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Performance Studies of Underwater Wireless Optical Communication Systems with Spatial Diversity: MIMO Scheme
Mohammad Vahid Jamali, Jawad A. Salehi, Farhad Akhoundi
TL;DR
UWOC performance analysis must jointly address turbulence-induced fading, absorption, and scattering. The paper derives BER expressions for MIMO OOK systems with optimal and equal gain combining, using quadrature, lognormal-sum approximation, and photon counting. Results show substantial spatial-diversity gains, especially under stronger turbulence, with close agreement between analytical results and simulations.
Problem
Prior UWOC studies did not comprehensively analytically evaluate BER while accounting for all major channel impairments, including turbulence, absorption, and scattering.
Method
The paper models absorption and scattering with Monte Carlo impulse responses, turbulence with lognormal fading, and derives BER expressions for MIMO OOK systems using optimal or equal gain combining.
Results
Spatial diversity considerably improves system performance, particularly for channels with higher turbulence, and analytical results closely match numerical simulations.
Takeaways & Limitations
MIMO spatial diversity can improve UWOC performance over longer links and under stronger fading, while equal gain combining offers performance close to optimal combining with lower complexity.
Abstract
from arXiv · showhide
In this paper, we analytically study the performance of multiple-input multiple-output underwater wireless optical communication (MIMO UWOC) systems with on-off keying (OOK) modulation. To mitigate turbulence-induced fading, which is amongst the major degrading effects of underwater channels on the propagating optical signal, we use spatial diversity over UWOC links. Furthermore, the effects of absorption and scattering are considered in our analysis. We analytically obtain the exact and an upper bound bit error rate (BER) expressions for both optimal and equal gain combining. In order to more effectively calculate the system BER, we apply Gauss-Hermite quadrature formula as well as approximation to the sum of lognormal random variables. We also apply photon-counting method to evaluate the system BER in the presence of shot noise. Our numerical results indicate an excellent match between the exact and upper bound BER curves. Also {a good match} between {the} analytical results and numerical simulations confirms the accuracy of our derived expressions. Moreover, our results show that spatial diversity can considerably improve the system performance, especially for channels with higher turbulence, e.g., a $3\times1$ MISO transmission in a $25$ {m} coastal water link with log-amplitude variance of $0.16$ can introduce $8$ {dB} performance improvement at the BER of $10^{-9}$.
I. INTRODUCTION
UWOC offers high-bandwidth optical transmission but is degraded by absorption, scattering, turbulence, and related channel effects. This paper develops a comprehensive analytical MIMO framework using spatial diversity to evaluate BER under these impairments.
- Optical turbulence is a major cause of fading in UWOC and results from random refractive-index variations driven mainly by temperature and salinity fluctuations.
- Existing studies often modeled only subsets of UWOC impairments or estimated BER numerically, leaving comprehensive analytical evaluation incomplete.
- The paper models absorption and scattering through Monte Carlo channel impulse responses and turbulence through multiplicative lognormal fading.
- MIMO repetition coding sends the same OOK symbol from all transmitters, enabling full channel diversity through non-destructive intensity addition.
- The analysis derives BER expressions for optimal and equal gain combining, using Gauss-Hermite quadrature, lognormal-sum approximation, and photon counting for shot noise.
- Absorption and scattering are characterized by wavelength- and water-dependent coefficients, with minimum effects reported over 400 nm < λ < 530 nm.
C. MIMO UWOC System Model
The MIMO UWOC model transmits OOK signals from multiple transmitters through channels combining fading-free absorption/scattering responses with multiplicative turbulence fading. Signals captured by multiple apertures are combined under controlled power and aperture assumptions.
- Each transmitter-to-receiver channel combines a fading-free impulse response with a multiplicative lognormal turbulence coefficient.
- The system uses intensity-modulation direct-detection with OOK signaling, transmitting an ON-state pulse P(t).
- For M transmitters, total ON-state power remains P, distributed across transmitter powers P_i for fair comparison with a single-transmitter system.
- At each receiver, optical signals from all transmitters are captured and summed after propagation through their individual channel impulse responses.
- Multiple receivers are constrained so their combined aperture area equals the aperture size of the corresponding single-receiver scheme.
- The analytical expressions support arbitrary transmitter power allocation, transmitted pulse shape, and channel model, while numerical results use equal power, rectangular OOK pulses, Monte Carlo responses, and lognormal fading.
