**1. Introduction**

Mobile cellular traffic has astoundingly increased during the last decade mainly due to the stunning expansion of smart wireless devices and bandwidth demanding applications (i.e., high-definition videos, gaming, social networking, etc.). The overall mobile data traffic is expected to grow up to 77 Exabyte's per month by 2022 which is about a seven-fold increase over 2017 data traffic [1]. In addition, the number of devices and connections will continue to grow exponentially. The fifth generation (5G) networks are aimed at meeting the requirements of mobile communications even beyond 2025. Current backhaul communication of cellular networks uses licensed microwave spectrum and wired copper/fiber based links. These two systems have several limitations (e.g., low data rates, security issues, and high cost of installation in urban canyons). Choosing a suitable technology in the design of the backhaul

network architecture plays a vital role in the performance of the next generation cellular networks [2]. Wireless free-space optics (FSO) and unlicensed millimeter wave (mmW) communications are being considered as the major technologies for high data rate backhaul traffic of next generation wireless networks of 5G and beyond [3]. Millimeter wave communications can achieve data rates up to 10 Gbits/s. This is because there is a large amount of bandwidth available in the mmW band (30– 300 GHz), and allocating this bandwidth is an efficient approach to enhance system capacity. FSO is a line-of-sight (LOS) technology that uses light emitting diodes (LEDs) or laser to transmit information through free space medium. FSO communication has attracted significant attention due to its unlimited bandwidth, as it operates in the unlicensed Tera Hertz spectrum band [4]. However, both the FSO and mmW links are affected by different weather conditions but are complementary in nature. Therefore, hybrid system comprising of FSO/mmW communication has the advantages of both the systems and overcomes the limitations of individual systems.

In wireless networks of 5G and beyond, a given macro cell will be covered by several small cells. In such cases, some of the small cell BSs which are located at the edges of that macro cell may not be able to communicate directly with it. Hence, there is a necessity of developing a cellular system that will be based on complex interactions between small cell base stations (SBSs) and cooperation between them is necessary in order to improve the overall performance (network coverage, reduced outage probability, spectral and power efficiency) of the system. For such scenarios, triple-hop (TH) hybrid FSO/mmW system has been proposed by considering neighboring small cell BSs as intermediate relays to forward the backhaul data. This chapter mainly focuses on the performance analysis of the proposed triple-hop hybrid FSO/mmW system, which improves the reliability and coverage between SBSs and macro cell BSs over weak and strong turbulence channel conditions. This chapter is organized as follows: Section 2 introduces the proposed system model and statistical characteristics of the system is presented in Section 3. The performance analysis of the proposed system over weak and strong turbulence channel conditions is discussed in Section 4. Section 5 presents the results and discussion, and finally conclusions are presented in Section 6.

### **2. Proposed system and channel models**

To improve the communication between a macro cell BS and small cell BS, a decode and forward (DF) relaying based triple-hop system is proposed by considering intermediate small cell BSs as relay nodes. In this, the first relay (R1) BS will decode the received signal from the source (S) BS and then forward it to the second relay (R2) BS. The second relay BS decodes and forwards the received signal from first relay BS to the destination (D) BS. The proposed DF relaying based triple-hop system comprising of source (S) BS, destination (D) BS, and two relay base stations (R1, R2) is shown in **Figure 1**. The FSO system communicates by using intensity modulation at the transmitter and direct detection scheme (IM/DD) at the receiver [5].

