# Mixed Topology of DF Relayed Terrestrial Optical Wireless Links with Generalized Pointing Errors over Turbulence Channels

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## Abstract

**:**

## 1. Introduction

## 2. System and Channel Model

#### 2.1. Signal and System Assumptions

_{{i,j}},at the jth hop in the ith path is expressed as [10,15]:

_{{i,j}}, is the normalized irradiance in the jth hop in the ith path, while x

_{{i,j}}and n

_{{i,j}}denote the corresponding modulated signal and the additive white Gaussian noise (AWGN) with zero mean and variance N

_{0}/2, respectively [3,4].

_{{i,j}}, is written as [3,4,10]:

_{a,}

_{{i,j}}and I

_{p,}

_{{i,j}}represent the normalized received irradiance in the jth hop in the ith path due to turbulence-induced and misalignment-induced fading, respectively, while I

_{l,}

_{{i,j}}stands for the corresponding deterministic weather-dependantpath loss parameter, [23]. However, the investigation of the path losses is not the scope of this work and thus, without loss of generality, it is assumed that is normalized to unity (I

_{l,m}= 1) [4,24].

#### 2.2. The Atmospheric Turbulence Effect

_{a,}

_{{i,j}}, is given as [3,20]:

_{ν}(.) denotes the modified Bessel function of the second kind of order ν (Equation (8.432.2) in [25]), Γ(.) represents the gamma function (Equation (8.310.1) in [25]), and the parameters α, β are estimated from each link’s parameters through the following expressions [22,26]:

_{n}

^{2}the refractive index structure parameter which is proportional to the atmospheric turbulence strength, and varies between 10

^{−17}m

^{−2/3}and 10

^{−13}m

^{−2/3}for weak to strong turbulence, respectively [27].

_{a,}

_{{i,j}}, is given as [21,22,28]:

#### 2.3. Generalized Pointing Errors

_{{i,j}}being the radial displacement in the jth hop in the ith path that is obtained as ${R}_{\left\{i,j\right\}}=\left|{\overrightarrow{R}}_{\left\{i,j\right\}}\right|=\sqrt{{R}_{x,\left\{i,j\right\}}^{2}+{R}_{y,\left\{i,j\right\}}^{2}}$, while ${\overrightarrow{R}}_{\left\{i,j\right\}}={\left[{R}_{x,\left\{i,j\right\}},{R}_{y,\left\{i,j\right\}}\right]}^{T}$ stands for the radial displacement vector, with R

_{x,}

_{{i,j}}and R

_{y,}

_{{i,j}}representing the corresponding offsets located along the horizontal and elevation axes at the detector plane, which are expressed as non-zero mean Gaussian distributed random variables, i.e., ${R}_{x,\left\{i,j\right\}}~N\left({\mu}_{x,\left\{i,j\right\}},{\sigma}_{x,\left\{i,j\right\}}^{2}\right)$ and ${R}_{y,\left\{i,j\right\}}~N\left({\mu}_{y,\left\{i,j\right\}},{\sigma}_{y,\left\{i,j\right\}}^{2}\right)$, where the parameters μ

_{x,}

_{{i,j}}, μ

_{y,}

_{{i,j}}, represent their corresponding mean values, and σ

_{x,}

_{{i,j}}, σ

_{y,}

_{{i,j}}, the corresponding jitters for horizontal and elevation displacements, respectively. The joint standard deviation of σ

_{x,}

_{{i,j}}, σ

_{y,}

_{{i,j}},in the jth hop in the ith path, can be obtained as [19],

_{{i,j}}correspond to a weaker amount of misalignment-induced fading, while, w

_{z,eq,}

_{{i,j}}denotes the equivalent beam radius in the jth hop in the ith path, which is given as ${w}_{z,eq,\left\{i,j\right\}}={\left[\sqrt{\pi}\mathrm{erf}({v}_{\left\{i,j\right\}}){w}_{z,\left\{i,j\right\}}^{2}/2{v}_{\left\{i,j\right\}}\mathrm{exp}(-{v}_{\left\{i,j\right\}}^{2})\right]}^{1/2}$, where ${v}_{\left\{i,j\right\}}=\sqrt{\pi}{r}_{a,\left\{i,j\right\}}/\sqrt{2}{w}_{z,\left\{i,j\right\}}$, with r

