*5.3.1. α* − *μ approximation to the sum of i.i.d.* G − G *variates*

Let *I* = ∑**<sup>N</sup>** *<sup>n</sup>*=<sup>1</sup> *In* be a sum of **N** i.i.d. G − G variates. We propose to approximate the PDF *fI*(*I*) and CDF *FI*(*I*) of I by the *α*-*μ* PDF and CDF given in [79]

$$f\_I(I) = \frac{\alpha \mu^{\mu} I^{a\mu - 1}}{\widehat{I}^{a\mu} \Gamma(\mu)} \exp\left(-\mu \frac{I^{a}}{\widehat{I}^{a}}\right) \tag{28}$$

$$F\_I(I) = 1 - \frac{\Gamma(\mu\_\prime \mu I^\alpha / \widehat{I}^\alpha)}{\Gamma(\mu)} \tag{29}$$

In (28), (29), *<sup>α</sup>*, *<sup>μ</sup>* <sup>&</sup>gt; 0 are the distribution parameters, *<sup>I</sup>* <sup>=</sup> **<sup>E</sup>**{*Iα*}1/*<sup>α</sup>* is a scale parameter and Γ(·, ·) is the incomplete gamma function [28, Eq. (8.350/2)]. The motivation behind this approximation is twofold: First, in a recently published work [3] it was shown that the gamma distribution can be used to approximate the sum of independent G − G variates. We feel that the use of a more generic distribution, which includes as special case the gamma distribution (in the *α*-*μ* case by setting *α* = 1), will result in a more accurate approximation. Second, as it will become evident, the estimation of the parameters of the resulting *α*-*μ* PDF requires the knowledge of the first, the second and the fourth moment of *I*, which can be easily evaluated given the moments of *In*. Therefore, the resulting PDF incorporates information regarding the mean, the variance *and* the kurtosis of *I*. In order to render (28) and (29) an accurate approximation, moment-based estimators for *α*, *μ* and *I* are used. These estimators are obtained as [79]

$$\frac{\Gamma^2(\mu + 1/\kappa)}{\Gamma(\mu)\Gamma(\mu + 2/\kappa) - \Gamma^2(\mu + 1/\kappa)} = \frac{\mathbb{E}^2\{I\}}{\mathbb{E}\{I^2\} - \mathbb{E}^2\{I\}}\tag{30}$$

$$\frac{\Gamma^2(\mu + 2/\mathfrak{a})}{\Gamma(\mu)\Gamma(\mu + 4/\mathfrak{a}) - \Gamma^2(\mu + 2/\mathfrak{a})} = \frac{\mathbb{E}^2\{I^2\}}{\mathbb{E}\{I^4\} - \mathbb{E}^2\{I^2\}}\tag{31}$$

$$\widehat{I} = \frac{\mu^{\frac{1}{\alpha}} \Gamma(\mu) \mathbb{E}\{I\}}{\Gamma\left(\mu + \frac{1}{\alpha}\right)}\tag{32}$$

The required moments **<sup>E</sup>**{*I*}, **<sup>E</sup>**{*I*2} and **<sup>E</sup>**{*I*4} can be evaluated using (13) and the multinomial identity as

$$\begin{aligned} \mathbb{E}\{I^{\mathsf{V}}\} &= \sum\_{j\_1=0}^{\mathsf{V}} \sum\_{j\_2=0}^{j\_1} \cdots \sum\_{j\_{\mathsf{N}-1}=0}^{j\_{\mathsf{N}-2}} \binom{\mathsf{v}}{j\_1} \binom{j\_1}{j\_2} \cdots \binom{j\_{\mathsf{N}-2}}{j\_{\mathsf{N}-1}} \\ &\times \mathbb{E}\{I\_1^{\mathsf{v}-j\_1}\} \mathbb{E}\{I\_2^{j\_1-j\_2}\} \cdots \mathbb{E}\{I\_{\mathsf{N}}^{j\_{\mathsf{N}-1}}\} \end{aligned} \tag{33}$$