#### **2.1 Modeling of received FSO signals at different BSs**

Mathematical formulation of the received signals for FSO and mmW links at R1, R2, and D base stations are presented in this section. In the case of FSO transmission, the signal received at base station R1 from base station S is given by

$$\mathbf{y}\_{\text{SR}\_1}^{\text{FSO}} = \mathbf{c}\_{\text{SR}\_1} I\_{\text{SR}\_1} \mathbf{g}\_{\text{SR}\_1} \mathbf{x} + \mathbf{n}\_{\text{SR}\_1} \tag{1}$$

*Triple-Hop Hybrid FSO/mmW Based Backhaul Communication System for Wireless Networks… DOI: http://dx.doi.org/10.5772/intechopen.104392*

**Figure 1.** *Schematic of proposed triple-hop hybrid FSO/mmW system.*

The signal received at base station R1 is decoded and forwarded to base station R2 and the signal received at base station R2 is given by

$$\mathcal{Y}\_{R\_1 R\_2}^{FSO} = \mathcal{c}\_{R\_1 R\_2} I\_{R\_1 R\_2} \mathbf{g}\_{R\_1 R\_2} \tilde{\boldsymbol{\kappa}} + \boldsymbol{\pi}\_{R\_1 R\_2} \tag{2}$$

The signal received at base station R2 is decoded and forwarded to base station D and the signal received at base station D is given by

$$\mathbf{y}\_{R\_2D}^{FSO} = \mathbf{c}\_{R\_2D} I\_{R\_2D} \mathbf{g}\_{R\_2D} \hat{\mathbf{x}} + \mathbf{n}\_{R\_2D} \tag{3}$$

where suffix SR1, R1R2, and R2D are source to relay1, relay1 to relay2, and relay2 to destination respectively, x is the symbol transmitted by source BS with average energy Es, *x*~is the estimate of x at R1, *x \_* is the estimate of *x*~ at R2, cj is the receiver's optical-to-electrical conversion coefficient, gj is the average gain of the jth FSO link, Ij represents the optical channel fading due to atmospheric turbulence of jth link which is modeled using Gamma-Gamma distribution [6], nj is the zero-mean circularly symmetric complex Gaussian noise of jth link, where *E* n *<sup>j</sup>*n<sup>∗</sup> *j* n o <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup> *<sup>n</sup>*, and *j*∈f g *SR*1, *R*1*R*2, *R*2*D* . The instantaneous electrical SNR at the output of the FSO receiver denoted by *γFSO <sup>j</sup>* is given as [7].

$$
\eta\_j^{FSO} = \overline{\eta}\_j^{FSO} I\_j^2 \tag{4}
$$

where *γFSO <sup>j</sup>* is the average SNR. By assuming perfect alignment between FSO transmitter and receiver apertures, the fading channel coefficient (Ij ) probability density function (PDF) can be written as

$$f\_{I\_j}^{GG}(\mathbf{I}\_j) = \frac{2(a\beta)^{\frac{a+\beta}{2}}}{\Gamma(a)\Gamma(\beta)} I\_j^{\frac{a+\beta}{2}-1} \mathcal{K}\_{a-\beta} \left(2\sqrt{a\beta I\_j}\right), \quad I \ge 0 \tag{5}$$

where α and β are the small scale and large scale parameters of the scattering environment, Γ(.) is the gamma function [8], and Ka (.) is the modified Bessel function of the second kind of order a. The expression for the PDF of instantaneous SNR of FSO link can be obtained from Eqs. (4) and (5) as

$$f\_{\overline{\mathcal{T}\_j^{\rm SO}}}^{\rm GG}(\boldsymbol{\chi}) = \frac{\left(\frac{a\boldsymbol{\beta}}{\sqrt{\overline{\mathcal{T}\_j^{\rm SO}}}}\right)^{\frac{a+\beta}{2}} (\boldsymbol{\chi})^{\frac{a+\beta}{4}-1}}{\Gamma(a)\Gamma(\beta)} \mathcal{K}\_{a-\beta} \left(\sqrt[2]{\frac{a\beta}{\sqrt{\overline{\mathcal{T}\_j^{\rm SO}}}}}(\boldsymbol{\chi})^{\frac{1}{2}}\right) \text{ for } \boldsymbol{\chi} \ge \mathbf{0} \tag{6}$$

The above PDF expression of *γFSO <sup>j</sup>* can be expressed in terms of Meijer Gfunction G (.) as [9].

