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Statistical characterization of kappa-mu shadowed fading

J. F. Paris

arXiv:1304.7435v1cs.ITmath.ITstat.AP

TL;DR

The paper examines a κ-µ fading generalization in which the line-of-sight component is shadowed. It derives closed-form PDF, CDF, MGF, and sum and maximum distributions, and the resulting model unifies several established fading distributions.

  • Problem

    A natural extension of the κ-µ distribution that captures shadowing in its line-of-sight component is the central modeling question.

    Source: §I · p. 2

  • Method

    The paper develops the extension by applying a line-of-sight shadow-fading model to the κ-µ distribution.

    Source: §I · p. 2

  • Results

    The κ-µ shadowed distribution’s PDF, CDF, and MGF, as well as its sum and maximum distributions, are obtained in closed form.

    Source: §I · p. 2

  • Takeaways & Limitations

    The κ-µ shadowed model unifies one-side Gaussian, Rayleigh, Nakagami-m, Ricean, κ-µ, and Ricean shadowed fading.

    Source: §I · p. 2

Abstract

from arXiv · show

This paper investigates a natural generalization of the kappa-mu fading channel in which the line-of-sight (LOS) component is subject to shadowing. This fading distribution has a clear physical interpretation, good analytical properties and unifies the one-side Gaussian, Rayleigh, Nakagami-m, Ricean, kappa-mu and Ricean shadowed fading distributions. The three basic statistical characterizations, i.e. probability density function (PDF), cumulative distribution function (CDF) and moment generating function (MGF), of the kappa-mu shadowed distribution are obtained in closed-form. Then, it is also shown that the sum and maximum distributions of independent but arbitrarily distributed kappa-mu shadowed variates can be expressed in closed-form. This set of new statistical results is finally applied to the performance analysis of several wireless communication systems.

I. INTRODUCTION

The paper introduces the κ-µ shadowed distribution by applying LOS shadowing to the κ-µ model, and establishes its physical interpretation, unification of fading models, and closed-form statistical characterizations.

I. INTRODUCTION

  • The related Ricean-shadowed model fits land mobile satellite data and has excellent experimental fit to underwater acoustic channels.Source: §I · p. 2
  • The κ-µ shadowed distribution is a physically motivated generalization of κ-µ fading that applies the Ricean-shadowed model’s multipath/shadowing scheme to LOS components.The paper investigates this model as a natural extension because κ-µ includes Ricean fading as a special case.Source: §I · p. 2
  • The distribution unifies one-sided Gaussian, Rayleigh, Nakagami-m, Ricean, κ-µ, and Ricean-shadowed fading as particular cases.Source: §I · p. 2
  • Closed-form PDF, CDF, and MGF are obtained, and sum and maximum distributions can also be expressed in closed form.Source: §I · p. 2

II. PHYSICAL MODEL · III. FUNDAMENTAL STATISTICS

The model extends κ-µ fading by allowing cluster dominant components to fluctuate under shadowing. The section derives the distribution’s PDF, CDF, and MGF and shows that parameter choices recover several other fading distributions.

II. PHYSICAL MODEL

  • The model represents a nonhomogeneous environment with clusters of multipath waves, equal-power scattered waves within each cluster, and relatively large intercluster delay spreads.Intracluster scattered waves have random phases and similar delay times; each cluster also has a dominant component of arbitrary power.Source: §II · p. 3
  • Unlike the κ-µ model’s deterministic dominant components, the shadowed model allows dominant components across clusters to fluctuate together under common shadowing.The cluster dominant components share the same common shadowing fluctuation.Source: §II · p. 3

III. FUNDAMENTAL STATISTICS

  • The κ-µ shadowed distribution’s PDF, CDF, and MGF are derived in closed form.The CDF is expressed using the bivariate confluent hypergeometric function Φ2.Source: §III · pp. 3–5
  • The natural cluster count n can be replaced by a nonnegative real extension µ, yielding a more general, flexible distribution, with κ defined as d²/(2σ²µ).Source: §III · p. 4
  • For γ distributed as Sκµ(¯γ; κ, µ, m), a probability density function is given, and the cited passage identifies it as the Ricean shadowed fading PDF when µ = 1.Source: §III · p. 4
  • The MGF of the κ-µ shadowed distribution is derived from its PDF.Source: §III · pp. 4–5
  • The CDF is given in closed form using the bivariate confluent hypergeometric function Φ2.Source: §III · pp. 4–5
  • Parameter choices recover one-side Gaussian, Rayleigh, Nakagami-m, Ricean, κ-µ, and Ricean shadowed fading distributions.The PDF reduces to Ricean shadowed fading when µ = 1.Source: §III · pp. 4–5

IV. SUM AND MAXIMUM DISTRIBUTIONS

The section derives the sum and maximum distributions of independent, non-identically distributed κ-µ shadowed random variables. It presents these results as evidence of the distribution’s analytical properties and potential for wireless-system modeling and analysis.

