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On Probabilistic Shaping of Quadrature Amplitude Modulation for the Nonlinear Fiber Channel

Tobias Fehenberger, Alex Alvarado, Georg Böcherer, Norbert Hanik

arXiv:1606.04073v2cs.IT

TL;DR

The paper studies how probabilistic shaping can improve spectral efficiency in multi-span optical fiber systems while accounting for nonlinear interference. It combines AWGN analysis, fiber simulations, and extended GN models to evaluate compact PMF choices and their effects. The results show that two PMFs per QAM format suffice over broad AWGN SNR ranges, while one 64QAM PMF provides large gains from 1,400 km to 3,000 km and remains a good choice after fiber propagation.

  • Problem

    Increasing spectral efficiency is practically important, but transceiver-limited SNR and the trade-off between shaping gain and increased nonlinear interference complicate probabilistic shaping for nonlinear fiber channels.

  • Method

    The paper combines AWGN numerical analysis, split-step fiber simulations, and modulation-dependent GN modeling to evaluate shaped QAM PMFs and propagated effective SNR.

  • Results

    Two shaped PMFs per QAM format remain within 0.1 dB SNR of full shaping gain over broad AWGN ranges, while one 64QAM PMF yields large gains from 1,400 km to 3,000 km and up to 0.65 dB sensitivity improvement.

  • Takeaways & Limitations

    An AWGN-optimized PMF is also a good practical choice for multi-span fiber systems because nonlinear shaping loss has relatively minor impact and channel-specific optimization adds no significant gain for the considered 64QAM system.

Abstract

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Different aspects of probabilistic shaping for a multi-span optical communication system are studied. First, a numerical analysis of the additive white Gaussian noise (AWGN) channel investigates the effect of using a small number of input probability mass functions (PMFs) for a range of signal-to-noise ratios (SNRs), instead of optimizing the constellation shaping for each SNR. It is shown that if a small penalty of at most 0.1 dB SNR to the full shaping gain is acceptable, just two shaped PMFs are required per quadrature amplitude modulation (QAM) over a large SNR range. For a multi-span wavelength division multiplexing (WDM) optical fiber system with 64QAM input, it is shown that just one PMF is required to achieve large gains over uniform input for distances from 1,400 km to 3,000 km. Using recently developed theoretical models that extend the Gaussian noise (GN) model and full-field split-step simulations, we illustrate the ramifications of probabilistic shaping on the effective SNR after fiber propagation. Our results show that, for a fixed average optical launch power, a shaping gain is obtained for the noise contributions from fiber amplifiers and modulation-independent nonlinear interference (NLI), whereas shaping simultaneously causes a penalty as it leads to an increased NLI. However, this nonlinear shaping loss is found to have a relatively minor impact, and optimizing the shaped PMF with a modulation-dependent GN model confirms that the PMF found for AWGN is also a good choice for a multi-span fiber system.

I. INTRODUCTION

The paper studies probabilistic shaping for QAM optical systems, focusing on achievable information rates and shaping effects in nonlinear WDM fiber channels. It combines AWGN analysis with auxiliary-channel AIR evaluation and fiber-channel modeling to assess practical shaped inputs.

  • Motivation: Optical systems require higher spectral efficiency as usable fiber bandwidth and transceiver-limited SNR constrain capacity growth.The paper motivates digital signal processing techniques that improve sensitivity and spectral efficiency while remaining robust to fiber nonlinearities.
  • Shaping approach: Probabilistic shaping uses a uniform-grid constellation with nonuniform symbol probabilities, achieving up to 1.53 dB shaping gain over the AWGN channel.Unlike geometric shaping, probabilistic shaping preserves constellation spacing while varying point probabilities.
  • Study scope and method: The study investigates probabilistic shaping for QAM over WDM fiber using a modulation-dependent GN model and full-field SSFM simulations.The GN model incorporates higher-order standardized modulation moments to analyze how the input distribution affects NLI.
  • System model: The transmitter uses probabilistic shaping, a bit-wise demapper based on qY |X, and a binary decoder, while fiber simulations omit the dotted FEC blocks to focus on AIRs.The nonuniform input distribution is used in the transceiver blocks marked with PX.
  • AIR framework: The analysis evaluates symbol-wise and bit-wise achievable information rates, with simulations using the BMD rate and an auxiliary channel instead of the unknown optical channel.The BMD rate accounts for dependent bit levels and is bounded above by symbol-wise mutual information.