$$f\_{\boldsymbol{\mathcal{T}}\_{j}^{\mathrm{FSO}} \mid \boldsymbol{G} \boldsymbol{G}}(\boldsymbol{\chi}) = P\_1(\boldsymbol{\chi})^{-1} \mathcal{G}\_{0,2}^{2,0} \left( \frac{a \beta(\boldsymbol{\chi})^{\frac{1}{2}}}{\sqrt{\overline{\boldsymbol{\mathcal{T}}\_{j}^{\mathrm{FSO}}}}} \Big|\_{\boldsymbol{a}, \boldsymbol{\beta}}^{-} \right) \tag{7}$$

where *<sup>P</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup>Γð Þ *<sup>α</sup>* <sup>Γ</sup>ð Þ *<sup>β</sup>* . The cumulative distributive function (CDF) of instantaneous SNR of FSO link is obtained by integrating the PDF of *γFSO <sup>j</sup>* and is given as

$$F\_{\mathcal{I}\_j^{\rm SO}}^{\rm GG}(\boldsymbol{\gamma}) = \int\_0^{\gamma\_j} f\_{\mathcal{I}\_j^{\rm SO}}^{\rm GG}(\boldsymbol{\gamma}) \, d\boldsymbol{\gamma} = P\_2 G\_{1,5}^{4,1} \left( \frac{(a\beta)^2 \boldsymbol{\gamma}}{16 \tilde{\boldsymbol{\gamma}}\_j^{\rm SO}} \Big|\_{\boldsymbol{Q}\_1} \right) \tag{8}$$

where *<sup>P</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*α*þ*β*�<sup>2</sup> *<sup>π</sup>*Γð Þ *<sup>α</sup>* <sup>Γ</sup>ð Þ *<sup>β</sup>* and *<sup>Q</sup>*<sup>1</sup> <sup>¼</sup> *<sup>α</sup>* <sup>2</sup> , *<sup>α</sup>*þ<sup>1</sup> <sup>2</sup> , *<sup>β</sup>* <sup>2</sup> , *<sup>β</sup>*þ<sup>1</sup> <sup>2</sup> , 0.

#### **2.2 Modeling of received mmW signals at different BSs**

The signal received by base station R1 from base station S, over the mmW link is given by

$$
\mathcal{Y}\_{\text{SR}\_1}^{mmW} = h\_{\text{SR}\_1} \mathfrak{x} + \mathfrak{n}\_{\text{SR}\_1} \tag{9}
$$

The decoded symbol at base station R1 is forwarded to base station R2 and the received signal at base station R2 is given as

$$
\mathcal{Y}\_{R\_1 R\_2}^{mmW} = h\_{R\_1 R\_2} \tilde{\mathbf{x}} + n\_{R\_1 R\_2} \tag{10}
$$

The signal received at base station R2 is decoded and forwarded to base station D and the signal received at base station D is given by

$$\mathcal{Y}\_{R\geq D}^{mm\,W} = h\_{R\geq D}\hat{\mathbf{x}} + \mathfrak{n}\_{R\geq D} \tag{11}$$

where *hSR*<sup>1</sup> , *hR*1*R*2*:* and *hR*2*<sup>D</sup>*, are S-R1, R1-R2, and R2-D fading channel coefficients respectively. The relation between instantaneous received SNR (*γmmW <sup>j</sup>* ) and average SNR (*γmmW <sup>j</sup>* ) of any given mmW link is given by

$$
\gamma\_j^{mmW} = \overline{\gamma}\_j^{mmW} \left| h\_j \right|^2 \tag{12}
$$

where hj is the mmW fading channel. The norm of the mmW fading channel is modeled as Nakagami-m distribution and its PDF is given by

$$f\_{\mathcal{I}\_j^{\text{new}}}(\boldsymbol{\chi}) = \left(\frac{m}{\mathcal{V}\_j^{\text{new}}}\right)^m \frac{\left(\boldsymbol{\chi}\right)^{m-1}}{\Gamma(m)} e^{-\frac{\overline{\mathcal{V}\_j^{\text{new}}}}{\mathcal{V}\_j^{\text{new}}}} \tag{13}$$

where m indicates fading severity of the channel, Γð Þ m is the standard Gamma function, and the CDF of *γmmW <sup>j</sup>* is obtained by integrating the PDF of *γmmW <sup>j</sup>* and is given by