_{a,}

_{{i,j}}being the radius of the circular detection aperture, and erf(.), the error function. Additionally A

_{0,}

_{{i,j}}describes the fraction of the collected power at r

_{a,}

_{{i,j}}= 0, with ${A}_{0,\left\{i,j\right\}}={\mathrm{erf}}^{2}({v}_{\left\{i,j\right\}})$ [7,19]. It is also worth mentioning that by setting in the jth hop in the ith path, the boresight displacement is equal to zero, i.e., ${s}_{\left\{i,j\right\}}=\sqrt{{\mu}_{x,\left\{i,j\right\}}^{2}+{\mu}_{y,\left\{i,j\right\}}^{2}}=0$, the Beckmann’s distribution reduces to the well-known Rayleigh’s distribution in [7], which accurately describes the corresponding zero boresight pointing errors effect [8]. Indeed, for s

_{{i,j}}= 0 it holds that μ

_{x,}

_{{i,j}}= μ

_{y,}

_{{i,j}}= 0 and σ

_{x,}

_{{i,j}}= σ

_{y,}

_{{i,j}}and thus, (6) reduces to Equation (10) in [7] and (8) reduces to Equation (11) in [7].

#### 2.4. Joint Turbulence and Pointing Errors Effects

_{{i,j}}, is obtained by the following integral [7,17]:

_{a,}

_{{i,j}}, either for G-G or NE distribution, Equations (3) or (5), respectively.

_{{i,j}}] is the corresponding expected normalized irradiance value, [15,17,22] which for both G-G and NE turbulence distributions is given according to [33,34,35], as:

## 3. Outage Probability Estimation

_{th}, which corresponds to the receiver’s sensitivity limit. Therefore, the OP inthe jth hop in the ith path is expressed as [10,12,15,17]:

_{{i,j},th}/μ

_{{i,j}}represents the normalized average electrical SNR, we conclude that (13) can be written as:

_{{i,j},th}/μ

_{{i,j}}that is assumed to obtain the same value for each link of the examined FSO system, i.e., γ

_{{i,j},th}/μ

_{{i,j} =}γ

_{th}/μ, under G-G and NE modeled turbulence conditions, respectively. Moreover, (14) and (15) refer to mixed DF relay FSO configurations, and thus, contrary to the ABER expressions in [4] they introduce the indices i and j.

## 4. Mean Outage Duration Estimation

_{od}for a specific period of time is calculated as [29,36]:

_{R}being the appropriately chosen reference time, e.g., T

_{R}= 3600 s for the case where the reference time is one hour. Additionally, the coherence time of the atmospheric turbulence, τ

_{0}, is in the range of milliseconds [37,38], and for T

_{od}, it should be verified that T

_{od}≥ τ

_{0}. Furthermore, note that for the simulation experiments conducted in the next section we assume a reference time of one hour.

_{GG,od}, for G-G:

_{NE,od}, for NE turbulent channels, respectively, as:

## 5. Numerical Results

_{n}

^{2}value is fixed at 2 × 10

^{−14}m

^{−2/3}, while for NE modeled, saturated turbulent channels, the parameter C

_{n}

^{2}obtains its maximum value, i.e., C

_{n}

^{2}= 10

^{−13}m

^{−2/3}.

