# The Rayleigh Birnbaum Saunders Distribution: A General Fading Model

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### The Birnbaum Saunders Distribution

## 3. The Proposed Fading Channel Model

#### Additional Features

## 4. The RBS Channel Phasor

#### Some Measures of Interest in the Setting of Fading Channel

**Proposition**

**1.**

**Proof.**

## 5. Some Measures for Comparing

#### Comparison with the RL, K and RIG Distributions

## 6. Simulating the Proposed Distribution

Algorithm 1: Simulating the RBS distribution. | |

Input: | |

Alpha: $\alpha $ parameter ($\alpha \in \mathbb{R}$) | |

Beta: $\beta $ parameter ($\beta \in \mathbb{R}$) | |

A: range of data (Amplitude $\in \mathbb{R}$) | |

NumberSamples: number of samples (NumberSamples $\in \mathbb{N}$) | |

Output: | |

S: set of data $\sim RBS(\alpha ,\beta )$ | |

Begin: | |

r ← CDF(Alpha, Beta) (0 → A) | ⊳CDF to a column vector for $i\leftarrow 1\mathrm{to}NumberSamples$ do |

end | |

$u\leftarrow \mathrm{rand}$ | ⊳Get a uniform random |

number | |

$S\left(i\right)\leftarrow \mathrm{max}\{r\le u\}$ | |

End |

#### The RBS Distribution for Modeling Fading Effects

Algorithm 2: Fading simulation. | |

Input: | |

Alpha: $\alpha $ parameter ($\alpha \in \mathbb{R}$) | |

Beta: $\beta $ parameter ($\beta \in \mathbb{R}$) | |

Rays: number of emitted signals (Rays $\in \mathbb{N}$) | |

Ref: number of reflections (Ref $\in \mathbb{N}$) | |

Speed: average speed of vehicle (Speed $\in \mathbb{R}$) | |

Freq: carrier frequency (Freq $\in \mathbb{R}$) | |

TimeF: simulation time (TimeF $\in \mathbb{R}$) | |

TimeS: sampling time (TimeS $\in \mathbb{R}$) | |

Output: | |

r: signal envelope | |

Begin: | |

A ← getPower(Alpha, Beta) | ⊳Get average signal strength and variance |

Phi ← rand(Rays, Ref) $\xb72\xb7\pi $ | ⊳Uniformly distributed phase |

Psi ← rand(Rays, Ref)$\xb72\xb7\pi $ | ⊳Uniformly distributed arriving signal |

D ←$2\xb7\pi \xb7$ Freq· 3.3e-03 | ⊳Maximum Doppler frequency shift |

w $\leftarrow D\xb7Speed\xb7cos\left(Psi\right)$ | ⊳Consider Doppler effects |

t ← 0: TimeS: TimeF | ⊳Time span |

X ← 0 | |

Y ← 0 | |

Df $\leftarrow w\xb7t-Phi$ | ⊳Obtain Doppler phase shift |

$X\leftarrow X+A\xb7cos\left(Df\right)$ | ⊳In-phase component |

$Y\leftarrow Y+A\xb7sin\left(Df\right)$ | ⊳Quadrature component |

end for | |

end for | |

$r\leftarrow \sqrt{{X}^{2}+{Y}^{2}}$ | ⊳Calculate the signal envelope |

End |

## 7. Final Comments

#### Data Availability

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of the cdf and pdf of the RBS Distribution

**Proposition**

**A1.**

**Proof.**

## References

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**Figure 2.**Probability density function of the $RBS(\alpha ,\beta )$ distribution for different parameter values.

**Figure 3.**Illustration of the hazard rate function of $RBS(\alpha ,\beta )$ distribution for a set of different parameter values.

**Figure 4.**Average BERs of DPSK and MSK for RLN, K, RIG, SR and RBS distributions assuming values for the parameters given in different settings provided in Table 1.

**Figure 5.**Graphics of the pdf of the different distributions for parameters given in Settings A (

**left**) and B (

**right**).

**Figure 6.**Comparison of the analytic RBS($\alpha =0.5$, $\beta =1$) magnitude pdf and the Monte Carlo simulated dataset (

**top**). Comparison of the analytic RBS ($\alpha =1$, $\beta =2$) magnitude pdf and the Monte Carlo simulated dataset (

**bottom**). Simulated samples (1,000,000) for both cases are also represented (

**right**column).

**Figure 7.**Samples simulated by using phasors of the RBS distribution ($\alpha $ = 0.5, $\beta =3$) (

**top**,

**left**) and a set of simulated samples (

**top**,

**right**). Samples simulated by using phasors of the RBS distribution ($\alpha $ = 1, $\beta =2$) (

**bottom**,

**left**) and a set of simulated samples (

**bottom**,

**right**). For both cases, the analytic RBS distribution is also shown, for the sake of comparison.

**Figure 8.**Simulated samples (using phasors) of the Rayleigh fading distribution (

**top left**), the SR fading distribution (

**top right**), the RLN fading distribution (

**bottom left**) and, the proposed RBS fading distribution. The parameters used in all cases are those providing similar mean and standard deviation values, and are represented spaced 0.1 wavelength apart for the 0 dB mean value.