*Triple-Hop Hybrid FSO/mmW Based Backhaul Communication System for Wireless Networks… DOI: http://dx.doi.org/10.5772/intechopen.104392*

$$F\_{\mathcal{I}\_j^{\text{unW}}}(\boldsymbol{\chi}) = \int\_0^\gamma f\_{\mathcal{I}\_j^{\text{unW}}}(\boldsymbol{\chi}) \, \mathrm{d}\boldsymbol{\chi} = \frac{1}{\Gamma(\mathbf{m})} \boldsymbol{\chi}\left(\boldsymbol{m}, \frac{\boldsymbol{\chi}\boldsymbol{m}}{\overline{\mathcal{I}\_j^{\text{unW}}}}\right) \tag{14}$$

where *γ*ð Þ a, x is the lower incomplete gamma function.

### **3. Statistical characteristics of the proposed system model**

In this section, the CDF and PDF for the proposed TH hybrid FSO/mmW system over Gamma-Gamma turbulence and Nakagami-m fading channel are derived.

#### **3.1 Derivation of CDF and PDF of TH system over G-G atmospheric channel**

The CDF of *γFSO TH* for TH FSO transmission over G-G turbulence can be derived as

$$\begin{split} F\_{\mathcal{I}\_{\mathcal{H}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) &= \mathbb{1} - \prod\_{j} \Big[ \mathbb{1} - F\_{\mathcal{I}\_{j}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) \Big] \\ &= F\_{\mathcal{I}\_{\mathcal{H}\_{1}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) + F\_{\mathcal{I}\_{\mathcal{H}\_{2}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) + F\_{\mathcal{I}\_{\mathcal{H}\_{2}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) - F\_{\mathcal{I}\_{\mathcal{H}\_{1}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) F\_{\mathcal{I}\_{\mathcal{H}\_{2}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) \\ &- F\_{\mathcal{I}\_{\mathcal{H}\_{1}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) F\_{\mathcal{I}\_{\mathcal{H}\_{2}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) - F\_{\mathcal{I}\_{\mathcal{H}\_{1}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) F\_{\mathcal{I}\_{\mathcal{H}\_{2}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) F\_{\mathcal{I}\_{\mathcal{H}\_{1}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) F\_{\mathcal{I}\_{\mathcal{H}\_{2}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) \end{split} \tag{15}$$

where, *FGG γFSO SR*1 ð Þ*<sup>γ</sup>* , *<sup>F</sup>GG γFSO R*1*R*2 ð Þ*<sup>γ</sup>* , *<sup>F</sup>GG γFSO R*2*D* ð Þ*γ* are CDF's of S-R1, R1-R2, and R2-D links respectively, and are given below.

$$F\_{\mathcal{I}\_{\text{SR}\_1}^{\text{FSO}}}^{\text{GG}}(\boldsymbol{\chi}) = P\_2 G\_{1,5}^{4,1} \left( \frac{(a\boldsymbol{\beta})^2 \boldsymbol{\chi}}{\mathbf{1} \mathbf{6} \overline{\gamma}\_{\text{SR}\_1}^{\text{FSO}}} \Big| \begin{matrix} \mathbf{1} \\ \mathbf{Q}\_1 \end{matrix} \right) \tag{16}$$

$$F\_{\mathcal{V}\_{\mathbb{R}\_1\mathbb{R}\_2}^{\mathrm{GG}}}^{\mathrm{GG}}(\boldsymbol{\gamma}) = P\_2 G\_{1,5}^{4,1} \left( \frac{(a\boldsymbol{\beta})^2 \boldsymbol{\gamma}}{16 \overline{\gamma}\_{R\_1 R\_2}^{\mathrm{FSO}}} \, \big|\, \begin{matrix} \mathbf{1} \\ \mathbf{Q}\_1 \end{matrix} \right) \tag{17}$$

$$F\_{\mathcal{V}\_{R\_2D}^{\rm GSO}}^{\rm GG}(\boldsymbol{\gamma}) = P\_2 G\_{1,5}^{4,1} \left( \frac{(a\beta)^2 \boldsymbol{\gamma}}{16 \overline{\gamma}\_{R\_2D}^{\rm FSO}} \Big| \begin{matrix} \mathbf{1} \\ \mathbf{Q}\_1 \end{matrix} \right) \tag{18}$$

