposed approach: The proposed relay uses parallel OHL banks for symbol-level M-PAM regeneration, avoiding AF noise accumulation and DF O/E/O conversion and buffering.
  • Proposed approach: A variable-gain EDFA compensates pointing-induced fluctuations while fixed OHL thresholds remain aligned with regenerated M-PAM levels.
  • Analysis and validation: Closed-form per-hop SER expressions and Markov transition matrices characterize reliability across modulation orders, noise conditions, gain limits, and hop counts.The framework is validated by Monte Carlo simulations and numerical integration with near-exact agreement.

III. POINTING-IMPAIRED INTER-SATELLITE CHANNEL AND OPTICAL NOISE MODEL

The channel model represents each inter-satellite hop through Gaussian-beam spreading, finite-aperture collection, and pointing-induced misalignment. It then forms the relay-input optical decision statistic after amplification and filtering, including signal and multiple noise contributions.

  • Gaussian beam propagation: Each hop is modeled as a Gaussian-beam FSO link whose beam radius depends on propagation distance and Rayleigh range.The receive aperture collects a distance-dependent fraction of the transmitted optical power.
  • Pointing-error statistics: Pointing misalignment is represented by radial beam displacement, with instantaneous channel gain determined by the displacement and residual angular pointing error.
  • Pointing-error statistics: The radial displacement follows a Rayleigh distribution, enabling derivation of the channel-gain CDF and PDF through the monotonic gain–displacement relation.The parameter ξ_i measures pointing fluctuations relative to the beam footprint.
  • Optical decision statistic: The relay-input statistic combines the propagated and amplified M-PAM signal with transmit-side ASE, receiver-side ASE, background noise, and implementation-dependent decision noise.The statistic is later compared with fixed OHL thresholds.
  • Optical decision statistic: Equations (17)–(19) define the optical-domain statistic used for threshold decisions at each regenerative relay.

D. EDFA ASE, Background, and Implementation Noise

The noise model accounts for amplifier ASE, ASE beat terms, background radiation, and implementation-dependent decision noise. Assuming mutual independence, these components form an aggregate decision-input variance used in a conditional Gaussian model.

  • Noise sources: The model includes transmit-side ASE, receiver-side ASE, background radiation after filtering, and implementation-dependent decision noise.The EDFA ASE power spectral density depends on the spontaneous-emission factor, Planck’s constant, optical carrier frequency, and gain.
  • Beat-noise terms: Receiver-side ASE contributes ASE–ASE beat noise, while signal–ASE beat noise depends on the transmitted symbol and conditional signal mean.Implementation coefficients capture filtering, polarization, and decision-stage effects.
  • Aggregate model: Mutually independent noise components are summed into an aggregate decision-input variance.
  • Aggregate model: The resulting optical decision statistic follows a conditional Gaussian model whose signal means depend on relay gain and channel conditions.

IV. VARIABLE-GAIN COMPENSATION AND ALL-OPTICAL M-PAM DETECTION

The relay compares amplified received power with fixed OHL thresholds for M-PAM detection. Variable-gain EDFA control stabilizes the optical scale when possible, while gain limits create a threshold-stable region and a low-channel-gain outage event.

  • All-optical detection: The OHL bank discriminates M-PAM symbols by comparing relay-input power with fixed equivalent input thresholds.In the absence of noise, each threshold must lie between adjacent conditional symbol means.
  • All-optical detection: Reliable slicing requires each fixed threshold to remain between the scaled optical powers of adjacent M-PAM levels.The condition is γ_i(h_i)P_{k−1} < ϑ_{i,k} < γ_i(h_i)P_k.
  • Threshold instability: Pointing fluctuations shift conditional means relative to fixed thresholds, producing random decision-boundary displacement when gain is fixed.
  • Variable-gain compensation: Receiver-side EDFA gain is adapted within G_min and G_max to keep the effective optical scaling near target γ_0, while thresholds remain fixed per frame.
  • Threshold-stable region and outage: The threshold-stable region consists of channel gains between h_L,i and h_U,i, where the EDFA can maintain the target scaling or reaches its minimum-gain boundary.For h_i below h_L,i, maximum gain is reached and a gain-limited outage occurs.

