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Impact of Pointing Errors on the Performance of Mixed RF/FSO Dual-Hop Transmission Systems
Imran Shafique Ansari, Ferkan Yilmaz, Mohamed-Slim Alouini
TL;DR
The paper addresses how pointing errors affect asymmetric dual-hop RF/FSO transmission systems, extending prior analysis that assumed no pointing errors and covered limited performance measures. It derives exact Meijer’s G-function expressions for SNR statistics and related performance metrics, with simulation examples validating and illustrating the formulations.
Problem
Prior analysis of asymmetric RF/FSO dual-hop systems assumed no pointing errors and was limited to CDF and outage probability.
Method
The paper derives exact closed-form CDF, PDF, MGF, moments, higher-order amount of fading, modulation error rates, and ergodic capacity using Meijer’s G functions.
Results
The derived analytical results are verified through computer-based Monte Carlo simulations, which also illustrate effects of atmospheric turbulence and pointing-error severity on system performance.
Takeaways & Limitations
The resulting expressions provide an analytical characterization of asymmetric mixed RF/FSO relay performance under pointing errors across distributional and communication-performance measures.
Abstract
from arXiv · showhide
In this work, the performance analysis of a dual-hop relay transmission system composed of asymmetric radio-frequency (RF)/free-space optical (FSO) links with pointing errors is presented. More specifically, we build on the system model presented in [1] to derive new exact closed-form expressions for the cumulative distribution function, probability density function, moment generating function, and moments of the end-to-end signal-to-noise ratio in terms of the Meijer's G function. We then capitalize on these results to offer new exact closed-form expressions for the higher-order amount of fading, average error rate for binary and M-ary modulation schemes, and the ergodic capacity, all in terms of Meijer's G functions. Our new analytical results were also verified via computer-based Monte-Carlo simulation results.
I. INTRODUCTION
The paper studies asymmetric dual-hop systems combining RF and FSO links, motivated by FSO bandwidth and cost advantages but challenged by atmospheric turbulence and pointing errors. It extends prior non-pointing-error analysis to characterize end-to-end performance under pointing errors.
- Motivation: FSO offers higher bandwidth and capacity than traditional RF systems while remaining license-free and potentially more cost-effective.These properties can help address expensive and scarce RF spectrum.
- Motivation: Atmospheric turbulence degrades FSO performance, while building sway can misalign transmitter and receiver beams through pointing errors.The paper identifies pointing errors as a serious concern for FSO equipment on high-rise buildings.
- System rationale: Asymmetric links are practical because different hops may use different communication systems or traverse physically different paths.A mixed RF/FSO relay can combine RF access with FSO transport in real-life environments.
- System rationale: The proposed model can multiplex RF-only mobile users into a shared FSO link to extend last-mile connectivity without equipping users with FSO hardware.The motivation includes reducing the need for new fiber infrastructure and avoiding unnecessary modifications to mobile devices.
- Contribution: Prior results assumed no pointing errors and were limited to CDF and outage probability, whereas this work analyzes broader performance effects.The analysis covers asymmetric RF/FSO dual-hop systems with fixed-gain relays.
- Contribution: The paper derives closed-form SNR statistics and performance metrics, including higher-order amount of fading, modulation error rates, and ergodic capacity.The expressions are given in terms of Meijer’s G functions.
- System model: The end-to-end SNR uses γ = γ1γ2/(γ2+C), where γ1 and γ2 represent the RF and FSO hop SNRs and C is a fixed relay gain.The RF hop is modeled with Rayleigh fading, while the FSO hop uses Gamma-Gamma fading with pointing-error impairments.
- System model: The FSO parameters α and β describe atmospheric turbulence, while ξ measures the ratio of equivalent beam radius to pointing-error jitter.Lower α and β indicate more severe turbulence.
III. CLOSED-FORM STATISTICAL CHARACTERISTICS
The paper develops exact Meijer’s G-function representations for the end-to-end SNR distribution, beginning with its CDF and validating the result against the non-pointing-error limit.
- A. Cumulative Distribution Function: The CDF is first introduced for the end-to-end SNR γ.The derivation starts from the established CDF expression before transforming it into the paper’s closed form.
- A. Cumulative Distribution Function: The initial CDF expression is rewritten into an equivalent form to prepare the subsequent special-function manipulations.This reformulation is stated before the final closed-form result.
- A. Cumulative Distribution Function: The exponential term in the CDF is rewritten using a tabulated identity.This step supports the conversion of the distribution into Meijer’s G-function form.
- A. Cumulative Distribution Function: Applying additional identities and algebra yields the CDF of γ in closed form.The resulting expression is obtained using Meijer’s G-function transformations.
- A. Cumulative Distribution Function: As ξ → ∞, the pointing-error CDF converges to the corresponding non-pointing-error expression.This limit represents the absence of pointing-error impairment.