III. BER ANALYSIS
The BER analysis averages conditional detection errors over noise, fading, and data sequences while accounting for intersymbol interference. Upper bounds and numerical integration methods make the resulting expressions tractable.
- The analysis derives exact and upper-bound BER expressions for SISO and MIMO systems using optimal or equal gain combining.
- Receiver noise is modeled as an equivalent Gaussian component combining background light, dark current, thermal noise, and signal-dependent shot noise.
- The average BER is obtained by averaging conditional errors over fading coefficients and all possible neighboring-bit sequences within the channel memory.
- With channel correlation times of 10^-5 to 10^-2 seconds, consecutive bits may share the same fading coefficient.
- The ISI-based BER upper bound uses worst-case surrounding-bit sequences: all ones around a transmitted zero and all zeros around a transmitted one.
- Gauss-Hermite quadrature efficiently evaluates the fading averages by replacing integrals with weighted finite sums.
B. MIMO UWOC Link with OC
For MIMO links using optimal combining, the BER calculation averages conditional ON/OFF errors over fading and data sequences. Transmitter-diversity cases are simplified through lognormal-sum moment matching and one-dimensional quadrature.
- The receiver integrated current combines transmitter contributions and Gaussian receiver noise for BER evaluation.
- Equal gain combining sums receiving branches, whereas optimal combining applies MAP detection, reducing to ML detection for equally likely symbols.
- MIMO average BER is obtained by averaging conditional ON/OFF error probabilities over an M × N-dimensional fading integral and 2^Lmax data sequences.
- The BER upper bound uses worst-case neighboring-bit sequences and evaluates the resulting multidimensional integrals with Gauss-Hermite series.
- For transmitter diversity, weighted sums of lognormal variables are moment-matched to an equivalent lognormal variable, reducing BER averaging to a one-dimensional integral.
C. MIMO UWOC Link with EGC
The MIMO UWOC system with EGC derives conditional and average BER expressions, using quadrature, lognormal-sum approximation, and photon-counting analysis for shot noise.
- The receiver selects its threshold using channel-state information to minimize the error probability.
- Photon-counting BER analysis models shot noise, dark current, and background light as Poisson-distributed components, while thermal noise is Gaussian.
- Saddle-point approximation evaluates BER through receiver-output MGFs under the two transmitted bit hypotheses.
- Exact and upper-bound BER expressions are developed for EGC, with Gauss-Hermite quadrature used to calculate multidimensional integrals.
A. SISO Configuration
The SISO configuration expresses the integrate-and-dump receiver output with thermal, background, dark-current, and signal-dependent noise, then derives BER by conditioning and averaging over fading.
- The SISO receiver models integrated thermal noise as a zero-mean Gaussian random variable.Its variance is specified through Boltzmann’s constant, receiver temperature, and load resistance.
- The receiver output includes Poisson-distributed photoelectron, background-radiation, and dark-current components.
- The conditional BER is obtained from the receiver-output MGF and then averaged over the fading coefficient.
B. MIMO Configuration with EGC
The MIMO EGC configuration combines signals across multiple receiver branches, models their mixed noise statistics, and reduces BER evaluation through approximations when direct integration is costly.
- Each receiving aperture collects all transmitter signals, while separate photodetectors add branch-specific noise.
- MIMO receiver-output MGFs are derived conditioned on the vector of fading coefficients, with SISO and SIMO obtained as special cases.
- Gaussian and saddle-point approximations evaluate conditional BER before averaging over the fading-coefficient vector.
- Direct (M × N)-dimensional integration can require excessive computational time, especially when the number of links is large.
- The weighted sum of M × N fading variables is approximated by an equivalent lognormal random variable, reducing the BER calculation to a two-dimensional integral.
MIMO +
The analysis links system performance to channel memory, intersymbol interference, and spatial diversity, while using lognormal approximations for MISO fading sums.
- The MISO fading term is modeled as the sum of M lognormal random variables and evaluated through a one-dimensional integral.
- Increasing channel memory raises averaging cost, making upper-bound BER evaluation more advantageous.
- More time spreading increases BER by increasing the zero-bit mean count and decreasing the one-bit mean count.
- Spatial diversity combines fading coefficients from different links and can be approximated by a lognormal variable with reduced log-amplitude variance.