_{x}/r

_{a},μ

_{y}/r

_{a}) = (0,0) and σ

_{x}= σ

_{y}, while ψ

_{1}= 5 for $\left({r}_{a},{w}_{z}/{r}_{a},{\mu}_{x}/{r}_{a},{\mu}_{y}/{r}_{a},{\sigma}_{x}/{r}_{a},{\sigma}_{y}/{r}_{a}\right)=\left(5\mathrm{cm},10,0,0,1,1\right)$. When Ν = 3 relays are considered, we assume for the remaining paths that $\left({r}_{a},{w}_{z}/{r}_{a},{\mu}_{x}/{r}_{a},{\mu}_{y}/{r}_{a}\right)=\left(5\mathrm{cm},10,1,2\right)$, while for weak to strong non-zero boresight pointing errors, the parameter ψ

_{2}takes the value 2.3 or 1.3 for (σ

_{x}/r

_{a},σ

_{y}/r

_{a}) = (2.1,1.5) or (σ

_{x}/r

_{a},σ

_{y}/r

_{a}) = (4,3), respectively. Finally, for Ν = 5, the non-zero boresight pointing errors effect is stronger and even more stronger, with ψ

_{3}= 0.9 or ψ

_{3}= 0.6, for (σ

_{x}/r

_{a},σ

_{y}/r

_{a}) = (5.8,4.8) or (σ

_{x}/r

_{a},σ

_{y}/r

_{a}) = (9,7), respectively.

_{j}= 2, under saturated turbulence conditions, modeled through the NE distribution, along with zero boresight pointing errors for the single path configuration, i.e., for Ν = 1, and additional weak to strong non-zero boresight pointing errors for multiple path configurations, i.e., for N = 3 or 5. It is clearly shown that even in saturated turbulence conditions, the OP performance is significantly improved by using parallel relaying method with dual-hop paths, and especially, by increasing the number of paths of the total system. Additionally, further OP performance enhancements are depicted as the normalized average electrical SNR obtains larger values and as the non-zero boresight misalignment-induced fading gets weaker, i.e., for larger ψ

_{2}or ψ

_{3}parameter values that refer to triple-path (N = 3) and quintuple-path (N = 5) dual-hop implementations, respectively. Moreover, it is worth mentioning that the performance comparison between the corresponding curves of Figure 2 and those which appear in Ref. [17] obtained by the analytical expression Equation (15) in [17], demonstrates significantly larger OP corresponding values due to the fact that Figure 2 refers to saturated NE modeled turbulence channels, while the results of [17] have been obtained for weak turbulence conditions with a different model, i.e. the Gamma distribution.

_{j}= 4. It can be seen that although qualitatively behavior is the same for both cases, in Figure 3 degraded corresponding OP performance results are obtained due to the larger number of hops in each path that lead to longer path lengths. However, we ought to mention that by increasing the hops number, the coverage area also increases. Thus, at the expense of an increase in OP values we can broaden the coverage FSO area, while considering our system’s OP specifications and demands, we can conclude whether it is wise to extend the total coverage area of our FSO system, under specific turbulence-induced and misalignment-induced fading conditions. Furthermore, by comparing Figure 3 and Figure 2 in [17] obtained by the analytical expression Equation (15) in [17], Figure 3 depicts significantly increased OP corresponding values due to the stronger turbulence conditions that are assumed.

_{j}= 2, under weak turbulence conditions, modeled through the G-G distribution, along with zero boresight pointing errors for Ν = 1, and additional weak to strong non-zero boresight pointing errors for N = 3 or 5. Thus, the performance comparison between Figure 2 and Figure 4 demonstrates that the obtained results of the latter outperformed the corresponding results of the former, in terms of OP. This is due to a weaker amount of turbulence strength that the FSO propagation has to deal with in the latter case. Consequently, the comparison between these two figures reveals the detrimental impact of turbulence-induced fading, on the outage FSO performance.

_{j}= 4. Once again, despite the identical qualitative behavior of the two figures, we now obtain increased corresponding OP values due to the larger number of the serially connected DF relays that are employed in each path. Furthermore, due to the weak turbulence conditions instead of the saturated we investigated before, we can observe the current performance comparison, i.e., between Figure 4 and Figure 5, which shows less significant outage performance degradation, by doubling the hops in each path, than the corresponding performance comparison between the Figure 2 and Figure 3.