where *ν* is positive integer. Using Maple, the command lines given in (34), can be utilized to obtain *<sup>α</sup>* and *<sup>μ</sup>* in a computationally efficient manner. In this case, *In* **<sup>E</sup>**{*In*}, *<sup>n</sup>* <sup>=</sup> 1, 2, 4. The parameter *I* can be finally obtained using (32). To demonstrate the accuracy of this analysis, Fig. 5 shows the exact and approximate CDFs of the sum of two and nine i.i.d. G − G variates with *I* = 1 for different values of parameters *k* and *m*. As it can be observed, in all considered test cases, the proposed approximation is highly accurate and practically indistinguishable from the exact CDF curves. A comparison of the proposed method with the one proposed in [3] reveals that our method performs equally well for both small and large values of **N**. Thus, a correcting factor, similar to the one introduced in [3] to obtain a sufficient approximation accuracy, is no longer required. Moreover, in [21], an approximate expression for the CDF of *I* in terms of Meijer-G functions [28, Eq. (9.301)] is provided. However, since the evaluation of Meijer-G functions can be sometimes laborious, (29) may be preferable to [21] in terms of computational complexity. Finally, our derived formulas are simpler than those presented in [14], since the latter are expressed as infinite series and require the computation of convolutional sums.

16 Will-be-set-by-IN-TECH

analytical expression for the statistical distribution of *I* is required. In the following it will be shown that when *In* are i.i.d G-G random variables, the distribution of *I* can be accurately

*<sup>I</sup>αμ*Γ(*μ*) exp

*FI*(*I*) = <sup>1</sup> <sup>−</sup> <sup>Γ</sup>(*μ*, *<sup>μ</sup>Iα*/*<sup>I</sup>α*)

In (28), (29), *<sup>α</sup>*, *<sup>μ</sup>* <sup>&</sup>gt; 0 are the distribution parameters, *<sup>I</sup>* <sup>=</sup> **<sup>E</sup>**{*Iα*}1/*<sup>α</sup>* is a scale parameter and Γ(·, ·) is the incomplete gamma function [28, Eq. (8.350/2)]. The motivation behind this approximation is twofold: First, in a recently published work [3] it was shown that the gamma distribution can be used to approximate the sum of independent G − G variates. We feel that the use of a more generic distribution, which includes as special case the gamma distribution (in the *α*-*μ* case by setting *α* = 1), will result in a more accurate approximation. Second, as it will become evident, the estimation of the parameters of the resulting *α*-*μ* PDF requires the knowledge of the first, the second and the fourth moment of *I*, which can be easily evaluated given the moments of *In*. Therefore, the resulting PDF incorporates information regarding the mean, the variance *and* the kurtosis of *I*. In order to render (28) and (29) an accurate approximation, moment-based estimators for *α*, *μ* and *I* are used. These estimators

*fI*(*I*) = *αμ<sup>μ</sup> <sup>I</sup>αμ*−<sup>1</sup>

Γ2(*μ* + 1/*α*)

Γ2(*μ* + 2/*α*)

*ν* ∑ *j*1=0

× **E**{*I*<sup>1</sup>

*j*1 ∑ *j*2=0 ···

*<sup>ν</sup>*−*j*<sup>1</sup> }**E**{*I*<sup>2</sup>

<sup>Γ</sup>(*μ*)Γ(*<sup>μ</sup>* <sup>+</sup> 2/*α*) <sup>−</sup> <sup>Γ</sup>2(*<sup>μ</sup>* <sup>+</sup> 1/*α*) <sup>=</sup> **<sup>E</sup>**2{*I*}

<sup>Γ</sup>(*μ*)Γ(*<sup>μ</sup>* <sup>+</sup> 4/*α*) <sup>−</sup> <sup>Γ</sup>2(*<sup>μ</sup>* <sup>+</sup> 2/*α*) <sup>=</sup> **<sup>E</sup>**2{*I*2}

Γ *μ* + <sup>1</sup> *α*

The required moments **<sup>E</sup>**{*I*}, **<sup>E</sup>**{*I*2} and **<sup>E</sup>**{*I*4} can be evaluated using (13) and the

*j***N**−<sup>2</sup> ∑ *j***N**−<sup>1</sup>=0

where *ν* is positive integer. Using Maple, the command lines given in (34), can be utilized to obtain *<sup>α</sup>* and *<sup>μ</sup>* in a computationally efficient manner. In this case, *In* **<sup>E</sup>**{*In*}, *<sup>n</sup>* <sup>=</sup> 1, 2, 4. The parameter *I* can be finally obtained using (32). To demonstrate the accuracy of this

*<sup>α</sup>* Γ(*μ*)**E**{*I*}

*ν j*1 *j*<sup>1</sup> *j*2 ···

*<sup>j</sup>*1−*j*<sup>2</sup> }··· **<sup>E</sup>**{*I***N***j***N**−<sup>1</sup> }

*<sup>I</sup>* <sup>=</sup> *<sup>μ</sup>* <sup>1</sup>

approximated with the so-called *α* − *μ* distribution [79].