- A. Cumulative Distribution Function: The limiting CDF agrees with the previously reported result in [1, Eq. (15)].The agreement provides a consistency check for the derived expression.
B. Probability Density Function
The PDF is obtained by differentiating the derived CDF and is expressed exactly using Meijer’s G functions, with a non-pointing-error convergence check.
- B. Probability Density Function: Differentiating the CDF with respect to γ produces the exact closed-form PDF in terms of Meijer’s G functions.The derivation uses the product rule, a tabulated identity, and algebraic simplification.
- B. Probability Density Function: As ξ → ∞, the pointing-error PDF converges to the non-pointing-error case.This provides a limiting consistency check for the PDF expression.
C. Moment Generating Function
The MGF is derived from the end-to-end SNR CDF and represented in Meijer’s G-function form, then checked against the non-pointing-error limit.
- C. Moment Generating Function: The MGF is defined as Mγ(s) = E[e^-γs] and related to the CDF.This relation provides the starting point for deriving the closed-form MGF.
- C. Moment Generating Function: Substituting the CDF into the MGF relation and applying a tabulated identity yields the MGF after algebraic manipulation.The resulting expression is presented in closed form using Meijer’s G functions.
- C. Moment Generating Function: When ξ → ∞, the derived MGF converges to the corresponding non-pointing-error expression.The limit checks the formula under vanishing pointing-error effects.
D. Moments
The moments E[γ^n] are obtained by expressing them through the complementary CDF and substituting the derived CDF into the resulting integral.
- The moments E[γ^n] are defined through the complementary CDF of the end-to-end SNR.
- Substituting the CDF into the moment expression and applying [21, Eq. (7.813.1)] yields the moments in closed form.
- For ξ →∞, the moments expression converges to the corresponding non-pointing-errors result.
A. Higher-Order Amount of Fading
The higher-order amount of fading characterizes the SNR distribution through normalized moments, with a classical amount of fading obtained as the second-order special case.
- The nth-order amount of fading is defined as AF^(n) = E[γ^n]/E[γ]^n − 1 for instantaneous SNR γ.
- For n = 2, the derived higher-order expression reduces to the classical amount of fading.
- When ξ →∞, the nth-order amount of fading converges to the non-pointing-errors expression.
- For n = 2 under non-pointing errors, the result gives the classical amount of fading.
B. Error Probability
The average BER for binary modulation schemes is derived from the SNR characterization, with modulation-specific parameters and a non-pointing-errors limiting case.
- 1) Average BER:: Substituting the derived SNR expression into the averaging formula yields the average BER of various binary modulations.
- 1) Average BER:: The parameters p and q represent different binary modulation schemes and can be selected using the listed modulation references or Table I.
- 1) Average BER:: When ξ →∞, the BER expression converges to the non-pointing-errors result.
2) Average SER:
Average SER expressions are obtained for M-PSK, M-AM, and M-QAM by substituting the derived SNR result into established conditional-SER formulas.
- 2) Average SER:: The conditional SER framework provides average SER formulas for M-PSK, M-AM, and M-QAM over generalized fading channels.
- 2) Average SER:: Substituting the derived SNR expression into the corresponding formulas produces the SER expressions for M-PSK, M-AM, and M-QAM.
- 2) Average SER:: The analytical SER expressions are exact and can be accurately estimated using the rapidly convergent Gauss-Chebyshev Quadrature formula.
C. Ergodic Capacity
The ergodic capacity is expressed through the CCDF of the end-to-end SNR and represented using the extended generalized bivariate Meijer’s G function. The expression has a non-pointing-error limit and can be evaluated using a Mathematica implementation.
- The ergodic capacity C ≜ E[log2(1 + γ)] is expressed in terms of the CCDF of γ.
- The capacity derivation uses an identity for (1 + az)^−b and an integral identity to obtain an EGBMGF representation.
- When ξ →∞, the ergodic-capacity expression converges to the non-pointing-errors case.
- The capacity expressions can be evaluated using the MATHEMATICA® implementation of the EGBMGF.
V. RESULTS AND DISCUSSION
The results compare analytical and simulated BER and examine how modulation choice, atmospheric turbulence, and pointing errors affect BER and ergodic capacity. The simulations match the analytical results, while the reported trends show performance dependence on ξ, α, and β.
- Simulation results provide a perfect match to the analytical BER results for the binary modulation schemes.
- CBPSK outperforms NBFSK in the average BER results.
- As ξ increases and pointing-error effects decrease, the BER deteriorates and vice versa.
- As α and β decrease, indicating more severe atmospheric turbulence, the BER deteriorates and vice versa.
- As atmospheric turbulence becomes more severe, ergodic capacity decreases; higher α and β correspond to higher ergodic capacity.
- As ξ increases and pointing-error effects decrease, ergodic capacity decreases.
- The simulations illustrate effects of atmospheric turbulence, pointing-error severity, and parameter imbalance on system performance.