The PDF of *γFSO TH* is obtained by differentiating *FGG γFSO TH* ð Þ*γ* with respect to *γ* and its expression is given by

*f GG γFSO TH* ð Þ¼ *γ f GG γFSO SR*1 ð Þþ *γ f GG γFSO R*1*R*2 ð Þþ *γ f GG γFSO R*2*D* ð Þ� *γ f GG γFSO SR*1 ð Þ*<sup>γ</sup> <sup>F</sup>GG γFSO R*1*R*2 ð Þ*γ* �*FGG γFSO SR*1 ð Þ*γ* f *GG γFSO R*1*R*2 ð Þ� *γ f GG γFSO R*1*R*2 ð Þ*<sup>γ</sup> <sup>F</sup>GG γFSO R*2*D* ð Þ� *<sup>γ</sup> <sup>F</sup>GG γFSO R*1*R*2 ð Þ*γ* f *GG γFSO R*2*D* ð Þ*γ* �*f GG γFSO SR*1 ð Þ*<sup>γ</sup> <sup>F</sup>GG γFSO R*2*D* ð Þ� *<sup>γ</sup> <sup>F</sup>GG γFSO SR*1 ð Þ*γ* f *GG γFSO R*2*D* ð Þþ *γ f GG γFSO SR*1 ð Þ*<sup>γ</sup> <sup>F</sup>GG γFSO R*1*R*2 ð Þ*<sup>γ</sup> <sup>F</sup>GG γFSO R*2*D* ð Þ*γ* <sup>þ</sup>*FGG γFSO SR*1 ð Þ*γ* f *GG γFSO R*1*R*2 ð Þ*<sup>γ</sup> <sup>F</sup>GG γFSO R*2*D* ð Þþ *<sup>γ</sup> <sup>F</sup>GG γFSO SR*1 ð Þ*<sup>γ</sup> <sup>F</sup>GG γFSO R*1*R*2 ð Þ*γ* f *GG γFSO R*2*D* ð Þ*γ* (19)

where *FGG γFSO SR*1 ð Þ*<sup>γ</sup>* , *<sup>F</sup>GG γFSO R*1*R*2 ð Þ*<sup>γ</sup>* , *<sup>F</sup>GG γFSO R*2*D* ð Þ*γ* are given by Eqs. (16)–(18) respectively. *f GG γFSO SR*1 ð Þ*γ* , *f GG γFSO R*1*R*2 ð Þ*γ* , *f GG γFSO R*2*D* ð Þ*γ* are PDF's of S-R1, R1-R2, and R2-D links respectively, and are given below.

$$f\_{\mathcal{I}\_{\text{SR}\_1}^{\text{FSO}}}^{\text{GG}}(\boldsymbol{\chi}) = P\_1(\boldsymbol{\chi})^{-1} G\_{0,2}^{2,0} \left( \frac{a \beta(\boldsymbol{\chi})^{\frac{1}{2}}}{\sqrt{\mathcal{I}\_{\text{SR}\_1}^{\text{FSO}}}} \Big|\_{\alpha, \beta}^{-} \right) \tag{20}$$

$$\left| f\_{\mathcal{T}\_{R\_1 R\_2}^{\rm SO}}^{\rm GG}(\boldsymbol{\chi}) = P\_1(\boldsymbol{\chi})^{-1} G\_{0,2}^{2,0} \left( \frac{a \theta(\boldsymbol{\chi})^{\frac{1}{2}}}{\sqrt{\widetilde{\mathcal{T}}\_{R\_1 R\_2}^{\rm SO}}} \boldsymbol{\chi}\_{a,\beta} \right) \right| \tag{21}$$

