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Unified Performance Analysis of Mixed Radio Frequency/Free-Space Optical Dual-Hop Transmission Systems
Jiayi Zhang, Linglong Dai, Yu Zhang, Zhaocheng Wang
TL;DR
Prior mixed RF/FSO analyses often used Rayleigh or Nakagami-m RF fading, which may not capture measured RF statistics well. This paper derives generalized end-to-end performance expressions for κ-µ or η-µ RF fading paired with ΓΓ FSO fading. Its analytical outage results agree with simulations, and its high-SNR approximation is tight in moderate- and high-SNR regimes in the reported setting.
Problem
Most prior mixed RF/FSO studies assumed Rayleigh or Nakagami-m RF fading, which may not fit experimental RF-link statistics well.
Source: §I · p. 1Method
The paper derives PDF, CDF, outage-probability, and average-BER expressions for dual-hop systems with κ-µ or η-µ RF fading and ΓΓ FSO fading.
Source: §I · p. 2Results
The analytical outage results match simulations, and the high-SNR approximation is tight in moderate- and high-SNR regimes for the reported setting.
Source: §V · p. 5Takeaways & Limitations
The framework includes Rayleigh, Rician, and Nakagami-m RF fading as special cases and recovers previous results as special cases.
Source: §I · pp. 1–2
Abstract
from arXiv · showhide
The mixed radio frequency (RF)/free-space optical (FSO) relaying is a promising technology for coverage improvement, while there lacks unified expressions to describe its performance. In this paper, a unified performance analysis framework of a dual-hop relay system over asymmetric RF/FSO links is presented. More specifically, we consider the RF link follows generalized $\kappa$-$\mu$ or $\eta$-$\mu$ distributions, while the FSO link experiences the gamma-gamma distribution, respectively. Novel analytical expressions of the probability density function and cumulative distribution function are derived. We then capitalize on these results to provide new exact analytical expressions of the outage probability and bit error rate (BER). Furthermore, the outage probability for high signal-to-noise ratios and the BER for different modulation schemes are deduced to provide useful insights into the impact of system and channel parameters of the overall system performance. These accurate expressions are general, since they correspond to generalized fading in the RF link and account for pointing errors, atmospheric turbulence and different modulation schemes in the FSO link. The links between derived results and previous results are presented. Finally, numerical and Monte-Carlo simulation results are provided to demonstrate the validity of the proposed unified expressions.
I. INTRODUCTION
The paper addresses the limited fit of common Rayleigh and Nakagami-m RF fading models by analyzing mixed RF/FSO relaying under more general fading. It derives generalized performance expressions and examines high-SNR outage and pointing-error effects.
I. INTRODUCTION
- Most earlier mixed RF/FSO studies assumed Rayleigh or Nakagami-m RF fading, which may not fit experimental RF-link statistics well.The paper identifies κ-µ and η-µ fading as models with a good fit to experimental data and a framework containing Rayleigh, Rician, and Nakagami-m as special cases.Source: §I · p. 1
- The analysis treats κ-µ or η-µ fading on the RF hop and ΓΓ fading on the FSO hop, with pointing errors and atmospheric turbulence included.In the mixed topology, RF and FSO transmission occupy different hops; it is distinct from a hybrid system with parallel RF and FSO links on the same path.Source: §I · pp. 1–2, Figure 1 · p. 2
- The paper derives generalized end-to-end outage probability and average BER expressions for mixed RF/FSO relaying across the stated fading models and modulation schemes.The formulations account for pointing errors and atmospheric turbulence in the FSO link.Source: §I · p. 2
- The contributions include an asymptotic closed-form outage expression at high SNR and an investigation of how FSO pointing errors affect BER across modulation schemes and fading conditions.Source: §I · p. 2
- For κ-µ/ΓΓ fading, fewer than ten terms of the infinite series provide satisfactory accuracy, such as error below 10^-6 in the considered cases.The derived PDF and CDF use Meijer’s G-functions, and prior Rayleigh and Nakagami-m results are shown to be special cases.Source: §I · p. 2
II. MIXED RF-FSO SYSTEM MODEL
The system is a dual-hop relay in which the source reaches the destination through an amplify-and-forward relay, with generalized RF fading and ΓΓ FSO fading.
