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On the Distribution of the Sum of Gamma-Gamma Variates and Applications in RF and Optical Wireless Communications

Nestor D. Chatzidiamantis, George K. Karagiannidis

arXiv:0905.1305v1cs.IT

TL;DR

MIMO analysis requires the distribution of sums of independent Gamma-Gamma variates, whose direct derivation is analytically difficult. The paper derives closed-form approximations using either a single Gamma-Gamma PDF or a finite weighted sum, with accurate approximative results across SNR, outage, and increasing diversity branches.

  • Problem

    The distribution of sums of independent Gamma-Gamma variates is required for MIMO analysis, but its direct derivation is analytically difficult.

  • Method

    The paper derives closed-form approximations for the sum PDF using a single Gamma-Gamma distribution for identical variates or a finite weighted sum for non-identical variates.

  • Results

    Excellent agreement is observed between simulation and approximative results across input SNR and normalized outage, with accuracy retained as diversity branches increase.

  • Takeaways & Limitations

    The proposed approximations provide accurate analytical expressions for evaluating systems involving sums of Gamma-Gamma variates.

Abstract

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The Gamma-Gamma (GG) distribution has recently attracted the interest within the research community due to its involvement in various communication systems. In the context of RF wireless communications, GG distribution accurately models the power statistics in composite shadowing/fading channels as well as in cascade multipath fading channels, while in optical wireless (OW) systems, it describes the fluctuations of the irradiance of optical signals distorted by atmospheric turbulence. Although GG channel model offers analytical tractability in the analysis of single input single output (SISO) wireless systems, difficulties arise when studying multiple input multiple output (MIMO) systems, where the distribution of the sum of independent GG variates is required. In this paper, we present a novel simple closed-form approximation for the distribution of the sum of independent, but not necessarily identically distributed GG variates. It is shown that the probability density function (PDF) of the GG sum can be efficiently approximated either by the PDF of a single GG distribution, or by a finite weighted sum of PDFs of GG distributions. To reveal the importance of the proposed approximation, the performance of RF wireless systems in the presence of composite fading, as well as MIMO OW systems impaired by atmospheric turbulence, are investigated. Numerical results and simulations illustrate the accuracy of the proposed approach.

I. INTRODUCTION

The paper motivates approximating sums of independent Gamma-Gamma variates because their distribution is required for MIMO analysis but is analytically difficult to derive. It proposes closed-form approximations and applies them to RF and optical wireless systems.

  • Gamma-Gamma statistics model composite and cascade multipath fading in RF systems and atmospheric-turbulence-induced irradiance fluctuations in optical wireless systems.
  • Although Gamma-Gamma models are analytically tractable for SISO systems, MIMO diversity analysis requires the distribution of sums of independent GG variates.
  • Directly deriving the GG-sum distribution is analytically infeasible because it involves the modified Bessel function of the second kind.
  • Earlier optical-wireless power-series methods were accurate at high SNR but became computationally unattractive as apertures increased or links became non-identically distributed.
  • The paper presents closed-form approximations for GG sums using either a single GG PDF or a finite weighted sum of GG PDFs.
  • The approximations are applied to RF MRC systems under KG fading and MIMO optical wireless systems under strong turbulence with EGC.

II. THE GG DISTRIBUTION

The GG distribution models communication-channel power or irradiance statistics and is constructed from products of independent Gamma variables. Its parameterization includes several established fading models as special or limiting cases.

  • A three-parameter GG random variable has shaping parameters k and m, mean Ω, and a PDF involving the modified Bessel function of the second kind.
  • As k tends to infinity, the GG distribution approximates the Gamma distribution, while m = 1 yields squared K-distributed statistics.
  • For k = 1 and m = 1, the GG model reduces to the power statistics of the Double-Rayleigh model used in cascade multipath fading channels.
  • The GG distribution is derived from the product of two independent Gamma random variables with parameters linked to k, m, and Ω.
  • The paper defines the sum of L GG variates using component Gamma variables with parameters (k_l, 1/k_l) and (m_l, Ω_l/m_l).

A. Identical Variates

For i.i.d. GG variates, the sum is approximated by a single GG distribution whose parameters are derived from the sum’s moments. The approximation error has zero mean but can increase with the number of summed variables, motivating an adjustment parameter.