$$f\_{\mathcal{I}\_{\mathcal{R}\_1 \mathcal{D}}^{\rm GG}}^{\rm GG}(\boldsymbol{\chi}) = P\_1(\boldsymbol{\chi})^{-1} G\_{0,2}^{2,0} \left( \frac{a \beta(\boldsymbol{\chi})^\ddagger}{\sqrt{\widetilde{\mathcal{V}}\_{R \circ D}^{\rm FSO}}} \, \Big|\, a \, \boldsymbol{\beta} \right) \tag{22}$$

#### **3.2 Derivation of CDF and PDF of TH system over Nakagami-m fading channel**

The cumulative distribution function of *γmmW TH* for mmW transmission is given by

$$F\_{\mathcal{I}\_{TH}^{\text{mmW}}}(\boldsymbol{\chi}) = \mathbf{1} - \prod\_{j} \left[ \mathbf{1} - F\_{\mathcal{I}\_{j}^{\text{mmW}}}(\boldsymbol{\chi}) \right] \tag{23}$$

*FγmmW TH* ð Þ*<sup>γ</sup>* is the CDF of *<sup>γ</sup>mmW TH* over Nakagami-m fading and is obtained by using Eq. (14) as

$$F\_{\mathcal{I}\_{\rm TH}^{\rm{mW}}}(\boldsymbol{\gamma}) = \mathbf{1} - \left[ \left( \frac{\Gamma\left(m, \frac{\rm{rm}}{\rm{\rm T}\_{\rm R}}\right)}{\Gamma(m)} \right) \left( \frac{\Gamma\left(m, \frac{\rm{rm}}{\rm{\rm T}\_{\rm R} \rm{p}\_2}\right)}{\Gamma(m)} \right) \left( \frac{\Gamma\left(m, \frac{\rm{rm}}{\rm{\rm T}\_{\rm R} \rm{p}}\right)}{\Gamma(m)} \right) \right] \tag{24}$$

where Γ (a, x) is the upper incomplete gamma function. The PDF of *γmmW TH* is denoted by *f <sup>γ</sup>mmW TH* ð Þ*γ* , which is obtained by differentiating *FγmmW TH* ð Þ*γ* with respect to *γ* and its expression is given by

$$\begin{split} f\_{\boldsymbol{\gamma}\_{\rm TH}^{\rm mmW}}(\boldsymbol{\chi}) &= \left(\frac{m}{\overline{\boldsymbol{\gamma}}\_{\rm SR\_1}^{\rm mmW}}\right)^{m} \frac{(\boldsymbol{\chi})^{m-1}}{\Gamma(m)} e^{-\frac{m\boldsymbol{\gamma}}{\overline{\boldsymbol{\gamma}}\_{\rm SR\_1}^{\rm mmW}}} + \left(\frac{m}{\overline{\boldsymbol{\gamma}}\_{\rm R\_1\boldsymbol{R}\_2}^{\rm mmW}}\right)^{m} \frac{(\boldsymbol{\chi})^{m-1}}{\Gamma(m)} e^{-\frac{m\boldsymbol{\gamma}}{\overline{\boldsymbol{\gamma}}\_{\rm R\_1\boldsymbol{R}\_2}^{\rm mmW}}} \\ &+ \left(\frac{m}{\overline{\boldsymbol{\gamma}}\_{\rm R\_2\boldsymbol{D}}^{\rm mmW}}\right)^{m} \frac{(\boldsymbol{\chi})^{m-1}}{\Gamma(m)} e^{-\frac{m\boldsymbol{\gamma}}{\overline{\boldsymbol{\gamma}}\_{\rm D}^{\rm mmW}}} \end{split} \tag{25}$$

