1. Introduction

In last few years, the development of atmospheric optical communications (AOC) is getting a growing interest mainly due to their advantages over the classical radio communication technologies such as high transmission capacity and low cost characteristics, which are specially useful in new generation all-optical communication systems [1]. However, in spite of these advantages, the transmitted optical signal in AOC systems is affected by fluctuations in both the intensity and the phase due to the random inhomogeneities in the medium refractive index. These inhomogeneities lead to fade the received optical signal intensity (phenomenon knowns as scintillation). To model this randomly fading characteristic of the atmospheric channel under variable turbulence conditions, extensive research have been carried out by the scientific community. As a result of this research, different mathematical models for the probability density function (pdf) of the received irradiance have been proposed so far [2–8]. Among them, the Log-normal [3] and the Gamma-Gamma [4] models are the most employed in the literature. The first one (the Log-normal) was proposed to model weak irradiance fluctuations, whereas the second one (the Gamma-Gamma) was proposed to model a wider range of turbulence conditions. On another note, Barakat [9] and Jakerman [10] proposed a generalization of the K distribution applicable to atmospheric laser transmissions under weak scattering conditions. This Generalized-K distribution accurately fits to other existing distributions such as Nakagami-m or Rayleigh-Lognormal and has been commonly used in radar applications [11, 12].

Recently, as an alternative to previous models, the authors proposed a new generalized statistical model named ℳ or Málaga [7, 13, 14]. This new model was validated by comparing its pdf with previously published experimental data [7]. The ℳ model is applicable to unbounded optical wavefront transmission conditions and unifies a great part of the existing statistical models such as Rice-Nakagami [2], Gamma [4], shadowed-Rician [15], K [16], exponential [4] or Gamma-Rician among others, as demonstrated in [7]. The proposal of this generalized model has stimulated new research from different relevant groups specialized in the field [17–22], allowing to study the irradiance behavior under any possible turbulence condition.

In this paper, an in-depth analysis of the mathematical expressions that define the M statistical model is presented. This analysis results in the proposal of new analytical tractable expressions for the pdf of the ℳ model that include the Generalized-K distribution. These new expressions are simpler than the equations of the original ℳ model and provide an alternative viewpoint that improves the physical interpretation of this model.

2. System model

Figure 1 shows the laser transmission scheme assumed in the ℳ model. In this scheme, the small-scale fading characteristic of the atmospheric channel which is primarily due to diffractive effects [2], is modeled by three different signal components [7]. Thus, the received irradiance results in the contribution of a line-of-sight (LOS) field component, U_L, and two scattered optical field components due to the small-scale fluctuations. The first one of these two scatter components, denoted as $U_S^C$, is the quasi-forward optical signal scattered by the eddies on the propagation axis, which is assumed to be coupled to the LOS component. The second one, denoted as $U_C^G$, is the classical scattering optical field due to the energy scattered by the off-axis eddies, which is statistically independent of the other two components. Thus, the received optical field can be described as

 

Fig. 1 Laser beam propagation scheme under a ℳ -distributed free space optical link. The three components received are: first, the line-of-sight (LOS) term, U_L; second, the novel coupled-to-LOS scattering term, $U_S^C$; third, the classic scattering term related to the off- axis eddies, $U_S^G$.

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In this expression, χ and S are real random variables (RVs) representing the log-amplitude and phase fluctuations of the optical field, respectively. Both RVs model the large-scale fading characteristic of the channel which is due to refractive effects [2]. In addition, the LOS and the coupled-to-LOS scattering components are defined as $U_L=\sqrt{G}\sqrt{\Omega }\exp (j{\varphi }_L)$, and $U_S^C=\sqrt{G}\sqrt{{\xi }_c}\exp (j{\varphi }_C)$, respectively, where G is a RV following a Gamma distribution with E[G] = 1 (here E[·] denotes the expectation operator) or, equivalently, with $\sqrt{G}$ following a Nakagami-m distribution. The constants ϕ_L and ϕ_C are the deterministic phases of the LOS and the coupled-to-LOS scatter components. The LOS average optical power is represented by Ω = E[|U_L|²], whereas the average power of the total scatter components is denoted by $\xi =\mathrm{E}[|U_S^C{|}^2+|U_S^G{|}^2]={\xi }_c+{\xi }_g$. The relationship between the two scattering components is defined by the parameter ρ that represents the amount of scattering power coupled to the LOS component and ranges from 0 to 1, i.e. 0 ≤ ρ ≤ 1. Thus, the average power of coupled-to-LOS and the classic scattering components are given by ξ_c = ρξ and ξ_g = (1−ρ)ξ, respectively, and the total average optical power is given by E[I] = Ω +ξ.

