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The Fluctuating Two-Ray Fading Model: Statistical Characterization and Performance Analysis

Juan M. Romero-Jerez, F. Javier Lopez-Martinez, José F. Paris, Andrea J. Goldsmith

arXiv:1611.05063v2cs.IT

TL;DR

The paper addresses the inability of conventional fading models to represent bimodal fluctuations in measured mmWave channels. It introduces and analytically characterizes the FTR model, then evaluates communications performance under FTR fading. FTR fits 28 GHz measurements better than Rician fading, and lighter fluctuations with more dissimilar specular components yield better BER and outage-capacity performance.

  • Problem

    Bimodal amplitude fluctuations in 28 GHz outdoor measurements are not accurately captured by conventional fading models.

  • Method

    The paper introduces FTR fading with two fluctuating random-phase specular components and a diffuse component, derives closed-form PDF, CDF, and MGF expressions, and analyzes BER and outage capacity.

  • Results

    FTR provides a much better fit than Rician fading for 28 GHz measurements, while lighter fluctuations and more dissimilar specular components yield better BER and outage-capacity performance.

  • Takeaways & Limitations

    FTR offers a flexible fading model spanning bimodal and classical fading behaviors while retaining closed-form statistical and performance analysis.

Abstract

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We introduce the Fluctuating Two-Ray (FTR) fading model, a new statistical channel model that consists of two fluctuating specular components with random phases plus a diffuse component. The FTR model arises as the natural generalization of the two-wave with diffuse power (TWDP) fading model; this generalization allows its two specular components to exhibit a random amplitude fluctuation. Unlike the TWDP model, all the chief probability functions of the FTR fading model (PDF, CDF and MGF) are expressed in closed-form, having a functional form similar to other state-of-the-art fading models. We also provide approximate closed-form expressions for the PDF and CDF in terms of a finite number of elementary functions, which allow for a simple evaluation of these statistics to an arbitrary level of precision. We show that the FTR fading model provides a much better fit than Rician fading for recent small-scale fading measurements in 28 GHz outdoor millimeter-wave channels. Finally, the performance of wireless communication systems over FTR fading is evaluated in terms of the bit error rate and the outage capacity, and the interplay between the FTR fading model parameters and the system performance is discussed. Monte Carlo simulations have been carried out in order to validate the obtained theoretical expressions.

I. INTRODUCTION

The paper introduces FTR fading to model bimodal amplitude fluctuations that conventional fading models cannot capture, while retaining analytical tractability and encompassing established special cases.

  • 28 GHz outdoor measurements exhibited bimodal empirical PDFs and CDFs that conventional and generalized fading models could not capture.
  • The FTR model generalizes TWDP by allowing two random-phase specular components to undergo random amplitude fluctuations.TWDP is recovered as a particular case when the specular amplitudes do not fluctuate.
  • FTR retains closed-form PDF, CDF, and MGF expressions despite its greater flexibility than TWDP.
  • FTR is inherently bimodal, includes Rician, Nakagami-m, Hoyt, and Rayleigh as special cases, and therefore covers more propagation conditions.
  • The channel model combines dominant specular waves with a diffuse complex Gaussian component whose independent random phases reflect multipath propagation.The diffuse component represents many weak, independently phased scattered waves.
  • For two fluctuating specular components, the resulting model is denoted FTR; the shared fluctuation is motivated by scatterers or disturbances affecting both components simultaneously.
  • The parameter K measures dominant-to-diffuse average power, while Δ measures similarity between the two specular average powers and ranges from 0 to 1.Δ = 1 corresponds to equal specular magnitudes, whereas Δ = 0 corresponds to one absent specular component and yields Rician shadowed fading.

A. MGF

This section establishes closed-form statistical characterization for FTR fading, including its MGF and, for positive integer m, its PDF and CDF, while relating special parameter settings to established models.

  • The FTR model admits a closed-form MGF for the received SNR γ.
  • FTR includes TWDP, Rician shadowed, Rician, Rayleigh, one-sided Gaussian, Nakagami-m, and Nakagami-q fading as particular cases.
  • With m = 1, FTR becomes Nakagami-q (Hoyt), and q = 0 or q = 1 further reduces to one-sided Gaussian or Rayleigh fading.
  • The relationship between FTR parameters K and Δ and the Hoyt parameter q is represented for m = 1, with larger K permitting the full q range.
  • For positive integer m, the FTR PDF and CDF can also be obtained in closed form from the MGF characterization.

