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Performance Analysis of Decode-and-Forward Relaying in Gamma-Gamma Fading Channels

Manav R. Bhatnagar

arXiv:1205.0326v1cs.ITmath.IT

TL;DR

The paper studies DF cooperative FSO communication under Gamma-Gamma fading, a model adopted for a wide range of atmospheric turbulence conditions. It derives CDF and PDF expressions for a Gamma-Gamma/Gaussian mixture and uses them to analyze BER. Simulations and analysis report a 14 dB SNR gain over direct transmission at BER=10^-4 in moderate turbulence.

  • Problem

    Cooperative FSO studies cited here used log-normal fading models commonly associated with weak turbulence, while Gamma-Gamma fading is adopted for a wider range of atmospheric conditions.

    Source: §I · p. 1

  • Method

    The paper derives the CDF and PDF of a random variable containing Gamma-Gamma and Gaussian variables and uses these expressions to analyze average BER for DF relaying.

    Source: §I · p. 1, §III · pp. 1–2

  • Results

    At BER=10^-4 in moderate turbulence, selective-relay DF achieves a 14 dB SNR gain over the non-cooperative scheme.

    Source: §V · p. 3

  • Takeaways & Limitations

    The analysis and simulation show that DF cooperative FSO has significantly lower BER at high SNR than direct non-cooperative transmission.

    Source: §VI · p. 3

Abstract

from arXiv · show

Decode-and-forward (DF) cooperative communication based on free space optical (FSO) links is studied in this letter. We analyze performance of the DF protocol in the FSO links following the Gamma-Gamma distribution. The cumulative distribution function (CDF) and probability density function (PDF) of a random variable containing mixture of the Gamma- Gamma and Gaussian random variables is derived. By using the derived CDF and PDF, average bit error rate of the DF relaying is obtained.

I. INTRODUCTION

The introduction motivates cooperative FSO communication and Gamma-Gamma fading as a model for a wide range of atmospheric turbulence. The paper analyzes DF system BER and derives the CDF and PDF needed for that analysis.

I. INTRODUCTION

  • FSO communication is motivated by its large bandwidth, low implementation cost, and security, while atmospheric turbulence complicates OOK detection-threshold selection.The introduction notes that SIM permits modulation schemes such as PSK and QAM.Source: §I · p. 1
  • Gamma-Gamma fading is adopted for FSO links because it models a wide range of turbulence conditions and fits experimental results.Earlier cooperative FSO studies cited here modeled turbulence with the log-normal distribution, commonly used for weak turbulence.Source: §I · p. 1
  • The paper derives the CDF and PDF of a random variable mixing Gamma-Gamma and Gaussian variables, then uses them to derive average BER for DF relaying.The system uses subcarrier intensity modulation with BPSK.Source: §I · p. 1

II. SYSTEM MODEL

The system uses a two-phase, BPSK subcarrier-intensity-modulated FSO link with one source, one relay, and one destination. The relay forwards only after correct decoding, and the links are modeled with Gamma-Gamma irradiance and additive Gaussian noise.

II. SYSTEM MODEL

  • The source transmits BPSK to the relay and destination in phase one, and the relay forwards its decoded symbols in phase two only after correct decoding.A CRC code lets the relay determine whether it decoded correctly.Source: §II · p. 1
  • The topology uses two source transmit apertures, two destination receive apertures, and a relay with one transmit and one receive aperture.The source apertures serve the relay and destination links; the destination apertures receive from the relay and source.Source: §II · p. 1
  • The S-R, S-D, and R-D links use Gamma-Gamma irradiance with unit mean and signal-independent, zero-mean complex AWGN.The model assigns each link its own irradiance and noise variance, with optical-to-electrical conversion coefficients for the links.Source: §II · p. 1

III. CHARACTERIZATION OF A RANDOM VARIABLE CONTAINING MIXTURE OF GAMMA-GAMMA AND GAUSSIAN RANDOM VARIABLES

This section characterizes a random variable formed from a Gamma-Gamma variable and a real Gaussian variable. It gives piecewise CDF and PDF expressions for that mixture, which underpins later analysis.

