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Unified Statistical Channel Model for Turbulence-Induced Fading in Underwater Wireless Optical Communication Systems

Emna Zedini, Hassan M. Oubei, Abla Kammoun, Mounir Hamdi, Boon S. Ooi, Mohamed-Slim Alouini

arXiv:1810.06314v1cs.IT

TL;DR

UWOC turbulence modeling must capture irradiance fluctuations caused jointly by air bubbles and temperature gradients in fresh and salty waters. The paper proposes a mixture EGG model based on experimental data, demonstrating perfect measured-data agreement across channel conditions and enabling closed-form performance analysis.

  • Problem

    Existing studies do not comprehensively model optical-beam irradiance fluctuations under both air bubbles and temperature gradients across underwater channels.

  • Method

    The paper proposes a mixture Exponential-Generalized Gamma distribution model for irradiance fluctuations and analyzes UWOC under IM/DD and heterodyne detection.

  • Results

    The mixture EGG model perfectly matches measured data collected under channel conditions ranging from weak to strong turbulence.

  • Takeaways & Limitations

    The EGG model provides a unified statistical description for turbulent UWOC channels involving air bubbles, temperature gradients, and salinity gradients.

Abstract

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A unified statistical model is proposed to characterize turbulence-induced fading in underwater wireless optical communication (UWOC) channels in the presence of air bubbles and temperature gradient for fresh and salty waters, based on experimental data. In this model, the channel irradiance fluctuations are characterized by the mixture Exponential-Generalized Gamma (EGG) distribution. We use the expectation maximization (EM) algorithm to obtain the maximum likelihood parameter estimation of the new model. Interestingly, the proposed model is shown to provide a perfect fit with the measured data under all channel conditions for both types of water. The major advantage of the new model is that it has a simple mathematical form making it attractive from a performance analysis point of view. Indeed, we show that the application of the EGG model leads to closed-form and analytically tractable expressions for key UWOC system performance metrics such as the outage probability, the average bit-error rate, and the ergodic capacity. To the best of our knowledge, this is the first-ever comprehensive channel model addressing the statistics of optical beam irradiance fluctuations in underwater wireless optical channels due to both air bubbles and temperature gradient.

I. INTRODUCTION

UWOC reliability is affected by absorption, scattering, and turbulence-induced irradiance fluctuations, while existing statistical models do not comprehensively capture combined air bubbles and temperature gradients. The paper proposes a unified mixture EGG model with strong measured-data agreement and tractable performance analysis.

  • Motivation: UOT arises from temperature fluctuations, salinity variations, and air bubbles, distorting optical-signal intensity and phase and degrading UWOC performance.Air bubbles can enhance scattering in underwater channels.
  • Research gap: The Lognormal distribution is inappropriate for turbulent-water irradiance fluctuations because underwater refractive-index spectra differ from atmospheric spectra.The introduction identifies a need for new accurate statistical models for turbulence-induced fading.
  • Research gap: Air-bubble measurements require a two-lobe statistical model because typical single-lobe distributions do not fit measured irradiance data across scintillation conditions.Prior work modeled bubble-induced irradiance using mixtures involving Exponential and Lognormal or Gamma components.
  • Research gap: Existing studies separately addressed air bubbles or temperature and salinity gradients, leaving no comprehensive statistical model for their combined effects.The paper targets fresh and salty waters under weak-to-strong turbulence conditions.
  • Contributions: The proposed mixture Exponential-Generalized Gamma model provides excellent goodness of fit under all channel conditions and supports closed-form outage, BER, and ergodic-capacity expressions.The paper also derives high-SNR asymptotic expressions in elementary functions.

A. Turbulent UWOC Channels with Gradient Temperature

The experiments measure intensity fluctuations in 1 m underwater channels using controlled air-bubble levels and temperature conditions in fresh and salty water. Optical transmission and photodiode detection provide samples for statistical analysis.