B. PDF and CDF

For positive integer m, the FTR model provides closed-form PDF and CDF expressions for the SNR using the confluent hypergeometric function, alongside finite-sum elementary approximations. These statistics can also be transformed to signal-envelope distributions and are evaluated across model parameters with Monte Carlo validation.

  • Closed-form expressions: For integer m, the MGF becomes a finite sum of elementary terms because the associated Legendre function reduces to a Legendre polynomial.This reduction supports finite-sum evaluation of the model statistics.
  • Signal-envelope statistics: The received signal-envelope PDF and CDF follow from the power-envelope statistics through fr(r) = 2r fγ(r^2) and Fr(r) = Fγ(r^2).The average-power parameter γ̄ is replaced by Ω = E{r^2}.
  • Closed-form expressions: For m ∈ Z+, the FTR SNR PDF and CDF are expressed using the confluent hypergeometric function Φ2.The function is established as a standard communication-theory function that can be efficiently evaluated.
  • Approximate expressions: The approximate FTR PDF and CDF use finite sums of exponential functions and powers, simplifying evaluation to operations comparable to the Nakagami-m Gamma distribution.The approximation is specified for integer m with M > ⌈K∆⌉.
  • Parameter evaluation: Figures 2–7 examine how K, ∆, and m affect signal- and power-envelope PDFs, comparing exact expressions with approximations and Monte Carlo simulations.For approximated results, M = ⌈K∆⌉ + 1 is used in every case.

IV. EMPIRICAL VALIDATION

The FTR model is validated against 28 GHz outdoor millimeter-wave measurements by comparing empirical and theoretical CDFs for LOS and NLOS scenarios. It achieves lower error factors than the best-fit Rician model in both cases.

  • Measurement validation: The validation uses empirical LOS and NLOS measurements from a 28 GHz outdoor millimeter-wave campaign.The comparison focuses on cross-polarized scenarios described in the referenced measurements.
  • Goodness of fit: The error factor ε is defined from a modified Kolmogorov–Smirnov statistic using logarithmic CDF differences.An error factor of ε = 1 corresponds to a one-order-of-magnitude difference between empirical and theoretical CDFs.
  • FTR fitting: The fitted FTR parameters are (K = 80, ∆ = 0.5873, m = 2) for LOS and (K = 32.7, ∆ = 0.8331, m = 10) for NLOS.The parameter m enables the CDF to change concavity and convexity to better adjust the empirical data.
  • Validation result: ε_FTR,LOS = 0.2246 and ε_FTR,NLOS = 0.2681, indicating a lower error factor than the corresponding Rician fit.The paper characterizes this as a remarkable improvement over the simpler Rician model.

V. PERFORMANCE ANALYSIS OF WIRELESS COMMUNICATIONS SYSTEMS

The paper uses the FTR model’s closed-form statistics to analyze wireless-system performance, deriving BER and outage-capacity expressions and high-SNR asymptotics. It examines how specular-component similarity and fluctuation strength affect these metrics under representative parameter settings.

  • Performance metrics: The FTR PDF, CDF, and MGF enable analysis of BER and outage capacity, including exact high-SNR asymptotic expressions.The BER analysis covers coherent modulations, while outage capacity is evaluated through a threshold probability.
  • Parameter effects: Figures 2 and 3 compare exact and approximate signal- and power-envelope PDFs as m varies with K = 15 and ∆ = 0.9.As m → ∞, the FTR distribution reduces to the TWDP fading distribution.
  • Average BER: Average BER is obtained by averaging the conditional error probability over the output SNR or, equivalently, by integrating with the SNR CDF.The resulting exact BER expression uses the Lauricella function F_D.
  • Parameter effects: Figures 4 and 5 compare exact and approximate signal- and power-envelope PDFs as ∆ varies with K = 15 and m = 5.The paired plots examine both envelope representations for the same parameter variation.
  • Parameter effects: Figures 6 and 7 compare exact and approximate signal- and power-envelope PDFs as K varies with m = 5 and ∆ = 0.9.The plots use Ω = 1 for signal-envelope results and γ̄ = 1 for power-envelope results.

B. Outage capacity

The section defines outage capacity probability as the chance that instantaneous capacity falls below a threshold and derives exact and high-SNR expressions from the FTR SNR CDF.