From Eq. (1), the received irradiance can be expressed as

In this regard, the large-scale fluctuations RV, X, follows a log-normal distribution [5]. Nevertheless, in order to strongly improve the mathematical tractability, among different alternative models, such as the inverse-Gaussian [8] or the Gamma [2] models, the Gamma pdf is chosen to approximate the log-normal distribution,

Likewise, the pdf of the small-scale fluctuations RV, Y, is obtained from the compound of a Nakagami-m distribution, related to the slow fluctuation of the LOS component, and a Rayleigh random phasor associated to the independent scattered signal [7]. In this sense, the pdf of the small-scale fluctuation is given by

Hence, from the Eq. (3) and Eq. (4), and by applying the algorithm described in [7], the ℳ model pdf of the received irradiance I is expressed as

A generalized pdf expression, with β ∈ ℝ, was also obtained in [7, 14]. This expression is written as

3. The ℳ scintillation model and the Generalized-K statistical distribution

Now, let us assume an optical field with a fading amplitude RV following a Generalized-K distribution, as described in [12, Eq. (1)]. This distribution was developed to model shadowed fading scenarios based on an initial Nakagami-m distributed amplitude RV, $\sqrt{x}$, whose mean value is considered to be a Gamma RV [11]. Then, the pdf of the associated optical irradiance, x, can be expressed, from [12, Eq. (2)], as

In the following subsections, some mathematical manipulations are applied to the ℳ statistical models, which reveal that the pdf expression of the Generalized-K distribution can be also included.

3.1. ℳ distribution pdf (β ∈ ℕ)

In order to express the Eq. (5) as a function of the Generalized-K distribution, the Eq. (9) is here rewritten as follows:

Note that by identifying x = I, a = α − k, b = α +k − 1, c = α, d = k, and $B=\frac{\alpha \beta }{{\xi }_g\beta +{\Omega }^{\prime }}$, the Eq. (12) can be also expressed as

Let us define a new parameter, p, as

This parameter depends on the AF parameter of the Gamma distribution that models the slow fading affecting both the LOS and the coupled-to-LOS components, given by 1/β [23], and also on the ratio between the total LOS optical power, Ω′ and the independent scattering power, ξ_g. The interpretation of this new parameter makes sense when is included in Eq. (15), which is now written as

It is worth noting that this expression corresponds to the pdf of a discrete Binomial distribution [24] and, thus, the coefficient m_k, with 1 ≤ k ≤ β, describes the probability of having (k 1) successes in (β − 1) independent Bernoulli experiments with an individual success probability p. From the definition of the success probability p given in Eq. (16), a direct relationship between p and the coupling factor ρ is derived. In this sense, for high values of ρ (optical power coupled to the LOS component), p tends to 1, whereas in contrast, for lower ρ the probability p tend to 0. Then, the parameter p could be interpreted as the probability for the optical power to be coupled to the LOS component. Note that, with this setting, ${\displaystyle {\sum }_{k=1}^{\beta }{m}_k=1}$.

Then, the ℳ model pdf when β ∈ ℕ can be described as a mixture of Generalized-K distribution and a discrete Binomial distribution. Thus, Eq. (14) can be physically interpreted as the superposition of β Generalized-K sub-channels, K_G(I;α,k, ℐ_k), corresponding each one to a different physical optical path, as shown in Fig. 2(a). Every sub-channel is weighted with a Binomial distribution m_k which depends on the inherent parameters of the ℳ distribution, i.e. (α,β,ρ,Ω,ξ). This m_k indicates the probability that the optical signal travels through the k-th optical path. For each of these sub-channels, the large-scale fading characteristic associated to each sub-channel is established by α and, as expected, it is common for every Generalized-K terms. In contrast, the small-scale fluctuations depend on the k value, corresponding more severe conditions for lower order sub-channels, as can be derived from Eq. (11), for which the probability of the optical power to be coupled to the LOS component is also lower, as shown in Eq. (17).

 

Fig. 2 Physical models defined by Eqs. (14) and (20). The subplot (a) represents the division of the atmospheric optical channel in a finite number β ∈ ℕ of sub-channels, each of them weighted with the corresponding m_k. The generalized version achieved with β ∈ ℝ is shown in the subplot (b), where there exist infinite optical paths between the transmitter and the receiver, weighted with the $m_k^{(G)}$ parameter.