  • Temperature conditions: Temperature-gradient experiments include a gradient of 0.22 ◦C.cm−1 and water temperatures of 17.3 ◦C and 39.2 ◦C.These temperatures were used to create a strong air-bubble effect.
  • Water conditions: Both fresh and salty waters are measured, with salinity produced by adding 118 g of table salt to the fresh-water tank.The tank interior is painted black to minimize sidewall reflections.
  • Air-bubble conditions: Five air-bubble levels are generated at BL=0 L/min, 2.4 L/min, 4.7 L/min, 7.1 L/min, and 16.5 L/min.Bubbles are introduced through a PVC pipe with 2 mm holes.

III. MODELING UNDERWATER TURBULENCE WITH THE MIXTURE EGG MODEL

The paper models received irradiance fluctuations caused by air bubbles and temperature-induced fading in fresh and salty water using a mixture EGG distribution. Parameters are estimated with EM from measured irradiance observations, supporting tractable UWOC performance analysis.

  • Channel scope: The model applies to air-bubble and temperature-induced fading in both fresh and salty UWOC channels.The salty-water measurements were obtained by adding salt to the fresh-water tank.
  • Mixture EGG model: The mixture EGG model represents irradiance fluctuations as a weighted combination of Exponential and Generalized Gamma distributions.The mixture coefficient satisfies 0 < ω < 1; λ belongs to the Exponential component, while a, b, and c parameterize the Generalized Gamma component.
  • Special case: For thermally uniform channels, the EG mixture is a special case of EGG obtained when c = 1.The EG model combines Exponential and Gamma distributions and has a simpler mathematical form.
  • Analytical tractability: The EGG model enables closed-form and mathematically tractable evaluation of UWOC performance metrics.The paper identifies outage probability, average BER, and ergodic capacity as target metrics for this analysis.
  • Parameter estimation: The EM algorithm estimates the mixture parameters by alternating hidden-variable expectation and parameter-maximization steps.Each irradiance observation is associated with a hidden binary variable indicating its component distribution.
  • Experimental parameter fitting: The fitted parameters vary with water temperature, salinity, and air-bubble level across the measured channel conditions.The experiments use 100000 irradiance realizations for each specified channel condition, with sampling at 100 S/s.

IV. GOODNESS OF FIT TESTS

The goodness-of-fit evaluation compares EGG with EG and Exponential-Lognormal distributions using MSE and R2 tests. Across fresh and salty channels and varying air-bubble and temperature-gradient levels, the EGG model provides analytical tractability and efficiently describes the measured fluctuations.

  • Evaluation design: The evaluation uses MSE and R2 tests to assess how well the distributions fit measured irradiance data.These tests are widely used to evaluate fading distributions against channel measurements.
  • Compared models: The comparison includes the proposed EGG distribution, the EG distribution, and the Exponential-Lognormal distribution.Estimated parameters and goodness-of-fit results are reported for different air-bubble levels under thermally uniform and gradient-based channels.
  • Overall finding: The model describes irradiance fluctuations under all channel conditions for both fresh and salty waters while remaining analytically tractable.This conclusion is based on the reported comparison with EG and Exponential-Lognormal distributions.

A. MSE Test

The MSE test measures the discrepancy between empirical and theoretical irradiance distributions, with lower values indicating better fit. The reported R2 measure provides a complementary goodness-of-fit assessment, but depends on histogram binning.

  • MSE definition: MSE evaluates how accurately the EGG model predicts measured irradiance fluctuations.It compares the empirical and theoretical distribution functions computed from fitted parameters.
  • MSE interpretation: MSE values approaching 0 indicate a better fit to the acquired experimental data.The paper treats lower MSE as evidence of a better model fit.
  • R2 test: R2 quantifies goodness of fit using the sum of squared errors and total squared deviations from the measured-data mean.Its range is 0 to 1, and values approaching 1 indicate better fit to measured intensity.
  • R2 limitation: R2 depends on the number of bins used in the acquired data histogram.This binning dependence is an explicit limitation of the reported measure.