  • Instantaneous channel capacity per unit bandwidth is expressed as C = log2(1 + γ).
  • Outage capacity probability is the probability that C falls below the predefined threshold RS.
  • The exact outage probability follows directly from the FTR CDF evaluated at x = 2^RS − 1.
  • A high-SNR approximation is obtained from the asymptotic CDF expression for γ.

VI. NUMERICAL RESULTS

The numerical results evaluate FTR performance analytically and validate the expressions with Monte Carlo simulations. They show that dissimilar specular components with lighter fluctuations improve BER and outage performance, while the model assumes fully correlated specular components.

  • Analytical evaluations and Monte Carlo simulations show excellent agreement for the reported performance results.
  • For K = 8, BPSK BER and outage probability are evaluated across average SNR for Δ = 0.9 or 0.1 and m = 8 or 2.
  • Dissimilar specular components with lighter fluctuations yield lower average BER and outage capacity probability.
  • For Δ = 0.1 and m = 8, both metrics exhibit an inflection point that virtually disappears as m decreases.
  • The presented FTR model assumes fully correlated specular components, whereas partial correlation would define a more general and analytically harder model.

APPENDIX I PROOF OF LEMMA 1

This appendix derives the FTR MGF by conditioning on the common fluctuation, relating the conditional model to TWDP fading, and averaging over the fluctuation variable in closed form.

  • Conditioning on a realization of the specular-component fluctuation produces a conditional fading model and conditional MGF.
  • The conditional model corresponds to classical TWDP fading with specular amplitudes scaled by the fluctuation realization.
  • The TWDP MGF is rewritten using the FTR K and Δ parameters to establish the correspondence between the models.
  • The FTR MGF is obtained by averaging the conditional MGF over all realizations of the fluctuation variable.
  • The required integral is evaluated in closed form using a Bessel-function integral identity, yielding the stated MGF.

APPENDIX II PROOF OF COROLLARY 1

This appendix rewrites the FTR MGF using a factorization of its polynomial term and applies inverse Laplace transforms to obtain the PDF and CDF.

  • For m = 1, the Legendre function in the FTR MGF reduces because P0(z) = 1.
  • The resulting expression is connected to the Nakagami-q (Hoyt) MGF to recover the corresponding q parameter.
  • The polynomial R(m, K, Δ; s) factorizes into two linear factors involving K and Δ.
  • Auxiliary parameters a2, a3, and a4 are introduced to compactly rewrite the MGF.
  • Inverse Laplace transforms of the MGF and its division by s produce the PDF and CDF, respectively.

APPENDIX IV PROOF OF LEMMA 3

The appendix derives approximate closed-form PDF and CDF expressions for the FTR fading power envelope by extending a finite Rician-mixture approximation and averaging over Gamma-distributed fluctuations. The added fluctuation preserves tractability rather than increasing mathematical complexity.

  • TWDP approximation: The TWDP power-envelope PDF is approximated as a mixture of 2M Rician distributions.The summation size is governed by K and ∆, with M > K∆ sufficient for a close match to the exact PDF.
  • FTR approximation: The FTR approximation averages the TWDP mixture over the Gamma-distributed random fluctuation of the specular components.This produces a mixture of 2M Rician shadowed PDFs.
  • Closed-form reduction: For integer m, the Kummer hypergeometric function is converted using Laguerre polynomials to obtain a finite-sum representation.The resulting PDF uses exponential functions and powers.
  • Closed-form reduction: Direct integration of the approximate PDF yields the corresponding approximate CDF.The CDF expression is given in closed form alongside the PDF approximation.
  • Tractability: The additional FTR fluctuation does not increase mathematical complexity and instead facilitates mathematical tractability.This contrasts with the integral-form exact TWDP power-envelope PDF.

APPENDIX V PROOF OF LEMMA 4

The appendix evaluates the double integral arising in the derivation by applying a similar approach to an earlier proof and using tabulated integral identities. This produces the stated closed-form result.

  • Derivation: The derivation uses the same general approach as the preceding appendix.The method is explicitly described as analogous to the approach used in Appendix I.
  • Derivation: The double integral is solved in closed form using two cited tabulated identities.These identities yield expression (35).
  • Result: The closed-form evaluation establishes the result stated as expression (35).The passage identifies (35) as the outcome of the integral calculation.
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