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A Unified MGF-Based Capacity Analysis of Diversity Combiners over Generalized Fading Channels
Ferkan Yilmaz, Mohamed-Slim Alouini
TL;DR
Existing unified exact average-capacity results for L-branch EGC and MRC receivers are lacking. This paper develops a generic MGF-based framework for generalized fading channels, derives results for Gamma-shadowed GNM fading, and validates the formalism numerically and through simulation.
Problem
Unified exact average-capacity results for L-branch EGC and MRC diversity receivers over generalized fading channels are not known.
Method
The paper develops a unified MGF-based capacity framework for EGC and MRC receivers and derives closed-form results for Gamma-shadowed GNM fading.
Results
MRC gives better capacity or performance than EGC in all figures, while numerical and simulation results are in perfect agreement.
Takeaways & Limitations
The framework provides a unified way to compute average capacity for EGC and MRC over generalized fading channels, including high-frequency fading environments.
Abstract
from arXiv · showhide
Unified exact average capacity results for L-branch coherent diversity receivers including equal-gain combining (EGC) and maximal-ratio combining (MRC) are not known. This paper develops a novel generic framework for the capacity analysis of $L$-branch EGC/MRC over generalized fading channels. The framework is used to derive new results for the Gamma shadowed generalized Nakagami-m fading model which can be a suitable model for the fading environments encountered by high frequency (60 GHz and above) communications. The mathematical formalism is illustrated with some selected numerical and simulation results confirming the correctness of our newly proposed framework.
I. INTRODUCTION
Average-capacity analysis for EGC and MRC diversity receivers is less developed than ASEP analysis, especially through unified MGF-based methods. The paper introduces such a framework for arbitrary branch correlation and fading channels, including exact results for Gamma-shadowed GNM fading.
- The paper addresses scarce average-capacity results for EGC and MRC diversity combiners over fading channels, despite extensive ASEP analysis.
- The approach overcomes earlier MGF-based methods that were limited to MRC capacity and not easily extendible to EGC.
- The authors develop a unified MGF-based approach for calculating ergodic capacity in arbitrarily correlated or uncorrelated fading channels.
- The framework covers both L-branch MRC and L-branch EGC receivers over a wide variety of fading channels and arbitrary numbers of diversity branches.
- The paper derives exact EGC and MRC average capacities for Gamma-shadowed generalized Nakagami-m fading channels, motivated by high-frequency communications at 60 GHz and above.
II. AN MGF-BASED CAPACITY ANALYSIS OF DIVERSITY COMBINERS
The paper develops an exact, unified MGF-based approach for average-capacity analysis of L-branch EGC and MRC receivers over generalized fading channels. The framework replaces difficult multidimensional PDF-based calculations with a generic single-integral expression applicable to correlated and independent branches.
- Motivation: The original capacity formulation requires an L-fold integral over the joint fading-envelope PDF, which becomes difficult to separate and evaluate as L increases.Generalized fading can additionally require multiple convolutions or integrals even when branch envelopes are independent.
- Unified framework: The proposed approach avoids finding the instantaneous-SNR PDF through an inverse Laplace transform by using the joint p-exponent MGF directly.This reduces the capacity calculation to a generic single integral rather than repeated PDF derivations.
- Unified framework: Theorem 1 gives exact average capacity for L-branch diversity combiners over correlated, not-necessarily identically distributed fading channels.The result uses a joint p-exponent MGF and supports both EGC and MRC through parameter choices.
- Independent branches: Corollary 1 specializes the exact capacity expression to mutually independent, not-necessarily identically distributed fading channels.The framework therefore covers both correlated and independent branch models.
- Computation: The auxiliary function is represented with Meijer’s G function, enabling numerical computation in standard mathematical software packages.The paper also notes that the integral can be accurately estimated with Gauss-Chebyshev quadrature using few terms.
- Special cases: The framework recovers known L-branch MRC capacity results when q = 1 and provides the corresponding EGC formulation when q = 2.The MRC expression agrees with a previously published result for unit transmitted power.
NAKAGAMI-m FADING CHANNELS
The paper introduces the Gamma-shadowed generalized Nakagami-m model and derives unified MGF expressions needed for exact EGC and MRC capacity analysis. The model encompasses several established fading distributions and is motivated by high-frequency propagation environments.
- Shadowing: The model represents shadowing by a Gamma-distributed local mean power, with larger msℓ corresponding to less severe shadowing.As msℓ tends to infinity, the local-mean distribution approaches a Dirac distribution and shadowing vanishes.
- Model validation: The derived PDF reduces to the generalized Nakagami-m distribution as msℓ tends to infinity and to the Gamma-shadowed Nakagami-m distribution when ξℓ = 1.These reductions provide analytical consistency checks for the proposed model.