B. Probabilistic Shaping with Maxwell-Boltzmann PMFs

The paper optimizes Maxwell-Boltzmann input distributions for QAM over an AWGN channel by jointly selecting constellation scaling and a shaping parameter to maximize RBMD.

  • The input distribution is chosen from a Maxwell-Boltzmann family, with probabilities controlled by a scaling factor ν.For each ν, a constellation scaling ρ enforces the average-power constraint.
  • The optimization jointly selects ρ and ν to maximize RBMD under the normalized average-power constraint.The channel SNR is defined using unit signal power and noise variance per dimension.
  • The AWGN results compare uniformly distributed QAM with QAM using optimized Maxwell-Boltzmann inputs across 16QAM, 64QAM, and 256QAM.The AWGN capacity 2 · log2(1 + SNR) is included as a reference.
  • RBMD is evaluated in bits per four-dimensional symbol to align the AWGN analysis with dual-polarization AIRs used later.The paper focuses on RBMD because it is more practical than RSYM and virtually achieves it.
  • Probabilistic shaping provides sensitivity improvements of up to 0.43 dB for 16QAM, 0.8 dB for 64QAM, and more than 1 dB for 256QAM over uniform input.

C. Shaping with Fixed PMFs

The study evaluates whether fixed shaped PMFs can retain most AWGN shaping gain across broad SNR ranges, reducing the need for transmitter-side SNR knowledge and frequent distribution optimization.

  • Transmitter-side shaping-SNR uncertainty arises from varying channel conditions, copropagating signals, DSP convergence, and component aging.Shaping without channel-SNR knowledge could simplify implementation.
  • 0.1 dB SNR loss is the accepted maximum relative to full shaping gain when selecting fixed PMFs for each QAM format.The selected PMF covers the largest SNR range satisfying this constraint.
  • A single fixed PMF covers a large SNR range for each QAM format, but the covered intervals are disconnected.An additional PMF is therefore needed to eliminate gaps.
  • Two input distributions per modulation format are sufficient to obtain a large shaping gain.The additional PMFs operate with an AWGN-capacity gap that is always larger than 0.1 dB.
  • The limited PMF set can facilitate shaping implementation when rate-adaptive FEC uses the shaping rate gain.The paper then examines the impact of probabilistic shaping on fiber nonlinearities.

III. SPM-XPM MODEL FOR THE NONLINEAR FIBER CHANNEL

The nonlinear-fiber analysis extends the classic GN perspective with modulation-dependent modeling and compares its implications with full-field SSFM simulations.

  • The effective SNR after optical-fiber propagation and receiver DSP combines amplifier noise with nonlinear-interference effects.ASE denotes amplifier spontaneous-emission noise, while NLI includes intra- and inter-channel distortions.
  • The classic GN model treats nonlinearities as additive memoryless circularly symmetric Gaussian noise independent of the channel input.This input independence is identified as an inaccurate simplification for modulation-dependent nonlinear effects.
  • Refined GN models incorporate channel-input properties into NLI variance modeling, enabling analysis of probabilistic shaping without expensive SSFM simulations.
  • The paper uses the frequency-domain SPM-XPM model, including self-phase and cross-phase modulation while omitting four-wave mixing.Four-wave mixing was found numerically negligible for the considered multi-span setup.

A. SPM-XPM Model

The SPM-XPM model separates modulation-independent noise from input-dependent nonlinear noise and expresses the latter through standardized input moments and fiber-nonlinearity coefficients.