III. CHARACTERIZATION OF A RANDOM VARIABLE CONTAINING MIXTURE OF GAMMA-GAMMA AND GAUSSIAN RANDOM VARIABLES

  • The section defines Y = aZ^2 + bZE, where a and b are positive constants, Z is Gamma-Gamma distributed, and E is a standard real Gaussian variable.The section also identifies the Gamma-Gamma parameters with atmospheric turbulence conditions.Source: §III · pp. 1–2
  • Theorem 1 gives the CDF of Y in piecewise form for positive, zero, and negative y.The expression uses separate function arguments for the positive and negative cases and a value at zero.Source: §III · p. 2
  • Theorem 2 gives the PDF of Y in piecewise form for positive, zero, and negative y.The expression uses corresponding positive, zero, and negative cases for its function arguments.Source: §III · p. 2

IV. BIT ERROR RATE OF DF RELAYING

This section formulates error events for the source-relay and destination receivers using the mixture random variable characterized earlier. It expresses destination error probability through a sum of two such variables and gives a series-form expression involving link turbulence parameters.

IV. BIT ERROR RATE OF DF RELAYING

  • The source-relay error setup considers transmitting 1 when the relay decides −1, with its probability related to the distribution of Y.The passage identifies the S-R turbulence parameters in the expression for this error probability.Source: §IV · p. 2
  • Equation (20) gives the probability term Pr(U + V ≤ 0), with random variables U and V having the same distribution as Y.Source: §IV · p. 2
  • The destination-error expression uses sums over link turbulence coefficients and an M function whose definition involves the hypergeometric function 2F1.The expression identifies turbulence parameters for the R-D and S-D links.Source: §IV · p. 2

V. NUMERICAL RESULTS

The numerical results compare cooperative SIM-BPSK DF relaying with direct transmission under moderate and strong Gamma-Gamma turbulence. Selective relaying improves BER over direct transmission and performs close to perfect relaying.

V. NUMERICAL RESULTS

  • S-R, S-D, and R-D links are assumed to share average SNR and turbulence conditions, with ηs,d = ηs,r = ηr,d = 1.Strong turbulence uses α = 4.2, β = 1.4; moderate turbulence uses α = 4.0, β = 1.9.Source: §V · p. 2
  • The comparison includes selective and perfect relays, plus direct transmission with the source transmitting at double the cooperative system's power.The SNR on the figure's vertical axis is the average SNR of the S-D link.Source: §V · pp. 2–3
  • 14 dB SNR gain over non-cooperative transmission at BER=10^-4 under moderate turbulence is reported for selective DF relaying.The comparison is with direct transmission, and Fig. 1 plots average BER against SNR.Source: §V · p. 3, Figure 1 · p. 3
  • Selective DF has much better diversity than direct transmission and operates close to perfect-relay DF under both turbulence conditions.The channels vary independently in every time interval.Source: §V · pp. 2–3

VI. CONCLUSIONS

The conclusion analyzes DF cooperative FSO communication over Gamma-Gamma fading channels. Analysis and simulation establish BER advantages over non-cooperative direct transmission, particularly at high SNR.

VI. CONCLUSIONS

  • The paper analyzes a DF cooperative FSO system over Gamma-Gamma fading channels.Source: §VI · p. 3
  • Analysis and simulation establish advantages of cooperative communication over non-cooperative direct transmission.Source: §VI · p. 3
  • At high SNR, the DF FSO system's BER significantly reduces compared with the non-cooperative system.Source: §VI · p. 3

APPENDIX I DERIVATION OF CDF AND PDF OF Y

The appendix derives the CDF and PDF of Y using conditional-probability and averaging steps, followed by differentiation and algebraic simplification. Its derivation uses Q-function and complementary-error-function relations.

APPENDIX I DERIVATION OF CDF AND PDF OF Y

  • The CDF of Y is expressed by averaging a conditional probability over W.The passage writes FY(y) as Pr(Y ≤ y) and uses EW{1 − Pr(Y ≥ y|W, a, b)}.Source: Appendix I · p. 3
  • The derivation uses the Q-function and complementary error function with integral relations to obtain intermediate expressions.Source: Appendix I · p. 3
  • The PDF expression is obtained by taking the partial derivative of the CDF expression with respect to y.The derivative is written in terms of B(α, β, a, b, y).Source: Appendix I · p. 3
  • Further expressions are simplified algebraically using stated identities and substitutions.The passage also states that equation (22) follows from earlier equations and a cited identity.Source: Appendix I · p. 3
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