*5.3.1. α* − *μ approximation to the sum of i.i.d.* G − G *variates*

and CDF *FI*(*I*) of I by the *α*-*μ* PDF and CDF given in [79]

*<sup>n</sup>*=<sup>1</sup> *In*. This integral is very difficult to be obtained in closed form since an

*<sup>n</sup>*=<sup>1</sup> *In* be a sum of **N** i.i.d. G − G variates. We propose to approximate the PDF *fI*(*I*)

 −*μ Iα Iα* 

<sup>Γ</sup>(*μ*) (29)

**<sup>E</sup>**{*I*2} − **<sup>E</sup>**2{*I*} (30)

**<sup>E</sup>**{*I*4} − **<sup>E</sup>**2{*I*2} (31)

(32)

*<sup>j</sup>***N**−<sup>2</sup> *j***N**−<sup>1</sup>

(28)

(33)

where *I* = ∑*<sup>N</sup>*

Let *I* = ∑**<sup>N</sup>**

are obtained as [79]

multinomial identity as

**E**{*I <sup>ν</sup>*} <sup>=</sup>

**Figure 5.** Exact and approximate CDF of the sum of **N** = 2 and **N** = 9 i.i.d G − G variates

In Fig. 6 the OP of the considered system is plotted as a function of the inverse normalized outage threshold *μ*/*μth* for *L* = 2km and *L* = 4km. The parameters *k* and *m* are obtained using (10) and (11) assuming *λ* = 1550nm, *C*<sup>2</sup> *<sup>n</sup>* <sup>=</sup> 1.7 <sup>×</sup> <sup>10</sup>−14m−2/3 and *<sup>D</sup>*/*<sup>L</sup>* <sup>→</sup> 0. Both numerically evaluated and computer simulation results are depicted. From the above mentioned plot, it is clear that the derived approximative expressions are highly accurate for every considered MIMO deployment and for all considered link distances.

To evaluate ABEP, the PDF of *I*, *fI*(*I*), at the combiner output, will be approximated by the PDF of a single channel given in (28) where the parameters *α* and *μ* are estimated as functions of *k* and *m*. Having obtained these parameters, the ABEP is easily obtained by substituting (28) into (21) and performing symbolic or numerical integration. In Fig. 7 the ABEP of the considered MIMO system is depicted as a function of the average electrical SNR, using the same parameters considered in the OP case. From the observation of Fig. 7, one can

**Figure 6.** Outage Probability of MIMO FSO systems employing EGC and operating over i.i.d G − G fading channels as a function of the inverse normalized outage threshold, (*λ* = 1550nm, *C*2 *<sup>n</sup>* <sup>=</sup> 1.7 <sup>×</sup> <sup>10</sup>−14m−2/3 and *<sup>D</sup>*/*<sup>L</sup>* <sup>→</sup> 0)

**Figure 7.** ABEP of MIMO FSO systems employing EGC and operating over i.i.d G − G fading channels as a function of the average electrical SNR, (*λ* = 1550nm, *C*<sup>2</sup> *<sup>n</sup>* <sup>=</sup> 1.7 <sup>×</sup> <sup>10</sup>−14m−2/3 and *<sup>D</sup>*/*<sup>L</sup>* <sup>→</sup> 0)

verify similar findings to that mentioned in Fig. 6 regarding the accuracy of the proposed approximation.

$$\begin{aligned} \text{XI} &:= \text{I}1^{\text{}} / (\text{I}2 - \text{I}1^{\text{}} \text{2}); \text{X2} := \text{I}2^{\text{}} / (\text{I}4 - \text{I}2^{\text{}} \text{2}); \\ \text{f1} &:= \text{X1} - (\text{GAMMA}(\text{mu} + 1/\text{alpha})) \text{2} / (\text{GAMMA}(\text{mu}) \* \text{GAMMA}(\text{mu} + 2/\text{alpha})) \\ &- (\text{GAMMA}(\text{mu} + 1/\text{alpha})) \text{2}) = 0; \\ \text{f2} &:= \text{X2} - (\text{GAMMA}(\text{mu} + 2/\text{alpha})) \text{2} / (\text{GAMMA}(\text{mu}) \* \text{GAMMA}(\text{mu} + 4/\text{alpha})) \\ &- (\text{GAMMA}(\text{mu} + 2/\text{alpha})) \text{2}) = 0; \\ \text{SOL} &:= \text{fsobve}(\{\text{f1}, \text{f2}\}, \{\text{alpha} \text{mu}\}, \{\text{alpha} = 0..1000\}); \end{aligned} \tag{34}$$