D. Parallel OHL Bank and Thermometer Decoding

The relay splits received power across M−1 parallel OHL branches, applies ordered thresholds, and decodes the exceeded-threshold pattern into one regenerated M-PAM level.

  • Parallel OHL bank: The input power is split across M−1 parallel OHL branches, with each branch receiving a scaled fraction of Pin,i.Branch coefficients satisfy 0 < ηi,k < 1 and account for splitter insertion loss and possible monitoring taps.
  • Threshold decisions: Each OHL compares branch power with a fixed threshold, equivalently imposing an ordered set of input thresholds θi,1 < θi,2 < ··· < θi,M−1.The equivalent input threshold is obtained by accounting for the branch splitting coefficient.
  • Thermometer decoding: The relay assigns symbol b by counting how many ordered thresholds the received input power exceeds.The interval rule maps Pin,i to symbol b when θi,b ≤ Pin,i < θi,b+1.
  • Threshold design: Midpoint slicing places each equivalent threshold between the stabilized means of adjacent M-PAM symbols.The physical threshold in branch k is τi,k = ηi,kθi,k.
  • Optical regeneration: After thermometer decoding, the relay regenerates the detected symbol as an optical power level from the same M-PAM alphabet.The regenerated level becomes the next-hop transmitted symbol, while the current analog noise waveform is not forwarded.

B. Conditional and Average Per-Hop SER

The per-hop SER is formed from conditional Gaussian decision errors, gain-limited outage, and pointing-induced channel averaging; a two-region approximation yields a compact closed form.

  • Gain-limited outage: Deep fades below hL,i force EDFA saturation at Gmax, preventing target scaling and motivating an outage-region SER model dominated by the lowest optical level.Outside outage, the conditional SER is evaluated with the gain-control law.
  • Average SER: The exact average per-hop SER is obtained by averaging the conditional SER over the pointing-induced channel gain.If hL,i > Ai, the gain target is unattainable over the full channel support and the average SER equals the outage probability.
  • Closed-form conditions: The closed-form approximation assumes a gain-controlled non-outage region hL,i ≤ h ≤ Ai with maintained target scaling γi(h) = γ0 and requires hU,i ≥ Ai.Otherwise, the exact numerical average SER should be used.
  • Two-region approximation: The approximation partitions integration into signal–ASE-dominant and constant-noise-dominant regions, producing closed forms involving the upper incomplete gamma function.The resulting expression retains outage, edge/interior symbol weighting, and gain- and pointing-dependent signal–ASE noise.

A. Markov State Representation

End-to-end regenerated symbols form a finite-state Markov chain over the shared M-ary alphabet, with each hop represented by a transition matrix that includes outage and non-outage behavior.

  • State representation: The regenerated symbol Xi is the state at node Ri, with X0 as the source symbol and XH as the destination decision.All relay nodes use the same M-ary intensity alphabet.
  • Hop evolution: Each hop depends on the previous regenerated symbol through its channel, noise, gain control, and OHL decision rule.After regeneration, previous analog noise samples are not forwarded as waveform components.
  • Transition matrices: The transition entry [Ti]a,b is the probability that relay Ri regenerates b when relay Ri−1 transmitted a.The symbol-probability row vector evolves as πi = πi−1Ti.
  • Outage decomposition: The per-hop transition matrix combines gain-limited outage and non-outage detection contributions averaged over the channel gain.The non-outage term integrates the exact conditional decision probability over [hL,i, Ai].
  • Validity conditions: If hL,i > Ai, outage occurs with probability one and the transition matrix reduces to the outage-conditioned matrix; otherwise, the exact transition model applies.The closed-form approximation is introduced only after this exact per-hop construction.

C. Closed-Form Non-Outage Transition Entries

The non-outage transition entries use the same two-region variance partition as the closed-form SER, integrating Gaussian decision probabilities over signal–ASE and constant-noise intervals.