  • Approximation construction: The unknown distribution of the sum of L i.i.d. GG variates is approximated by a single GG-distributed random variable.The construction uses a product of two Gamma-distributed variables whose resulting product is GG distributed.
  • Error analysis: The approximation error has mean 0, while its variance depends on k, m, and Ω.The variance therefore characterizes the statistical behavior of the approximation error.
  • Error analysis: The variance of the approximation error increases as the number of summed RVs increases for certain k, m, and Ω combinations.This increase causes the approximating distribution to lose accuracy.
  • Accuracy adjustment: An adjustment parameter modifies the shaping parameters to improve approximation accuracy.The parameter is obtained through a nonlinear optimization problem and an approximate numerical solution based on nonlinear regression.
  • Parameter derivation: The approximating GG parameters are defined from moments of the original sum and the approximating distribution.The moments are obtained from the GG model and multinomial expansion, then used to define the single-GG parameters.
  • Accuracy adjustment: The resulting single-GG construction accurately approximates the distribution of the sum of L i.i.d. GG variates.The adjustment parameter is expressed as a function of L, k, and m.

B. Non-Identical Variates

When GG variates are independent but non-identical, a single-GG approximation remains available when one shaping parameter is common, while a finite weighted sum of GG PDFs handles the broader non-identical case.

  • Single-GG approximation: For non-identical GG variates sharing one shaping parameter, the sum can be approximated by a single GG-distributed random variable.The approximation is defined by the same product-based construction used for the identical case.
  • Weighted-sum approximation: The weights in the nested sum are evaluated recursively from the parameters of the non-identical Gamma variates.The recursive formula provides the weights needed for the Gamma-PDF representation.
  • Weighted-sum approximation: When one shaping parameter is common across all variates, the sum’s PDF can be approximated by a nested finite weighted sum of GG PDFs.The construction uses a finite weighted sum of Gamma PDFs for an intermediate variable.
  • RF application: The RF application considers L-branch macrodiversity with independent shadowing and multipath effects and maximum ratio combining.Each branch SNR is GG distributed, and the output SNR is the sum across branches.
  • Symmetry: The approximation also applies when the common shaping parameter is m rather than k by interchanging the two shaping parameters.This follows from the symmetry of the GG representation.
  • RF application: The RF model assumes independent diversity branches, common shadowing parameter k, and identical AWGN power spectral density across branches.The common-k assumption reflects shadowing over large geographical areas.

B. Error Analysis

The RF error analysis averages conditional BER over the output-SNR PDF for BPSK and DBPSK. For i.i.d. branches, the sum approximation reduces the calculation to a single-GG BER evaluation.

  • BER evaluation: Average BER is obtained by averaging the modulation-dependent conditional BER over the PDF of the total output SNR.The analysis considers BPSK and DBPSK modulation schemes.
  • Identical branches: For i.i.d. diversity branches, the total-SNR PDF is approximated by a single GG variate with parameters derived from the sum approximation.This approximation is then used in the BER integrals.
  • BPSK: The BPSK diversity-system BER is approximated using the corresponding closed-form BER expression after substituting the single-GG output-SNR approximation.The derivation uses the BPSK conditional-error expression and Whittaker-function representation.
  • DBPSK: The DBPSK diversity-system BER is evaluated analogously using its corresponding conditional BER expression.The same single-GG output-SNR approximation supports the DBPSK calculation.

2) Independent, but not Necessarily Identically Distributed Diversity Branches:

For independent, non-identically distributed diversity branches, the total-SNR PDF is approximated by a nested finite weighted sum of GG PDFs. This yields tractable BER and outage calculations.

  • PDF approximation: When diversity branches are independent but not identically distributed, the sum of their GG variates is approximated by a nested finite weighted sum of GG PDFs.The approximation applies the non-identical GG-sum construction to the total output SNR.
  • BER analysis: The resulting approximation is substituted into the BPSK and DPSK average-BER expressions.The required integrals are evaluated using the corresponding BER expressions for a SISO system.
  • BER analysis: The BPSK and DBPSK diversity-system BER expressions are thereby approximated from the weighted GG representation of the total SNR.The method retains the branch-specific parameters through the nested weighted sum.
  • Outage analysis: Outage probability is defined as the probability that the output SNR falls below a threshold γ_th representing the minimum satisfactory channel SNR.The outage probability is evaluated using the approximated total-SNR distribution.
  • Outage analysis: For the considered diversity system, the outage probability can be related to the outage probability of a SISO system with equivalent GG parameters.The equivalent representation uses a GG-distributed variate with parameters associated with the total SNR.