  • The section derives the sum distribution of independent, non-identically distributed κ-µ shadowed random variables.Source: §IV · p. 5
  • It also derives the maximum distribution for these random variables.Source: §IV · p. 5
  • The results indicate that the distribution has good analytical properties and potential for modeling and analyzing wireless communication systems.Source: §IV · p. 5

A. Sum distribution

The section characterizes the sum of independent κ-µ shadowed random variables, covering arbitrarily distributed variables, the i.i.d. case, and asymptotic PDF and CDF behavior.

A. Sum distribution

  • The sum of mutually independent, arbitrarily distributed κ-µ shadowed variables is characterized by its PDF and CDF.Proposition 1 specifies the distributions for variables with potentially different parameters.Source: A. Sum distribution · pp. 6–7
  • For identically distributed, mutually independent κ-µ shadowed variables, Corollary 1 gives the sum’s PDF.The variables share the same average SNR and κ, µ, and m parameters.Source: A. Sum distribution · pp. 6–7
  • Corollary 2 gives asymptotic behavior for the PDF and CDF of sums of mutually independent, arbitrarily distributed κ-µ shadowed variables.The result applies to variables with potentially different parameters.Source: A. Sum distribution · pp. 6–7

B. Maximum distribution · V. PERFORMANCE ANALYSIS OF WIRELESS COMMUNICATION SYSTEMS

The maximum of independent κ-µ shadowed variates has closed-form PDF and CDF expressions, and the model’s statistics support closed-form outage and error analysis under MRC or SC.

B. Maximum distribution

  • The maximum’s CDF and PDF are derived from the corresponding marginal CDFs and PDFs of the independent variates.The paper gives the general characterization in terms of fγk and Fγk for k = 1, ..., M.Source: B. Maximum distribution · p. 8
  • The maximum of M independent κ-µ shadowed variates has closed-form PDF and CDF expressions.These expressions follow by substituting the corresponding marginal distributions into (14).Source: B. Maximum distribution · p. 8

V. PERFORMANCE ANALYSIS OF WIRELESS COMMUNICATION SYSTEMS

  • Because it encompasses the κ-µ and Ricean shadowed models, the κ-µ shadowed distribution is presented as a general tool for characterizing fading channels.Source: §V · p. 8
  • The κ-µ shadowed model adds parameter m, physically related to shadowing, and its experimental-data fit must be at least as good as the κ-µ model’s.Source: §V · p. 8
  • The derived closed-form statistics yield outage probability and/or error probability expressions for κ-µ shadowed channels using MRC or SC.The analysis considers maximal ratio combining (MRC) and selection combining (SC).Source: §V · p. 8
  • These expressions generalize literature results for κ-µ fading, Ricean shadowed fading, and the fading distributions encompassed by both models.Source: §V · pp. 8–9

A. Selection combining with κ-µ shadowed fading

The section derives the outage probability for selection combining over mutually independent κ-µ shadowed branches. It reports that shadowing can significantly affect performance and that simulations validate the analytical derivation.

A. Selection combining with κ-µ shadowed fading

  • The outage probability for selection combining is derived for L mutually independent κ-µ shadowed fading branches.The derivation uses (14) and Lemma 3.Source: A. Selection combining with κ-µ… · p. 9
  • As all branch SNRs approach zero, Φ2 tends to unity, which is used to characterize the outage probability’s behavior.The passage introduces this limit before presenting the behavior of Po.Source: A. Selection combining with κ-µ… · p. 9
  • Simulations of the selection-combining outage probability validate the analytical derivations.The results are shown in Fig. 2.Source: A. Selection combining with κ-µ… · p. 9
  • Shadowing significantly affects selection-combining performance in the illustrated three-branch scenario, even though each branch’s κ is below 5 dB.As m approaches infinity, the results correspond to selection combining under κ-µ fading.Source: A. Selection combining with κ-µ… · p. 9

B. Maximal ratio combining with κ-µ shadowed fading

For independent κ-µ shadowed branches combined by MRC, the section derives closed-form outage and bit-error probabilities. Simulations validate the outage analysis, and the shadowing parameter m strongly affects both outage and bit-error probability.