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An additional correspondence can be established between Eq. (10) and [4, Eq. (15)], i.e., between the amount of fading of the Generalized-K distribution and the total scintillation index in a Gamma-Gamma distribution. As indicated above, α represents the effective number of large-scale cells of the scattering process. Similarly, k can be seen now as a subset of the total effective number of small-scale cells (represented by β) that are affecting the optical signal which travels through the k-th sub-channel. Id est, each sub-channel k will be affected by k small-scale cells. The total small-scale contributions to scintillation will be obtained by including the contribution of all the k sub-channels (from 1 to β). Consequently, different optical sub-channels may be associated to a different turbulence condition. In this regard, we must recall that the condition required to consider that turbulence induced fading is uncorrelated at each of the optical sub-channels is that the spacing between such optical sub-channels should be greater than the fading correlation length, $d_0=\sqrt{\lambda L}$. Depending on the power distribution in each sub-channel, different statistical models already proposed in the literature can be modeled. So, for instance, as discussed in Section 4, for the Gamma-Gamma distribution m_k = 0 ∀k ≠ β and m_β = 1 so that it is clear that, in that case, k = β represents the total effective number of small-scale cells. But in the rest of cases, there will be a number of sub-channels (k ≤ β) contributing to the small-scale scintillation and representing the different optical paths between a transmitter and a receiver.

Finally, it must be noted that the amount of optical irradiance that travels through each sub-channel, ${\overline{I}}_k$, is different and can be obtained from the statistical average of I given by

3.2. Generalized ℳ distribution pdf (β ∈ ℝ)

Here, the same procedure described in the previous subsection is applied to the generalized pdf expression given in Eq. (7). In this case, from Eqs. (7) and (12) and applying the same Generalized-K parameters identification with $B=\frac{\alpha }{{\xi }_g}$, the following identity is derived

If the Pochhammer symbol equivalence with the Gamma function [25, Eq.(06.10.02.0001.01)] is considered, then, the Eq. (21) is rewritten as

Finally, by including the definition given in Eq. (16) of the p parameter, this coefficient is expressed as

Now, unlike the case with β ∈ ℕ, the resultant equation corresponds to the pdf of a discrete Negative Binomial distribution [24]. Thus, here, the number of total independent Bernoulli experiments is unknown and $m_k^{(G)}$, with 1 ≤ k < ∞, represents the probability of having (k 1) successes and β failures until the experiments sequence is stopped, with an individual success probability, p. Note that, although the Negative Binomial distribution is commonly defined for an integer number of failures, the definition can be also extended to real values. In this case, the Negative Binomial is also named Pólya distribution and models the dependence between contagious or non-independent Bernoulli experiments [26]. In practice, this statistic distribution is applied to many natural random processes such as spreading diseases, climatological probabilities, or to the initial assumptions in Generalized-K definitions presented in [9, 10]. Obviously, the sum of all the coefficients is ${\displaystyle {\sum }_{k=1}^{\infty }{m}_k^{(G)}=1}$.

Note that by using the same physical interpretation achieved when β ∈ ℕ, but for the case of β ∈ ℝ, the ℳ model, given by Eq. (20), assumes that the atmospheric optical channel is a superposition of an infinite number of sub-channels, each one modeled with a Generalized-K distribution, K_G(I;α,k, ℐ_k), as shown in Fig. 2(b). These sub-channels correspond to different optical paths associated to all the possible small-scale fluctuation sources. Every sub-channel is again weighted with a coefficient calculated, in this case, as a Negative Binomial distribution which depends on the set of parameters (α, β, ρ, Ω, ξ). Again, $m_k^{(G)}$ is the probability that certain portion of the optical signal travels through the k-th path, for which the probability of the optical power to be coupled to the LOS component is established by p. In the same way as for β ∈ ℕ, the shape parameters of the Generalized-K distributions are the large-scale fading parameter, α, for every sub-channels, the small-scale fading parameter, k, associated to lower fluctuation conditions as k increases, and the mean value of each term, given in this case by ℐ_k. Hence, the optical power distribution between sub-channels is given by the expression ${\overline{I}}_k=m_k^{(G)}{ℐ}_k$.