  • The NLI variance is expressed using the standardized input moments µ̂4 and µ̂6 together with coefficients χ0, χ4, χ′4, and χ6.The coefficients represent contributions from fiber nonlinearities.
  • The model assumes all WDM channels use the same modulation format and average launch power Ptx.
  • The coefficients are rearranged to present the relationship between nonlinear-interference contributions and the input moments more clearly.
  • Table II compares µ̂4 and µ̂6 across complex modulation formats and input distributions.Constant-modulus PSK minimizes both moments, while shaped Maxwell-Boltzmann QAM has larger moments than uniform input.
  • The total noise is split into modulation-independent and modulation-dependent terms.The first term depends on system and fiber parameters, whereas the second depends on µ̂4 and µ̂6 and therefore on the channel input.

B. Standardized Moments

The channel input's standardized moments characterize modulation-dependent nonlinear interference, with shaped Maxwell–Boltzmann inputs having larger moments than uniform inputs.

  • Role in nonlinear interference: The effective SNR depends on the channel input's standardized moments ˆµ4 and ˆµ6.
  • Modulation dependence: Constant-modulus PSK minimizes both ˆµ4 and ˆµ6, while uniform QAM moments increase with modulation order.
  • Shaping dependence: Shaped Maxwell–Boltzmann QAM inputs have larger ˆµ4 and ˆµ6 than uniform input distributions.

C. NLI Increase due to Shaping

Probabilistic shaping changes modulation-dependent NLI through higher-order moments, creating a trade-off between shaping gains and increased nonlinear interference.

  • Moment dependence: Small ˆµ4 decreases (ˆµ4−2) and therefore reduces the associated NLI contribution.
  • Moment dependence: PSK minimizes ˆµ4 and ˆµ6, whereas AWGN-suited shaped PMFs have increased moments and can induce more NLI.
  • Shaping trade-off: Shaping improves ASE and modulation-independent NLI contributions while simultaneously increasing modulation-dependent NLI.
  • Shaping trade-off: The optimal trade-off between shaping gain and shaping penalty is not obvious and is studied numerically for QAM.

IV. PROBABILISTIC SHAPING OF 64QAM FOR A MULTI-SPAN FIBER SYSTEM

The multi-span fiber analysis evaluates achievable information rates with uniform and shaped 64QAM, focusing on shaping's effect on fiber nonlinearities and effective SNR.

  • Scope: The analysis numerically evaluates AIRs for a multi-span fiber link using uniform and shaped 64QAM inputs.
  • Scope: The study focuses on how probabilistic shaping affects fiber nonlinearities and the resulting effective SNR.
  • Scope: Transceiver impairments and further effects requiring or resulting from advanced DSP are excluded.

A. Fiber Simulations

The fiber simulations propagate shaped and uniform 64QAM through a multi-span WDM system, then evaluate effective SNR and bit-metric achievable rates while comparing model and simulation behavior.

  • Simulation setup: 64QAM symbols are distribution-matched, pulse-shaped, propagated span by span with SSFM, amplified by EDFAs, and corrupted by added ASE noise.
  • Simulation setup: All WDM channels use the same PMF with statistically independent symbol sequences, while the center channel is filtered and digitally processed at reception.
  • Metrics: The channel SNR is averaged over both polarizations, and RBMD is calculated per polarization before summing across polarizations.
  • NLI evaluation: At 2,000 km, χ0 and χ4 are the dominant NLI contributions used to compute effective SNR.
  • NLI evaluation: For the considered parameters, increasing ˆµ4 produces negative terms involving χ′4 and the modulation-dependent NLI contribution.

C. Reach Increase from Shaping

For 64QAM over multi-span WDM fiber, probabilistic shaping improves achievable rate and reach, while one fixed PMF can match distance-dependent shaping across much of the studied range.