V. EXPERIMENTAL VALIDATION

Experimental validation shows that the EGG distribution matches measured irradiance data across weak-to-strong turbulence caused by air bubbles and temperature gradients. It also achieves the strongest reported goodness-of-fit and supports simpler analytical performance evaluation.

  • Temperature-gradient effects: As temperature-induced turbulence increases, the measured histogram becomes more left-skewed, stretched, and wider at the peak.The EG and Exponential-Lognormal distributions fit well at low temperature gradients but fail to follow the stretching shape as the gradient increases.
  • Distributional validation: The EGG model perfectly matches measured data under all channel conditions from weak to strong turbulence.The comparison covers irradiance histograms measured under air bubbles and temperature-induced turbulence.
  • Distributional validation: The EGG distribution remains suitable for irradiance fluctuations caused by both air bubbles and temperature-induced turbulence.The reported agreement holds across the tested turbulence conditions.
  • Scintillation index: The EGG scintillation index is closest to the value calculated from measured data.Increasing air-bubble level or temperature gradient increases turbulence strength and the scintillation index.
  • Goodness of fit: The EGG distribution gives the best overall quality of fit to the measured data.Its scale parameter increases as channel turbulence increases, indicating the stretching nature of the histogram.
  • Goodness of fit: The EGG model has the smallest MSE and highest R2 values under all tested turbulence conditions.These results support EGG as a more accurate and simple alternative to the Exponential-Lognormal model for UWOC turbulence-induced fading.

B. Turbulent UWOC Channels with Uniform Temperature

Under uniform temperature, the EGG distribution is evaluated against measured irradiance data for fresh and salty waters across air-bubble turbulence conditions. It provides the best fit and supports tractable performance analysis for thermally uniform UWOC channels.

  • Model fit: The EGG model matches measured data across different air-bubble levels and turbulence conditions for both fresh and salty waters.The comparisons include weak, moderate, and strong turbulence conditions.
  • Model fit: EG and Exponential-Lognormal distributions also fit the measured data well, with nearly indistinguishable plots under uniform temperature.This makes EG attractive for modeling turbulence-induced fading in thermally uniform UWOC channels.
  • Water salinity: Adding 118g of table salt to the fresh-water tank does not significantly affect the scintillation index.The salinity comparison is reported for the uniform-temperature experiments.
  • Goodness of fit: The EGG distribution achieves the highest R2 values under all turbulence conditions and gives the best measured-data fit.The comparison uses estimated parameters and R2 goodness-of-fit results for EGG, EG, and Exponential-Lognormal distributions.
  • Performance analysis: The mathematically simple EGG model yields closed-form, analytically tractable expressions for outage probability and average BER.Its PDF also simplifies analytical calculations of UWOC performance metrics, with analytical accuracy verified by Monte Carlo simulations.

A. Probability Density Function

The paper transforms the mixture EGG irradiance model into a unified SNR-domain PDF for heterodyne detection and IM/DD. The resulting expression includes the EG model as a uniform-temperature special case and supports tight high-SNR approximations.

  • SNR model: The instantaneous SNR uses detection parameter r, with r = 1 for heterodyne detection and r = 2 for IM/DD.The average electrical SNR is defined from the normalized irradiance moments and the detection-dependent parameter.
  • SNR-domain PDF: The mixture EGG irradiance distribution is transformed into a unified SNR PDF expressed using the Fox’s H function.The derivation applies random-variable transformations and special-function identities to the EGG model.
  • Special case: Setting c = 1 reduces the resulting PDF to the EG fading model for uniform temperature.This establishes the uniform-temperature case as a special case of the unified formulation.
  • Asymptotic analysis: A tight high-SNR asymptotic expression for the CDF is derived from the unified SNR-domain model.The expression is used to derive high-SNR asymptotic approximations for ergodic capacity.

D. Applications to Performance Analysis

The EGG channel model is applied to derive analytical expressions for outage probability, average BER, and ergodic capacity. The formulations cover both detection techniques and multiple modulation schemes, with simpler special cases under uniform temperature.