_{j}= 2 or 4, under weak gamma-gamma modeled turbulence and different amounts of non-zero boresight pointing errors. As we can see, apart from increasing the electrical SNR value and mitigating the strength of pointing errors effects, the increase of the number of paths can lead to significant outage performance enhancements of the total system, and therefore, N = 5 configurations are shown to outperform N = 3 configuration, in terms of MOD metric. However, by increasing also the number of hops in each path, the total MOD value increases too. Additionally, the performance comparison between Figure 6 and Figure 7 concludes that the MOD results of the latter outperform the corresponding MOD results of the former. This is due to the weaker amount of turbulence-induced fading that is addressed in the latter case. In fact, given that γ/μ = 18 dB for N = 5 with H

_{j}= 4 and strong non-zero boresight pointing errors, due to the weaker turbulence conditions we now obtain a strongly reduced MOD value almost of 1.5 ms per hour instead of the almost 2.9 s per hour we obtained before for the same system’s characteristics, except for saturated turbulence conditions. Additionally, given that γ/μ = 18 dB for N = 5 with H

_{j}= 4 and weak non-zero boresight pointing errors this time, the MOD value is now shown to be almost 4.8 μs per hour instead of the corresponding MOD value of almost 1.6 s per hour of Figure 6.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Total outage probability (OP) of a dual-hop relay-aided system with varrying path configurations as a function of a wide range of γ/μ, for negative exponential (NE) modeled saturated turbulent channels along with different amounts of pointing mismatch.

**Figure 3.**Total outage probability (OP) of a quadruple-hop relay-aided system with varying path configurations as a function of a wide range of γ/μ, for negative exponential (NE) modeled saturated turbulent channels along with different amounts of pointing mismatch.

**Figure 4.**Total outage probability (OP) of a dual-hop relay-aided system with varying path configurations as a function of a wide range of γ/μ, for gamma-gamma (G-G) modeled weak turbulent channels, along with different amounts of pointing mismatch.

**Figure 5.**Total outage probability (OP) of a quadruple-hop relay-aided system with varying path configurations as a function of a wide range of γ/μ, for gamma-gamma (G-G) modeled weak turbulent channels along with different amounts of pointing mismatch.

**Figure 6.**Total mean outage duration (MOD) of a quadruple-hop relay-aided system with varying path configurations as a function of a wide range of γ/μ, for negative exponential (NE) modeled saturated turbulent channels along with different amounts of pointing mismatch.

**Figure 7.**Total mean outage duration time (MOD) of a relay-aided system with both varying path and hop configurations as a function of a wide range of γ/μ, for gamma-gamma (G-G) modeled weak turbulent channels along with different amounts of pointing mismatch.

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## Share and Cite

**MDPI and ACS Style**

Varotsos, G.K.; Nistazakis, H.E.; Stassinakis, A.N.; Volos, C.K.; Christofilakis, V.; Tombras, G.S.
Mixed Topology of DF Relayed Terrestrial Optical Wireless Links with Generalized Pointing Errors over Turbulence Channels. *Technologies* **2018**, *6*, 121.
https://doi.org/10.3390/technologies6040121

**AMA Style**

Varotsos GK, Nistazakis HE, Stassinakis AN, Volos CK, Christofilakis V, Tombras GS.
Mixed Topology of DF Relayed Terrestrial Optical Wireless Links with Generalized Pointing Errors over Turbulence Channels. *Technologies*. 2018; 6(4):121.
https://doi.org/10.3390/technologies6040121

**Chicago/Turabian Style**

Varotsos, George K., Hector E. Nistazakis, Argyris N. Stassinakis, Christos K. Volos, Vasileios Christofilakis, and George S. Tombras.
2018. "Mixed Topology of DF Relayed Terrestrial Optical Wireless Links with Generalized Pointing Errors over Turbulence Channels" *Technologies* 6, no. 4: 121.
https://doi.org/10.3390/technologies6040121