  • C. Reach Increase from Shaping: 300 km reach improvement, from 2,000 km to 2,300 km, is achieved at an AIR of 8.86 bit/4D-sym with distance-matched shaping.Shaping also provides sensitivity improvements of up to 0.65 dB over uniform input.
  • C. Reach Increase from Shaping: From 1,400 km to 3,000 km, fixed PMF d) provides identical gains to distance-matched shaping.The comparison uses uniform input, distance-dependent shaped PMFs, and fixed PMF d) for 64QAM.
  • C. Reach Increase from Shaping: At 2,000 km, shaping improves AIR by 0.35 bit/4D-sym over uniform input at the optimum launch power.The SPM-XPM model and SSFM simulations show good agreement across launch powers.
  • C. Reach Increase from Shaping: In the highly nonlinear regime, shaping gains are significantly reduced and disappear at very high launch powers because of shaping’s adverse nonlinear effects.These effects limit the benefit of shaping at excessive transmit powers.

E. Sensitivity Analysis of Probabilistic Shaping

The analysis links probabilistic shaping’s nonlinear penalty to input-dependent NLI, then evaluates whether channel-specific PMF optimization improves performance beyond AWGN-oriented shaping.

  • 1) Shaping Decreases SNR:: Decreasing SNR mismatch increases moments µ̂4 and µ̂6, which increases NLI and decreases effective channel SNR.The SPM-XPM model predicts the simulation trend within 0.05 dB, while AWGN behavior is input-independent.
  • 2) Shaping Increases AIR:: 0.4 bit/4D-sym is the peak shaping gain, with gains above 0.3 bit/4D-sym from approximately −2 dB to 4 dB mismatch.The gain curve is relatively flat around its maximum.
  • 2) Shaping Increases AIR:: ∆opt = 1.06 dB for fiber simulations, whereas ∆= 0 dB is optimal for the AWGN reference.The positive fiber optimum reflects the trade-off between shaping and increased NLI.
  • 2) Shaping Increases AIR:: The shaping penalty relative to AWGN is approximately 0.07 bit/4D-sym at the optimum mismatch and disappears as the input approaches uniformity.The penalty is attributed to increased NLI from shaping and is relatively small.
  • F. Optimized Shaping for the Nonlinear Fiber Channel: At Ptx = −8 dBm and Ptx = 3 dBm, optimized PMFs have virtually identical effective SNRs, but the higher-power PMF is less shaped to avoid increased NLI.The optimization uses the SPM-XPM model and compares 1D and 2D PMFs.
  • F. Optimized Shaping for the Nonlinear Fiber Channel: The optimized 1D and 2D PMFs yield identical RBMD, while fixed PMF d) effectively achieves the maximum shaping gain for the considered system.There is virtually no benefit to the additional degrees of freedom of 2D optimization.

V. CONCLUSIONS

The paper finds that fixed probabilistic shaping remains effective for multi-span 64QAM despite shaping-induced NLI, and that this penalty is minor near optimal launch power. It also identifies nonlinear-link settings where further shaping optimization may be worthwhile.

  • V. CONCLUSIONS: One fixed 64QAM PMF provides large shaping gains from 1,400 km to 3,000 km in the studied multi-span system.The conclusion states that fiber-channel PMF optimization is unnecessary for this considered system.
  • V. CONCLUSIONS: The NLI penalty caused by shaping is relatively minor around the optimal launch power in a multi-span system.The conclusion attributes this penalty to modulation-dependent nonlinear effects.
  • V. CONCLUSIONS: Further shaping optimization may benefit links where χ4 contributes significantly, including inline dispersion-managed or high-power single-span links.Such optimization may simultaneously increase shaping gain and reduce NLI.
  • V. CONCLUSIONS: Shaping across several time slots and four-dimensional optimization are proposed as future directions for exploiting temporal or polarization-related structure.The stated targets are increased spectral efficiency and possible benefits in highly nonlinear polarization-multiplexed links.
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