II. MIXED RF-FSO SYSTEM MODEL
- The source cannot communicate directly with the destination because of distance or a blocking obstacle, so transmission proceeds through relay R.Source: §II · p. 2
- The S–R RF hop follows κ-µ or η-µ fading, which includes Rayleigh, Rician, and Nakagami-m as special cases; the R–D FSO hop follows ΓΓ fading.Source: §II · p. 2
- Relay R uses fixed-gain amplify-and-forward, amplifying its received signal before forwarding it to the destination.The FSO model assumes high-energy on-off keying and approximates background-radiation and thermal noise as AWGN with variance N0/2.Source: §II · p. 2
- The notation defines c as G²/N₀, with γ₁ denoting the RF-hop (S-R link) SNR and γ₂ denoting the electrical SNR of the FSO hop (R-D link).Source: §II · p. 2
III. STATISTICAL CHARACTERISTICS
The section derives the mixed RF/FSO system’s CDF first and obtains its PDF by differentiation.
III. STATISTICAL CHARACTERISTICS
- The CDF is derived before the PDF, which is obtained by differentiating the CDF expression.The section states these expressions support the subsequent analysis.Source: §III · p. 2
A. Cumulative Distribution Function
The section derives end-to-end SNR CDFs for mixed RF/FSO links using generalized RF fading and an FSO channel with pointing errors and atmospheric turbulence. It also establishes accuracy and special-case relationships for the resulting expressions.
A. Cumulative Distribution Function
- The end-to-end CDF uses the relay-node SNR PDF and destination electrical-SNR CDF, with the FSO model accounting for detection type and pointing errors.The detection parameter t is 1 for heterodyne and 2 for IM/DD; smaller ξ means stronger pointing errors.Source: A. Cumulative Distribution Function · pp. 2–3
- For η-μ fading, the derivation uses Format 1 with integer μ, while separate procedures yield CDF expressions for η-μ/ΓΓ and κ-μ/ΓΓ fading.The η-μ case uses a binomial expansion and an integral identity; the κ-μ case uses a finite-series representation of the incomplete gamma function.Source: A. Cumulative Distribution Function · p. 3
- The κ-μ/ΓΓ CDF can be truncated to fewer than ten terms with satisfactory accuracy, such as error below 10^-6.The stated accuracy applies to the considered cases.Source: A. Cumulative Distribution Function · p. 3
- The generalized CDF reduces to earlier Nakagami-m/ΓΓ and Rayleigh/ΓΓ results under the stated parameter choices.The Rayleigh/ΓΓ case is specified by η→0, κ→0, and μ=1, and agrees with [5, Eq. (2)].Source: A. Cumulative Distribution Function · pp. 3–4
B. Probability Density Function
The section obtains analytical PDFs for both generalized mixed-link fading cases by differentiating their CDFs. The expressions recover earlier Nakagami-m/ΓΓ and Rayleigh/ΓΓ results under IM/DD.
B. Probability Density Function
- The analytical PDFs follow by differentiating the two end-to-end CDF expressions with respect to γ.The resulting cases reduce to prior Nakagami-m/ΓΓ and Rayleigh/ΓΓ PDFs when IM/DD is used.Source: B. Probability Density Function · p. 4
IV. PERFORMANCE ANALYSIS
The performance analysis derives exact expressions for end-to-end outage probability and average BER, and assesses high-SNR outage and BER without pointing errors.
IV. PERFORMANCE ANALYSIS
- The section derives analytical expressions for end-to-end outage probability and average BER in mixed RF/FSO dual-hop systems.Source: §IV · p. 4
- It assesses the asymptotic outage probability in the high-SNR regime.Source: §IV · p. 4
- It also assesses BER performance without pointing errors.Source: §IV · p. 4
A. Outage Probability
Outage probability is the chance that the end-to-end SNR falls below a threshold, and its high-SNR approximation shows how outage changes as the FSO-hop SNR increases.