  • Threshold offsets: Under midpoint thresholding and target scaling, the difference between each threshold and the corresponding conditional mean is independent of channel gain.This gain independence follows because the non-outage region maintains γi(h) = γ0.
  • Region-specific integration: The closed-form transition calculation uses auxiliary functions for Gaussian-CDF integrals over signal–ASE-dominant and constant-noise-dominant regions.The integrals are evaluated using clipped bounds b1,i,a and b2,i,a.
  • Closed-form entries: The resulting entries approximate the exact non-outage transition contribution using the same signal–ASE and constant-noise partition as the closed-form per-hop SER.When Di,a = 0, only constant-noise terms remain for row a.

D. End-to-End Transition Matrix and SER

The paper models end-to-end symbol errors by multiplying per-hop transition matrices, yielding conditional and average SER expressions for arbitrary hop chains. For statistically identical hops, the resulting error behavior is governed by the eigenstructure of the common per-hop matrix.

  • Transition-matrix formulation: The conditional probability of final symbol b given source symbol a is the corresponding entry of the total transition matrix.The model then derives conditional and equiprobable average end-to-end SER expressions from these transition probabilities.
  • Closed-form extension: For the two-region closed-form model, the end-to-end transition matrix uses the corresponding closed-form per-hop matrix in the same product structure.The closed-form transition model therefore supports the same end-to-end SER calculation framework.
  • Transition-matrix formulation: The end-to-end transition matrix is formed from the ordered product of the per-hop symbol-transition matrices.This converts multi-hop analysis from waveform-level noise accumulation into discrete-state error propagation.
  • Identical-hop specialization: With statistically identical hops, every per-hop matrix equals a common matrix T, simplifying the chain-wide transition matrix and average SER expressions.This specialization assumes identical channel statistics and hardware parameters across hops.
  • Identical-hop specialization: As hop count H increases, dominant eigenmodes govern how the regenerated symbol distribution evolves along the relay chain.The eigenstructure of the per-hop transition matrix determines the multi-hop error behavior for statistically identical hops.

VII. SIMULATION RESULTS AND DISCUSSION

Simulations validate the proposed regenerative relay and show that reliability depends strongly on pointing jitter, hop partitioning, beamwidth, modulation order, and threshold scaling. The analytical model closely matches simulation and numerical integration while revealing operating tradeoffs and limits.

  • Analytical validation: MC simulation, full-noise numerical integration, and the proposed closed form nearly overlap across the power range for M = 4, 8, and 16.The agreement validates the one-hop SER analysis under the modeled optical noise and pointing conditions.
  • Analytical validation: About 7×10−4 is the common high-power SER floor for M = 4, 8, and 16, with floors emerging near 10, 17, and 24 dBm, respectively.Deep pointing fades and gain-limited outage limit further improvement from transmit power.
  • Relay comparison: At 25 dBm, AF SER is about 10−3, 10−2, and 3 × 10−2 for H = 4, 8, and 12, while the proposed relay has floors of about 2 × 10−5, 5 × 10−5, and 8 × 10−5.The proposed relay retains all-optical operation while suppressing AF-like noise accumulation.
  • Pointing and hop effects: Increasing pointing jitter raises the high-power floor, from about 1.5 × 10−4 at 2 µrad to 1.8 × 10−2 and 9 × 10−2 at 3 and 4 µrad for M = 4.For M = 8, the waterfall shifts right by about 7 dB and the floors are slightly higher.
  • Pointing and hop effects: At H = 10, end-to-end SER reaches about 4 × 10−6, 1 × 10−5, and 9 × 10−4 for M = 4, 8, and 16, respectively.With fixed total distance, increasing hop count shortens each hop and reduces pointing-induced fading; lower-order PAM benefits more strongly.
  • Parameter tradeoffs: For M = 4, minimum SER improves from about 5 × 10−3 at Pmax = 5 dBm to 2 × 10−4 and 2 × 10−5 at Pmax = 10 and 15 dBm, respectively.Beamwidth has a U-shaped effect because aperture coupling trades off against pointing robustness; higher M also narrows the reliable region.
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