### **4. Performance analysis of the proposed TH hybrid system**

The performance of the proposed TH system is evaluated by using outage probability and average BER. In this section, the expressions for outage probability and average BER of the proposed TH hybrid FSO/mmW system between source and destination BSs over Nakagami-m and Gamma-Gamma turbulence channels are derived. The outage probability expression of the proposed TH system is given by

$$P\_{out}^{TH-GG} = F\_{\mathcal{I}\_{TH}^{SO} \text{GG}}\left(\mathcal{Y}\_{th}^{SO}\right) F\_{\mathcal{I}\_{TH}^{mmW}}\left(\mathcal{Y}\_{th}^{mmW}\right) \tag{26}$$

where *FγFSO TH GG <sup>γ</sup>FSO th* � � is obtained by replacing *<sup>γ</sup>* with *<sup>γ</sup>FSO th* in Eq. (15) and *FγmmW TH <sup>γ</sup>mmW th* � � is obtained by replacing *<sup>γ</sup>* with *<sup>γ</sup>mmW th* in Eq. (24).The transmitted data *Triple-Hop Hybrid FSO/mmW Based Backhaul Communication System for Wireless Networks… DOI: http://dx.doi.org/10.5772/intechopen.104392*

is mapped by using BPSK modulation and then transmitted through the FSO link or the mmW link and it is assumed that both the links operate at the same data rate. The BER for BPSK modulation as a function of the instantaneous SNR is given by (Usman M., Yang H. C., and Alouini M. S., 2014)

$$p(\mathbf{e}/\gamma) = \mathbf{0}.\mathbf{5}\mathbf{erfc}(\sqrt{\gamma})\tag{27}$$

The average BER during non-outage period can be calculated in terms of the average BER of individual FSO and mmW links, and is given as

$$
\overline{P}\_b^{TH-GG} = \frac{B\_{TH}^{GG} \left(\chi\_{th}^{FSO}\right) + \mathbf{F}\_{\chi\_{TH}^{NO}}^{GG} \left(\chi\_{th}^{FSO}\right) \mathbf{B}\_{TH}^{mmW} \left(\chi\_{th}^{mmW}\right)}{\mathbf{1} - P\_{out}^{TH-GG}} \tag{28}
$$

where, *PTH*�*GG out* is given by Eq. (26)and F*GG γFSO TH γFSO th* � � is given by Eq. (15). The average BER of FSO link (*BGG TH γFSO th* � �) and mmW link (*BmmW TH γmmW th* � �) are given below. The average BER of FSO link is obtained by averaging the conditional BER of BPSK signal over the probability density function of *γFSO TH* based on the condition *γFSO TH* >*γFSO th* and is given as,

$$B\_{\rm TH}^{\rm GG}(\chi\_{th}^{\rm FSO}) = \int\_{r\_{th}^{\rm FSO}}^{\infty} p(\mathbf{e}/\boldsymbol{\chi}) \mathbf{f}\_{r\_{\rm TH}^{\rm FSO}}^{\rm GG}(\boldsymbol{\chi}) d\boldsymbol{\chi} \tag{29}$$

where *<sup>p</sup>*ð Þ <sup>e</sup>*=<sup>γ</sup>* and f*GG γFSO TH* ð Þ*γ* are given by Eqs. (27) and (19) respectively. Similarly, the average BER of mmW link is obtained by averaging the conditional BER of BPSK signal over the probability density function of *γmmW TH* based on the condition *γmmW TH* >*γmmW th* and is given as,

$$B\_{TH}^{mmW} \left( \boldsymbol{\gamma}\_{th}^{mmW} \right) = \int\_{\boldsymbol{\gamma}\_{th}^{mmW}}^{\boldsymbol{\bullet}} \boldsymbol{p} (\mathbf{e}/\boldsymbol{\gamma}) \boldsymbol{\mathbf{f}}\_{\boldsymbol{\gamma}\_{TH}^{mmW}} (\boldsymbol{\gamma}) d\boldsymbol{\up} \tag{30}$$

where, *p*ð Þ e*=γ* and f *<sup>γ</sup>mmW TH* ð Þ*γ* are given by Eqs. (27) and (25) respectively. By substituting Eqs. (29) and (30) in Eq. (28), the average BER of the TH hybrid FSO/ mmW system without direct link between source and destination BSs can be obtained.

