# Closed-Form Hyper-Rayleigh Mode Analysis of the Fluctuating Double-Rayleigh with Line-of-Sight Fading Channel

^{*}

## Abstract

**:**

## 1. Introduction

- The closed-form expression is derived for the moment-function of the fluctuating double-Rayleigh with line-of-sight channel model, which helps to evaluate the exact expression for the amount of fading.
- Novel exact expressions for the cumulative distribution function and the outage probability of the fluctuating double-Rayleigh with line-of-sight channel model are derived, extending the existing result, and are valid for an arbitrary positive fading parameter m.
- A hyper-Rayleigh map, which associates the set of system parameter values with the specific propagation conditions, is obtained. The presented results are valid not only for an “asymptotic” definition of the hyper-Rayleigh map (as it is defined in classic technical literature), but for all possible values of the system and channel parameters.
- The performed analysis is exemplified with a problem of physical layer security analysis. A thorough numeric analysis of the strictly positive secrecy capacity is performed for all possible hyper-Rayleigh scenarios (i.e., channel parameters).
- The asymptotic parameters’ regions, where either the amount of fading, or the outage probability, or the ergodic capacity are nearly insensible to the channel parameter change are identified.

## 2. Materials and Methods

#### 2.1. Physical Channel Model

#### 2.2. Statistical Channel Model

#### 2.3. Hyper Rayleigh Mode

- amount of fading is greater than AoF for the Rayleigh model, i.e., 1:$$AoF>Ao{F}^{Ray}=1$$
- Or for equal values of the asymptotic (in sense of ${\gamma}_{th}\ll \overline{\gamma}$) outage probability the average SNR is greater than the Rayleigh average SNR
- Or for equal average SNRs, the asymptotic (in sense of $\overline{\gamma}\to \infty $) average capacity is less than that for the Rayleigh model.

#### 2.4. Secrecy Communication Metrics

## 3. Results

#### 3.1. Amount of Fading

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 3.2. Outage Probability

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Proof.**

#### 3.3. Channel Capacity

## 4. Discussion

- For the simulation, the shadowing parameter m is upper-bounded by $m=20$ (although the plots are presented for $m<10$), since numerous studies [26,27] showed that the values $m>5\xf710$ (depending on the problem under consideration) can be assumed as ”almost asymptotic”. This generally means that its increase does not deliver significant changes in the result.
- The lower bound of the shadowing parameter was set to 0.5, although, $0<m<0.5$ are also possible (since the shadowing is introduced via Gamma-distributed model). But numerous simulations demonstrated that $m<0.5$ do not make any contribution to the current research, thus were omitted.
- The Rician K-factor was chosen in order to be consistent with practical scenarios of real-life measurements (i.e., from $-30$ dB to 30 dB).
- The expression (16) is given in terms of an infinite series, thus for further numerical calculations it was truncated to 200 terms to yield at least 4-digit accuracy compared with the brute-force numerical integration. Moreover, if an extra precision (for a fixed number of summands) is required, well-established techniques for convergence increase can be applied (for example, Richardson extrapolation or Shanks’ transformation (see [69])).

#### 4.1. Hyper-Rayleigh Regime Numerical Analysis

#### 4.2. Illustrative Example: SPSC in Hyper-Rayleigh Regime

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

## Appendix B. Proof of Theorem 2

## References

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**Figure 1.**Hyper-Rayleigh mode metrics for the fdRLoS channel model with various k and m: (

**a**) Amount of Fading map (yellow dashed line denotes the contour of the Rayleigh model, i.e., $Ao{F}^{Ray}$); (

**b**) outage probability map (in logarithmic scale) with ${\gamma}_{th}=0$ dB and $\overline{\gamma}=10$ dB (yellow dashed line denotes the contour of the Rayleigh model, i.e., ${P}_{out}^{Ray}$); (

**c**) ergodic capacity map with $\overline{\gamma}=10$ dB (yellow dashed line denotes the contour of the Rayleigh model, i.e., ${\overline{C}}^{Ray}$).

**Figure 2.**The joint hyper-Rayleigh map of the fdRLoS channel: red region—full HRM, orange—strong HRM, yellow—weak HRM, green—no HRM.

**Figure 3.**$\mathbb{P}\{\overline{C}>0\}$ for the fdRLoS model as a function of the ${\overline{\gamma}}_{D}$ for different ${\overline{\gamma}}_{E}$: green lines denote no HRM, red—weak HRM, blue—strong HRM, and black—full HRM.

Parameter | Parameter Value |
---|---|

Shadowing coefficient (m) | $\frac{1}{2}\dots 20$ |

Rician K-factor (k) | ${10}^{-3}\dots {10}^{3}$ |

Average SNR ($\overline{\gamma}$), dB | $-20\dots 50$ |

Threshold SNR (${\gamma}_{th}$), dB | $-10\dots 30$ |

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**MDPI and ACS Style**

Gvozdarev, A.S.; Artemova, T.K.; Alishchuk, A.M.; Kazakova, M.A.
Closed-Form Hyper-Rayleigh Mode Analysis of the Fluctuating Double-Rayleigh with Line-of-Sight Fading Channel. *Inventions* **2023**, *8*, 87.
https://doi.org/10.3390/inventions8040087

**AMA Style**

Gvozdarev AS, Artemova TK, Alishchuk AM, Kazakova MA.
Closed-Form Hyper-Rayleigh Mode Analysis of the Fluctuating Double-Rayleigh with Line-of-Sight Fading Channel. *Inventions*. 2023; 8(4):87.
https://doi.org/10.3390/inventions8040087

**Chicago/Turabian Style**

Gvozdarev, Aleksey S., Tatiana K. Artemova, Aleksandra M. Alishchuk, and Marina A. Kazakova.
2023. "Closed-Form Hyper-Rayleigh Mode Analysis of the Fluctuating Double-Rayleigh with Line-of-Sight Fading Channel" *Inventions* 8, no. 4: 87.
https://doi.org/10.3390/inventions8040087