# RSSI Probability Density Functions Comparison Using Jensen-Shannon Divergence and Pearson Distribution

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Atmopsheric Turbulence

_{0}, i.e., outer scale and end at the microscale l

_{0}, i.e., inner scale. An important assumption of this theory is that eddies smaller than L

_{0}are statistically homogeneous and isotropic [13]. Due to this characteristic, the covariance of the refractive index reduces to a function of solely a scalar distance R = |

**R**

_{1}−

**R**

_{2}|

^{2}, where

**R**

_{1}and

**R**

_{2}are two random points in space. Then, the structure function within the inertial subrange, i.e., l

_{0}<< R << L

_{0}, is [13],

_{n}

^{2}is the refractive index structure parameter. A point estimate of C

_{n}

^{2}can be easily obtained by measuring the temperature difference within two points with known separation distance. This can be done by using a set of thermocouples, with various numbers of fine wires. Another important parameter is the temperature structure parameter, C

_{T}

^{2}, which is related to C

_{n}

^{2}as follows [13],

_{n}

^{2}measurements can also be taken using an instrument called scintillometer. A large number of experimental research works have led to the construction of many empirical models, for any kind of terrain, range and geometry of measurements, which can give good estimates for the refractive index structure parameter along the path, assuming constant values. A comprehensive list of C

_{n}

^{2}predictive models can be found in Tab. 2.5 [12].

#### 1.2. Channel Modeling

^{2}is the log-irradiance, which depends on the aperture diameter, the wavelength and turbulence strength. On the other hand, the GG distribution is obtained as the product of two independent gamma modeled distributions, modelling small and large scale fluctuations. The great advantage of the GG distribution is that its two parameters, a and b, are directly related to C

_{n}

^{2}. The PDF of the GG distribution is given as [4],

_{q}(.) is the modified Bessel function of the second kind of order q. The parameters a and b are directly related to turbulence and system parameters for a spherical wave as follows [4],

_{th}[4]. Mathematically, this is expressed as,

_{γ}is the corresponding cumulative distribution function (CDF) of the SNR. The outage probability can now be expressed in accordance with the model that has been used to model the turbulent channel. Using the relation between irradiance, instantaneous and average SNR [4], i.e.,

_{out}can be expressed as [4],

## 2. Data Acquisition

## 3. Results

#### 3.1. Kullback–Leibler Divergence

_{KL}, the less the distance between p and q, which ultimately becomes zero if the two distributions are identical. Additionally, if p

_{i}> 0 and q

_{i}= 0 for a given i, the KL divergence goes to infinity. Cover and Thomas give a technical interpretation of the KL divergence, as the “coding penalty” associated with a distribution q selected to compare with a distribution p [22].

^{−2}, followed by the Weibull distribution yielding a value of 3.07 × 10

^{−2}.

#### 3.2. Jensen–Shannon Divergence

#### 3.3. Pearson Distribution Family

_{1}and β

_{2}. Any valid solution of Equation (22) defines a Pearson type PDF [26].

_{0}= b − c

^{2}(4d)

^{−1}and A

_{1}= c(2d)

^{−1}, yields the best fit. The Pearson Type IV cumulative distribution function (CDF) against the RSSI data is plotted in Figure 8 and a very good agreement is observed.

## 4. Conclusions

^{−2}and 6.53 × 10

^{−3}, respectively. The Weibull and EV distributions also exhibited a comparable accuracy in contrast with the Gamma and Lognormal, whose “distance” from real data was an order of magnitude higher. The Pearson distribution family of continuous probability functions was also used and experimental results showed that the type IV distribution yield the best fit. Both methods offer a straightforward process of comparing among different PDFs for accurately fitting real data. The two aforementioned methods follow a different approach to select the most appropriate PDF to fit to a dataset. The KL and JD metrics allow for the calculation of the performance metric of each candidate PDF, thus designating the best one as the one with the lowest divergence. From a practical point of view, the KL and JD metrics are more useful to apply in a real-world dataset, since they directly provide a certain output that allows any PDF to be examined. On the other hand, the selection of the best Pearson distribution using a moment matching method is preferable when a limited number of options for statistical modeling are available.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**The maritime free-space optical (FSO) link propagation path (

**left**) and the MRV TS5000/155 FSO terminal on the lighthouse of Psitalia Island (

**right**). On the upper left back side of the system, the received signal strength indicator (RSSI) indicator can be seen.

**Figure 2.**The Kullback–Leibler divergence values for Gamma, Weibull, Lognormal, Burr and Extreme-Value distribution.

**Figure 3.**Probability density functions fits to RSSI data for (

**a**) Burr distribution, (

**b**) Extreme-Value (left) and Gamma (right) distribution, (

**c**) Log-Normal (left) and Weibull (right) distribution.

**Figure 7.**The Jensen–Shannon divergence values for Gamma, Weibull, Lognormal, Burr and Extreme-Value distribution.

**Table 1.**Descriptive statistics for the RSSI measurements from the 30 November 2019 to the 27 October 2020.

Statistic | Value |
---|---|

Mean | 420.385927 |

Standard Error | 0.084275342 |

Median | 425 |

Mode | 445 |

Standard Deviation | 32.06917633 |

Kurtosis | 1.481024233 |

Skewness | −0.798160493 |

Maximum | 187 |

Minimum | 517 |

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**MDPI and ACS Style**

Lionis, A.; Peppas, K.P.; Nistazakis, H.E.; Tsigopoulos, A.
RSSI Probability Density Functions Comparison Using Jensen-Shannon Divergence and Pearson Distribution. *Technologies* **2021**, *9*, 26.
https://doi.org/10.3390/technologies9020026

**AMA Style**

Lionis A, Peppas KP, Nistazakis HE, Tsigopoulos A.
RSSI Probability Density Functions Comparison Using Jensen-Shannon Divergence and Pearson Distribution. *Technologies*. 2021; 9(2):26.
https://doi.org/10.3390/technologies9020026

**Chicago/Turabian Style**

Lionis, Antonios, Konstantinos P. Peppas, Hector E. Nistazakis, and Andreas Tsigopoulos.
2021. "RSSI Probability Density Functions Comparison Using Jensen-Shannon Divergence and Pearson Distribution" *Technologies* 9, no. 2: 26.
https://doi.org/10.3390/technologies9020026