B. Maximal ratio combining with κ-µ shadowed fading

  • For independent κ-µ shadowed branches, the MRC outage probability is evaluated from Proposition 1, and simulations validate the analytical derivations.The example uses three branches with equal average SNR per branch and plots outage against average branch SNR for different m values.Source: B. Maximal ratio combining with κ-µ… · pp. 9–11
  • The bit-error probability for MRC under κ-µ shadowed fading is derived in closed form using a multivariate Lauricella function.The expression follows by substituting the fading density and evaluating the resulting integral.Source: B. Maximal ratio combining with κ-µ… · pp. 10–11
  • The shadowing parameter m has a great influence on MRC outage probability.The comparison varies m while plotting outage against average SNR per branch.Source: B. Maximal ratio combining with κ-µ… · p. 11
  • For BPSK with MRC, the shadowing parameter m has a great impact on bit-error probability.The figure compares analytical and simulated bit-error results across m values as a function of average SNR per branch.Source: B. Maximal ratio combining with κ-µ… · p. 11

VI. CONCLUSIONS · distributions. The three basic statistical characterizations, i.e. probability density function (PDF),

The paper derives the statistics of the κ-µ shadowed fading model, a physically interpretable generalization of κ-µ fading that unifies several familiar fading distributions. It obtains closed-form characterizations and applies the results to wireless-system performance analysis.

VI. CONCLUSIONS

  • The κ-µ shadowed model is a natural generalization of κ-µ fading that incorporates shadowing and has a clear physical interpretation and good analytical properties.Source: §VI · p. 11
  • The distribution unifies one-side Gaussian, Rayleigh, Nakagami-m, Ricean, κ-µ, and Ricean shadowed fading distributions.Source: §VI · p. 11

distributions. The three basic statistical characterizations, i.e. probability density function (PDF),

APPENDIX I PROOF OF LEMMA I · APPENDIX II PROOF OF LEMMA II

The appendices complete the proofs of Lemmas I and II by transforming the integral in Θ(γ) and deriving the MGF from the Laplace transform of fγ. The derivations use standard integral identities and Laplace-transform properties, then simplify to the cited expressions.

APPENDIX I PROOF OF LEMMA I

  • The quadratic substitution t = ξ2 transforms the integral appearing in Θ(γ).This is the opening transformation in the proof of Lemma I.Source: Appendix I · p. 12
  • Applying identities [8, eq. 4.16.20] and [7, eq. 9.220-2] expresses the transformed integral.Source: Appendix I · p. 12
  • The proof of Lemma I concludes by substituting (25) into (22) and simplifying algebraically.Source: Appendix I · p. 12

APPENDIX II PROOF OF LEMMA II

APPENDIX III PROOF OF PROPOSITION I · APPENDIX IV PROOF OF LEMMA IV

Appendix III proves Proposition I by deriving the sum distribution’s MGF, PDF, and CDF. Appendix IV proves Lemma IV using an ancillary function, its Laplace transform, and evaluation at t = 1.

APPENDIX III PROOF OF PROPOSITION I

APPENDIX IV PROOF OF LEMMA IV

APPENDIX V NUMERICAL COMPUTATION OF THE FUNCTIONS Φ2 AND FD · and MATHEMATICA; however it can be efficiently computed by inverting its one-dimensional

The appendix explains how to compute the special functions used in the paper’s results: some are available in standard software, while others can be evaluated through series, integration, or Laplace-transform inversion.

APPENDIX V NUMERICAL COMPUTATION OF THE FUNCTIONS Φ2 AND FD

  • The appendix treats Φ2 and FD separately because these bivariate functions feature in many of the paper’s results.This motivates the numerical-computation guidance provided for each function.Source: Appendix V · p. 14
  • The Appell function F1, equivalent to the bivariate hypergeometric function FD, is implemented in MATLAB and Mathematica, making its computation straightforward.The paper notes that many of its results involve Φ2 or FD.Source: Appendix V · pp. 14–15
  • Although the bivariate confluent hypergeometric function Φ2 is not yet implemented in MATLAB, it can be expressed as a 1F1 series suited to numerical computation.Φ2 is defined in the popular mathematical handbook edited by Gradshteyn and Ryzhik.Source: Appendix V · p. 15
  • The Φ2 series uses terms involving 1F1(b′; c + k; z).This term is given in equation (32).Source: Appendix V · p. 15
  • The multivariate hypergeometric function FD^(N) is not implemented in MATLAB or Mathematica, but can be computed using its Euler-type representation and standard numerical integration.The stated condition ℜ[c] > ℜ[a] > 0 holds for the FD^(N) function appearing in (21).Source: Appendix V · p. 15

and MATHEMATICA; however it can be efficiently computed by inverting its one-dimensional

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