A. Outage Probability
- Outage probability is the probability that the end-to-end SNR falls below a predetermined threshold.Source: A. Outage Probability · p. 4
- At high SNR, the derived asymptotic outage probability becomes smaller as the average electrical SNR of the FSO link increases.The high-SNR expressions are given in elementary functions and agree with findings in –.Source: A. Outage Probability · p. 4
B. Bit Error Rate
The BER analysis uses a unified expression parameterized by modulation scheme and relates the resulting expressions to prior special cases.
B. Bit Error Rate
- Parameters p and q select the binary modulation scheme, including NBFSK and CBFSK as specified examples.NBFSK uses p = 1 and q = 0.5; CBFSK uses p = 0.5 and q = 0.5.Source: B. Bit Error Rate · p. 4
- Substituting the derived end-to-end distributions into the unified BER expression yields analytical BER expressions.Source: B. Bit Error Rate · p. 4
- For mixed Rayleigh/ΓΓ fading with IM/DD, the BER expressions reduce to the result in [5, Eq. (14)].Source: B. Bit Error Rate · p. 5
- As pointing errors vanish, the BER expressions converge to non-pointing-error forms, which reduce to [5, Eq. (15)] for IM/DD.The paper also states that the same approach can obtain average BER expressions for non-binary modulation schemes.Source: B. Bit Error Rate · p. 5
V. NUMERICAL RESULTS
Numerical and Monte Carlo results examine outage and BER across RF/FSO fading and turbulence conditions, comparing analytical expressions with simulations and high-SNR approximations.
V. NUMERICAL RESULTS
- For η-µ/ΓΓ fading, analytical and simulated outage results match, while the high-SNR approximation is tight at moderate and high FSO-hop SNRs.Increasing µ and/or η helps overcome fading under the stated weak-turbulence and large-pointing-error settings.Source: §V · p. 5, Figure 2 · p. 5
- Analytical and simulated outage results match well for κ-µ/ΓΓ fading, and ten terms suffice for accurate numerical results in the selected Fig. 3 settings.Increasing κ or µ helps overcome fading, with µ having the more pronounced effect.Source: Figure 3 · p. 5, §V · p. 6
- Under adverse weather or strong turbulence, the mixed RF/FSO system loses substantial outage performance.Source: §V · p. 6, Figure 4 · p. 6
- For η-µ fading and moderate turbulence, analytical BER matches simulation; CBFSK has lower BER than NBFSK, and pointing errors matter especially when ξ is small.Fig. 5 varies the average electrical SNR of the RF hop.Source: §V · p. 6, Figure 5 · p. 6
VI. CONCLUSIONS
The paper derives unified performance expressions for mixed RF/FSO dual-hop relaying and uses them to characterize outage and BER across generalized RF fading and FSO impairments.
VI. CONCLUSIONS
- The authors derive exact end-to-end SNR CDF/PDF, outage probability, and average BER expressions for η-μ/ΓΓ and κ-μ/ΓΓ mixed links.The expressions use finite sums of Meijer G-functions and account for pointing errors, atmospheric turbulence, and different modulation schemes.Source: §VI · p. 6
- Fewer than ten terms yield error below 10^-6 for the κ-μ/ΓΓ results in all considered cases.The corresponding infinite series converges quickly and steadily.Source: §VI · p. 6
- At high SNR, outage probability decreases as the FSO link’s electrical SNR increases.The conclusion identifies this relationship as an insight from the high-SNR asymptotic outage probability.Source: §VI · p. 6
- Pointing errors have an obvious effect on BER performance.Source: §VI · p. 6
APPENDIX
The appendix evaluates an integral involving power, exponential, and Meijer G-functions through variable changes and cited identities.
APPENDIX
- The appendix frames the integral in terms of power, exponential, and Meijer G-functions.Source: APPENDIX · p. 7
- A change of variables, x^-1 to x, and identity [24, Eq. (9.31.2)] are used to transform the integral.Source: APPENDIX · p. 7
- The derivation uses identity [33, Eq. (2.24.3.1)] and applies [24, Eq. (9.31.2)] again to express the analytical result.Source: APPENDIX · p. 7