2) Independent but not Necessarily Identically Distributed Diversity Branches:

For independent, non-identically distributed diversity branches, the total-SNR PDF is approximated by a nested finite weighted sum of GG PDFs, enabling analytical BER and outage evaluation. Numerical comparisons show close agreement with simulations across branch counts and parameter settings.

  • The PDF of the total SNR is approximated by a nested finite weighted sum of GG PDFs.
  • The resulting outage expression is equivalent to a nested finite weighted sum of SISO outage probabilities.
  • The analytical BER and outage expressions are evaluated against Monte Carlo simulations for arbitrary diversity-branch counts and selected shaping parameters.
  • Excellent agreement is observed between simulations and approximations across input SNR and normalized-outage values, including increasing diversity-branch counts.
  • The BER approximation differs from simulation results by no more than 3 dB at target BER 10^-5, even as the decaying factor δ increases.
  • For outage probability, analytical and simulation results remain within 3 dB in all examined cases, while the approximation acts as a lower bound at high SNR.

A. System Model

The modeled MIMO optical wireless system uses repetition coding across transmit apertures, equal-gain combining across receive apertures, and GG fading for strong atmospheric turbulence. Its BER is formulated from the aggregate irradiance or its PDF under independent channel assumptions.

  • Information is transmitted through M apertures and received by N apertures under strong atmospheric turbulence.
  • On-Off keying with repetition coding is used, while large receiver fields of view motivate an AWGN approximation to photon counting.
  • Each optical link is modeled by a GG distribution under strong turbulence, with k = 1, m = a_pq, and Ω = E[I_pq].
  • The received signals are combined using equal-gain combining, with scaling factors preserving total transmit power and aggregate receive-aperture area.
  • The BER formulation assumes perfect CSI and can be evaluated using the aggregate irradiance PDF and Meijer’s G-functions.

2) Independent and Not Identically Distributed OW Links:

For independent but non-identically distributed optical links, the aggregate irradiance PDF is approximated by a nested weighted sum of GG PDFs. This yields BER and outage expressions for the MIMO optical system.

  • With independent, non-identically distributed optical links, the aggregate irradiance PDF is approximated by a nested finite weighted sum of GG PDFs.
  • The approximation represents L = MN underlying links using link-specific GG parameters derived from a_pq and E[I_pq].
  • Substitution into the BER formulation produces an analytical error-probability expression evaluated through Meijer’s G-functions.
  • The corresponding outage probability is the probability that the combined output SNR falls below the normalized threshold h_th/N_0, a parameter relevant to OW network design.

1) Independent and Identically Distributed OW Links:

For identically distributed OW links, the proposed GG-sum approximation yields analytical BER and outage results that closely match Monte Carlo simulations across investigated deployments. For nonidentical links, it remains useful but becomes less accurate as link count increases.

  • Identically distributed OW links: The MIMO OW outage probability can be approximated through a SISO GG turbulence-model outage expression.For identically distributed links, the summed irradiance is represented by an equivalent GG model.
  • Nonidentically distributed OW links: For nonidentical OW links, the MIMO outage probability is represented by a finite nested weighted sum of SISO GG-link outage probabilities.The formulation extends the GG-sum approximation beyond identically distributed links.
  • Identically distributed OW links: The approximation remains accurate across SNR regimes and investigated MIMO deployments, irrespective of transmit and receive aperture counts.This agreement is reported for both evaluated performance metrics.
  • Identically distributed OW links: Analytical BER and outage results closely match Monte Carlo simulations for the investigated MIMO OW deployments.The reported difference for practical average BER and outage values is not greater than 2 dB.
  • Nonidentically distributed OW links: The approximation acts as a lower bound and becomes less accurate as the number of nonidentical OW links increases.Despite this limitation, the closed-form expressions provide alternatives to time-consuming Monte Carlo simulations when numerical evaluation becomes difficult.

APPENDIX

The appendix derives moment-based expressions for the approximation error when summing identically distributed squared-Nakagami variates. It computes the error's first moment and variance using independence and expansions of the relevant sums.

  • Error-moment derivation: The appendix models the summed variables as identically distributed squared-Nakagami variates with specified parameters and moments.The index l ranges over the L summed variables.
  • Error-moment derivation: Independence allows the first moment of the approximation error to be calculated directly.The derivation explicitly uses the independence of the summed variables.
  • Error-moment derivation: The variance of the approximation error is obtained from its second moment.The subsequent derivation expands and evaluates the resulting terms using earlier moment relations.
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