  • Outage probability: Outage probability is obtained by evaluating the SNR CDF at the specified threshold γth.It is defined as the probability that instantaneous SNR γ falls below γth.
  • Average BER: A unified closed-form average-BER expression covers OOK, BPSK, M-QAM, and M-PSK under heterodyne detection and IM/DD.The modulation- and detection-dependent parameters are summarized in Table III.
  • Average BER: The average-BER derivation is expressed with the Fox’s H function and simplifies to a Meijer’s G representation for uniform temperature.The high-SNR average BER is also expressed asymptotically.
  • Ergodic capacity: The ergodic capacity is expressed in closed form for the EGG-based UWOC system.The derivation substitutes the EGG-domain expressions into the capacity definition and uses special-function representations.
  • Ergodic capacity: For c = 1, the capacity expression becomes the uniform-temperature UWOC capacity and can be simplified using the Meijer’s G function.A high-SNR asymptotic approximation is also derived from moments of γ.

VII. NUMERICAL RESULTS

Numerical results validate the EGG-based analytical expressions for outage probability, average BER, and ergodic capacity. Increasing air bubbles or temperature gradients strengthens turbulence and degrades reported system metrics, while high-SNR asymptotics closely match analytical results.

  • Outage probability: The analytical and simulation outage-probability results match closely under IM/DD across different turbulence conditions.The high-SNR asymptotic outage results also perfectly match the analytical results.
  • Channel conditions: Higher air-bubble levels or temperature gradients increase the scintillation index, indicating stronger turbulence and performance deterioration.The reported temperature-gradient comparison uses a fixed air-bubble level of BL=2.4 L/min.
  • Outage probability: At SNR=30 dB, Pout = 3.075850 × 10−2 for a 0.05◦C.cm−1 temperature gradient and Pout = 3.422170×10−2 for 0.15◦C.cm−1.The corresponding scintillation index values are I = 0.1484 and I = 0.1915, respectively, at BL=2.4 L/min.
  • Average BER: For salty water under uniform temperature, increasing air-bubble levels increases average BER for IM/DD with OOK.The analytical BER results match simulations, and the same behavior is observed for fresh water.
  • Average BER: BPSK outperforms the other tested modulation techniques, while 16-QAM outperforms 16-PSK under heterodyne detection in strong turbulence.The comparison includes 64-QAM, 16-QAM, 16-PSK, and BPSK at a bubbles level of 23.6 L/min and temperature gradient of 0.22 ◦C.cm−1.
  • Ergodic capacity: Analytical ergodic-capacity results agree perfectly with Monte Carlo simulations under IM/DD for varying air-bubble levels and temperature gradients.The high-SNR asymptotic results are reported as accurate and tight.

VIII. CONCLUSION

The paper proposes a unified mixture EGG model for irradiance fluctuations in turbulent UWOC channels, covering air bubbles and temperature gradients across fresh and salty waters. The model fits measured data across channel conditions and supports tractable performance analysis.

  • The mixture EGG model characterizes irradiance fluctuations across turbulent UWOC channels in both fresh and salty waters.It is intended to capture combined effects from air bubbles and temperature-related gradients.
  • The model perfectly matches measured data collected under weak-to-strong turbulence conditions in fresh and salty waters.
  • Under uniform water temperature, the EG distribution is identified as the best description of received laser-beam intensity and is a special case of EGG.
  • The paper derives exact closed-form expressions for outage probability, average BER across modulation schemes, and ergodic capacity under varied turbulence conditions.The analysis covers IM/DD and heterodyne detection in the presence of temperature gradients and air bubbles.
  • High-SNR asymptotic results for ergodic capacity are derived using a moments-based method and expressed with simple functions.
  • UWOC performance degrades as the temperature gradient or air-bubble level increases.The paper frames the model and analysis as relevant to improving underwater-link reliability.
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