## **5. Results and discussion**

In this section, the performance of the proposed systems are compared by evaluating the analytical expressions which are derived for outage probability and average BER over weak and strong turbulent channel conditions. The procedure followed in the implementation of the proposed TH hybrid FSO/mmW backhaul communication system over weak and strong turbulent channels is shown in **Figure 2**. The FSO link is modeled for weak and strong turbulence conditions using Gamma-Gamma distribution. The typical values of α, β for strong turbulence are 2.064, 1.342 and for weak turbulence are 2.902, 2.51 respectively. Further, mmW channel is modeled by using Nakagami-m distribution with m = 5.

**Figure 2.**

*Implementation of the TH hybrid FSO/mmW system over weak and strong turbulence conditions.*

#### **5.1 Outage probability vs. SNR of TH system over G-G atmospheric channel**

The variation in the outage probability with respect to average SNR for FSO -TH and TH hybrid FSO/mmW systems at fixed threshold value of *γmmW th* <sup>¼</sup> *<sup>γ</sup>FSO th* ¼ 5 *dB* is shown in **Figure 3**. As can be seen from **Figure 3**, the TH hybrid FSO/mmW system has better outage performance when compared to FSO-TH system, particularly when a high quality mmW link γmmW av <sup>¼</sup> 10 dB is used. Even with a low quality mmW link *γmmW av* <sup>¼</sup> <sup>5</sup>*dB* , the TH hybrid system shows improvement over FSO-TH system. For instance, at 10 dB of average SNR, FSO-TH system achieves an outage probability of 8.6 � <sup>10</sup>�<sup>1</sup> , whereas TH hybrid FSO/mmW system achieves outage probabilities of 7.9 � <sup>10</sup>�<sup>1</sup> and 5.7 � <sup>10</sup>�<sup>2</sup> with low quality and high quality mmW links respectively. It is observed that from above results, TH hybrid system has low outage probability when compared to FSO-TH system and it is mainly due to adaptive transmission nature of TH hybrid system as it selects the transmission path based on threshold value.

The outage performance of the TH hybrid FSO/mmW system with respect to average SNR for different values of α, β with a high quality mmW link is shown in **Figure 4**. As can be seen from **Figure 4**, TH hybrid FSO/mmW system has better outage performance when compared to FSO-TH system. However, the outage performance deteriorates for strong turbulence conditions when compared to weak turbulence conditions, which is as expected. **Table 1** shows outage probability values of the FSO-TH and TH hybrid FSO/mmW system for different values of α, β with a high quality mmW link.

*Triple-Hop Hybrid FSO/mmW Based Backhaul Communication System for Wireless Networks… DOI: http://dx.doi.org/10.5772/intechopen.104392*

**Figure 3.** *Outage probability of the TH hybrid FSO/mmW system over G-G atmospheric channel.*

**Figure 4.**

*Outage probability of the TH hybrid FSO/mmW system with different values of α, β over G-G atmospheric channel.*

For instance, at 10 dB average SNR, the TH hybrid FSO/mmW system achieves outage probabilities of 5.1 <sup>10</sup><sup>2</sup> and 5.7 <sup>10</sup><sup>2</sup> , while FSO-TH system achieves outage probabilities of 7.7 <sup>10</sup><sup>1</sup> and 8.6 <sup>10</sup><sup>1</sup> , respectively, for weak and strong turbulence channel conditions.



For instance, at 20 dB average SNR, FSO-TH system achieves ABER of 2.3 <sup>10</sup><sup>2</sup> and 8.4 <sup>10</sup><sup>2</sup> , while TH hybrid FSO/mmW system achieves ABER of 4.3 <sup>10</sup><sup>4</sup> and 5.9 <sup>10</sup><sup>4</sup> , respectively for weak and strong turbulence conditions.