V. NUMERICAL RESULTS
Numerical results validate the analytical BER expressions and show that spatial diversity improves UWOC performance, particularly under stronger turbulence, while absorption, scattering, noise, and ISI constrain gains across configurations and conditions.
- Validation: Analytical BER results closely match numerical simulations across the evaluated UWOC configurations.This agreement confirms the accuracy of the derived system BER expressions.
- Transmitter diversity: At σX = 0.1, additional transmitters provide little improvement because scattering and absorption remain substantial while fading has a minuscule effect.The result illustrates that spatial diversity primarily combats fading rather than non-fading channel impairments.
- Receiver combining: EGC performs very close to OC, making EGC more practically interesting because of its lower receiver complexity.The comparison covers 1 × 2 SIMO, 1 × 3 SIMO, and 2 × 2 MIMO configurations.
- Receiver diversity: Receiver diversity can trade aperture area and increased dark-current and thermal noise against improved fading mitigation, with the preferred configuration depending on SNR and turbulence.More receivers help at higher SNR under notable turbulence, whereas fewer receivers can perform better at low SNR or very weak turbulence.
- BER evaluation: Upper-bound BER curves remain tight to exact curves, while Gaussian approximation closely matches saddle-point approximation and photon-counting comparisons support the analytical results.The evaluated approximations and photon-counting methods are applied for a 25 m coastal water link with Rb = 1 Gbps and σX = 0.4.
- ISI and data rate: Increasing data rate worsens BER through stronger ISI, although MISO remains better than SISO at the examined rates.The effect is especially severe for multiple-receiver schemes; at Rb = 0.5 Gbps, direct and indirect-link delay spreads do not considerably degrade SIMO and MIMO performance.
VI. CONCLUSION
The paper derives BER results for MIMO UWOC with OOK under lognormal turbulence and evaluates both combining strategies, quadrature-based calculations, and shot-noise effects. Analytical results agree closely with photon-counting evaluations, while EGC offers lower complexity with performance close to optimal combining.
- The study analyzes MIMO UWOC with OOK using equal-gain and optimal combining under a channel model incorporating the relevant impairments.
- Closed-form BER expressions for lognormal underwater fading use Gauss-Hermite quadrature and an approximation for sums of lognormal random variables.
- Photon-counting evaluation in the presence of shot noise closely matches the analytical BER results, supporting the derivation assumptions.
- Equal-gain combining is more practically interesting because it has lower complexity and performance close to optimal combining.
- The study evaluates both exact BER expressions and an upper bound for the system BER.
APPENDIX A DECISION RULE FOR MIMO UWOC SYSTEM WITH OC
This appendix develops optimal-combining BER evaluation for MIMO UWOC using maximum-likelihood detection, conditional branch statistics, and multidimensional averaging over lognormal fading coefficients.
- Decision rule: The optimal-combining decision rule is derived from maximum-likelihood detection assuming perfect channel state information.
- Decision rule: The receiver uses integrated received-current vectors and fading-coefficient vectors to formulate conditional ON and OFF probabilities.
- Decision rule: Substituting the conditional probabilities into the detection metric simplifies the optimal-combining decision rule.
- BER averaging: Gauss-Hermite quadrature converts the M × N-dimensional fading average into a finite series for BER calculation.
- BER averaging: The quadrature expression is verified by induction for functions of M × N lognormal fading coefficients.
APPENDIX C MGF OF THE RECEIVER OUTPUT IN SISO SCHEME
The appendix derives receiver-output moment-generating functions by conditioning on fading and exploiting independence among random components and MIMO branches.
- SISO receiver output: The SISO receiver output is represented as the sum of two independent random variables, enabling its conditional MGF derivation.
- SISO receiver output: Independent Bernoulli data symbols simplify the expectation in the conditional SISO MGF.
- SISO receiver output: Multiplying the derived conditional MGF by the threshold-noise MGF yields the SISO receiver-output MGF.
- MIMO receiver output: For MIMO EGC, the combined receiver output is formed from branch outputs whose conditional MGFs multiply because the branches are independent given fading.
- MIMO receiver output: Assuming equal channel memory across links permits evaluation of the branch-output MGF, while zero-padding handles shorter-memory links.
- MIMO receiver output: The MIMO receiver-output MGF is obtained after combining the branch-level expressions.
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