**Figure 9.**Fading signal for the RBS distribution ($\alpha =0.66$, $\beta =1.74$); carrier frequency = 1 GHz and speed = 50 Km/h (

**left**) and speed = 120 km/h (

**right**).

Setting | ||||
---|---|---|---|---|

Model | Parameters | A | B | C |

RLN | $\mu $ | 0.63 | 0.51 | −1.57 |

$\lambda $ | 0.85 | 1.21 | 1.56 | |

K | a | 1.00 | 1.00 | 1.00 |

b | 0.35 | −0.37 | −0.65 | |

RIG | $\kappa $ | 2.71 | 1.29 | 0.11 |

$\delta $ | 2.69 | 3.47 | 0.70 | |

GR | $\vartheta $ | 4.76 | 24.07 | 155.48 |

$\omega $ | 7.33 | 25.86 | 21.61 | |

SR | q | 3.45 | 2.80 | 2.48 |

$\sigma $ | 1.14 | 0.36 | 0.14 | |

RBS | $\alpha $ | 0.94 | 1.54 | 2.52 |

$\beta $ | 0.53 | 0.63 | 5.96 |

Setting | ||||
---|---|---|---|---|

Measure | Model | A | B | C |

JSD | K | 0.001 | 0.049 | 0.032 |

RIG | 3.0 $\times {10}^{-5}$ | 2.9 $\times {10}^{-4}$ | 0.001 | |

GR | 5.5 $\times {10}^{-4}$ | 9.6 $\times {10}^{-4}$ | 0.005 | |

SR | 0.188 | 0.050 | 0.025 | |

RBS | 8.5 $\times {10}^{-5}$ | 1.0 $\times {10}^{-3}$ | 0.007 | |

ISE | K | 0.002 | 0.090 | 0.328 |

RIG | 3.9 $\times {10}^{-5}$ | 5.1 $\times {10}^{-4}$ | 0.007 | |

GR | 3.5 $\times {10}^{-4}$ | 1.5 $\times {10}^{-3}$ | 0.030 | |

SR | 5.2 $\times {10}^{-3}$ | 0.068 | 0.122 | |

RBS | 8.1 $\times {10}^{-5}$ | 1.7 $\times {10}^{-3}$ | 0.047 |

**Table 3.**Mean and variance values for the analytic RBS distribution and those estimated from the simulated samples for two sets of parameters.

RBS ($\mathit{\alpha}$ = 0.5, $\mathit{\beta}=1$) | RBS ($\mathit{\alpha}$ = 1, $\mathit{\beta}=2$) | |
---|---|---|

Mean (Analytic) | 1.2909 | 0.9831 |

Mean (simulated data) | 1.2772 | 0.9685 |

Relative error | 1.05% | 1.48% |

Variance (analytic) | 0.5836 | 0.5335 |

Variance (simulated data) | 0.5698 | 0.5078 |

Relative error | 2.37% | 4.82% |

**Table 4.**Means and variances for the analytic $RBS\phantom{\rule{0.166667em}{0ex}}(\alpha ,\beta )$ distribution and, the values estimated using phasors for two-parameter sets.

RBS$(0.5,3)$ | RBS$(1,2)$ | |
---|---|---|

Mean (Analytic) | 0.7453 | 0.9831 |

Mean (simulated data) | 0.7213 | 0.9533 |

Relative error | 3.21% | 3.03% |

Variance (analytic) | 0.1945 | 0.5335 |

Variance (simulated data) | 0.1937 | 0.5080 |

Relative error | 0.42% | 4.78% |

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**MDPI and ACS Style**

Gómez-Déniz, E.; Gómez, L.
The Rayleigh Birnbaum Saunders Distribution: A General Fading Model. *Symmetry* **2020**, *12*, 389.
https://doi.org/10.3390/sym12030389

**AMA Style**

Gómez-Déniz E, Gómez L.
The Rayleigh Birnbaum Saunders Distribution: A General Fading Model. *Symmetry*. 2020; 12(3):389.
https://doi.org/10.3390/sym12030389

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, and Luis Gómez.
2020. "The Rayleigh Birnbaum Saunders Distribution: A General Fading Model" *Symmetry* 12, no. 3: 389.
https://doi.org/10.3390/sym12030389