- Channel model: The Gamma-shadowed generalized Nakagami-m model provides a unified representation for envelope statistics across many known wireless and optical communication channels.Its limiting and special cases include Rayleigh, Nakagami-m, Weibull, lognormal, AWGN, and related distributions.
- Unified MGF: Theorem 2 derives a unified closed-form MGF for the Gamma-shadowed generalized Nakagami-m envelope distribution.Selecting p = 1 or p = 2 yields the branch MGF needed for EGC or MRC, respectively.
- Computational considerations: Meijer’s G representations require rational shaping parameters and can become less efficient as the rationalization accuracy increases.The paper therefore notes that the Fox’s H representation may be computationally preferable in this situation.
B. Unified Average Capacity of Diversity Combiners
Using the unified branch MGFs and their derivatives, the paper obtains exact single-integral average-capacity expressions for EGC and MRC over Gamma-shadowed generalized Nakagami-m fading. The expressions reduce to established special-case results and admit rapidly convergent numerical evaluation.
- MGF derivative: Theorem 3 supplies the derivative of the unified Gamma-shadowed generalized Nakagami-m MGF required by the capacity formulas.The derivative is stated over the convergence region Re{s} ∈ R+.
- Unified capacity: The paper derives new exact single-integral expressions for L-branch diversity-combiner capacity over Gamma-shadowed generalized Nakagami-m fading.The expressions use the unified MGF and derivative together with the auxiliary function from Theorem 1.
- Numerical evaluation: The numerical integral can be converted into a finite N-term Gauss-Chebyshev sum that converges rapidly and steadily with few terms.This provides a practical numerical route for evaluating the derived capacity expressions.
- Combiner unification: The unified expression covers both MRC and EGC by selecting q = 1 for MRC and q = 2 for EGC.This same formulation also spans commonly used channel-fading models represented by the Gamma-shadowed generalized Nakagami-m family.
- MRC special case: For MRC, the unified capacity reduces to the L-branch MRC result over Gamma-shadowed generalized Nakagami-m fading.Further specialization recovers the average capacity of independent identically distributed Nakagami-m channels.
- EGC special case: For EGC, the unified capacity reduces to the L-branch EGC result over Gamma-shadowed generalized Nakagami-m fading and to the generalized-K case under further specialization.The resulting expression can be evaluated using the same Gauss-Chebyshev quadrature rule.
IV. NUMERICAL RESULTS
The numerical study examines average capacity across branch count, fading parameters, shadowing, and combining schemes for Gamma-shadowed GNM channels. Numerical and simulation results agree, while MRC generally outperforms EGC.
- MRC gives better capacity/performance than EGC across all figures, with the smallest performance difference for two-branch combining.
- Increasing the number of branches increases average capacity, although the diversity gain from each added branch decreases as L increases.
- Numerical and simulation results are in perfect agreement across the selected scenarios.
- Increasing the fading figure from 0.5 to 2.0 produces a large diversity gain, while values above 2 gradually and linearly increase average capacity.
- Average capacity approaches zero as the common shaping factor ξ approaches zero, but increases gradually and linearly for large ξ.
- Average capacity becomes insensitive to shadowing as ms approaches infinity because local-mean-power variation diminishes.
V. CONCLUSION
The conclusion presents a unified capacity framework for EGC and MRC over generalized fading channels and introduces Gamma-shadowed GNM fading for high-frequency environments. Closed-form MGF results and selected simulations support the framework's accuracy.
- The paper presents a unified framework for computing average capacity of EGC and MRC diversity schemes over fading channels.
- Gamma-shadowed GNM fading is proposed to characterize fading environments in communications at 60 GHz and above.
- The paper derives novel closed-form expressions for the MGF of Gamma-shadowed GNM fading and its special cases.
- Selected simulations verify the framework, with numerical and simulation results in perfect agreement.
APPENDIX A
Appendix A derives the average-capacity expression for linear diversity receivers by specializing the framework to MRC and EGC and applying integral-transform identities.
- For q=1 in MRC and q=2 in EGC, substituting the auxiliary result into the capacity expression yields the average capacity of both linear diversity receivers.
- The derivation uses differentiation of exp(−sX), Leibniz's rule, and an integral identity involving the Mittag-Leffler function.
- A Mellin-Barnes representation of the Mittag-Leffler function transforms the derivation into a Fox H-function form.
APPENDIX B
Appendix B represents the auxiliary function through a Mellin-Barnes integral and applies algebraic and special-function identities to derive its closed form.
- The auxiliary function Cq(s) is represented as a Mellin-Barnes integral within the convergence region ℜ{C}∈(−1,0).
- Substitution and algebraic manipulation, followed by an identity applied to the parenthesized term, derive equation (12) and prove Corollary 2.