4. Results and interpretation

In this section, the analytical expressions proposed in this paper are analyzed, modifying the parameters of the ℳ distribution for various turbulence intensity conditions. First, the influence of these parameters on the Binomial and Negative Binomial distributions is studied. Then, the analysis of the ℳ model pdf is performed, modifying the large-scale and small-scale parameters, i.e. α and β, as well as the coupling parameter ρ.

4.1. Binomial and Negative Binomial distributions

As explained previously, in the new multichannel interpretation of the ℳ -model, the coefficients that follow the Binomial and Negative Binomial distributions, m and $m_k^{(G)}$, which are given by Eq. (17) and Eq. (23), respectively, define the number of relevant sub-channels of the model and the optical power distribution among them.

To clarify this idea, in Fig. 3, both coefficients are presented for different values of the ℳ distribution parameters β and ρ. In particular, Fig. 3(a) shows the Binomial distributed coefficients, m_k, for values of ρ ranging from 0 to 1, with 0.1 step size, whereas Fig. 3(b) depicts the Negative Binomial distributed coefficients, $m_k^{(G)}$, for values of ρ ranging from 0 to 0.9 (note that here, the case ρ = 1 is not plotted due to the analytical singularity when p = 1). A typical turbulence parameter β = 14 [7] and a normalized optical power, i.e. Ω +ξ = 1, are assumed in both figures.

 

Fig. 3 Generalized-K sub-channels coefficients for: (a) Binomial distributed m_k for ρ = 0: 0.1: 1 and (b) Negative Binomial distributed $m_k^{(G)}$ for ρ = 0 : 0.1 : 0.9. Typical values of large-scale and small-scale parameters and normalized optical power Ω +ξ = 1 have been assumed.

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From Fig. 3(a), it follows that, when the coupling factor ρ increases, the maximum values of m_k shift to sub-channels with higher k index. Note that, as explained before, in this case where β ∈ ℕ, the total number of sub-channels is precisely β. Thus, when the maximum coupling factor is achieved (ρ = 1), the weight is concentrated in a single Generalized-K sub-channel, the highest one (k = 14), i.e. m_k = 0∀k ≠ β and m_β = 1. It is worth noting that this special situation corresponds to the approximation of the ℳ model as a Gamma-Gamma distribution [7]. Therefore, for this case, the physical interpretation of Gamma-Gamma distribution parameters can be also applied to the ℳ distribution. As a consequence, there is a direct correspondence between the β parameter in the ℳ model and the small-scale β shape parameter in the Gamma-Gamma distribution, which is related to the effective number of small-scale turbulent eddies [4]. Likewise, it would be reasonable to extend this idea and to assume that the k parameter of each Generalized-K sub-channel is associated to the effective number of small-scale eddies modeled in each Generalized-K optical path. Note that, as occurs with the Gamma-Gamma distribution, the higher values of k, the lower scintillation intensity for the k-th sub-channel.

In Fig. 3(b), the results obtained from the general case of the ℳ model where β ∈ ℝ are shown. Here, the same behavior of the $m_k^{(G)}$ coefficients is observed. Again, as ρ increases, the maximum values of $m_k^{(G)}$ move to sub-channels with higher index k but, unlike in Fig. 3(a), the $m_k^{(G)}$ coefficients may range from 1 to infinity. However, despite this fact, as clearly shown in the figure, the values of $m_k^{(G)}$ for higher sub-channel always tends to zero, regardless of β and ρ values. Thus, a maximum coefficient, k_max, can be defined to transform the infinity sum of Eq. (20) to a finite one. In this way, a new approximated closed-form expression for the pdf of the optical channel in the generalized case when β ∈ ℝ is achieved. This new expression is specially valuable because it simplifies the conventional equation given by Eq. (7). It is written as

In order to describe in more detail the dependence of the Binomial and Negative Binomial distributions on the ℳ -model parameters, the Fig. 4 shows the expected sub-channel indexes as well as the associated variances as functions of the small-scale shape parameter β and the scattering coupling factor ρ. Note that if the sub-channel index k is considered as a RV following a Binomial or a Negative Binomial distribution with pdfs m_k or $m_k^{(G)}$, respectively, then, the average E[k] corresponds to the expected index of the most representative Generalized-K sub-channel, whereas the variance Var[k] provides information about the number of relevant sub-channels involved in the optical signal propagation; Figs. 4(a) and 4(b) show the values of E[k] and Var[k] for the possible discrete values of β, with β ∈ ℕ. From Fig. 4(a), it can be observed that when increasing β (corresponding to lower turbulence intensities), the expected sub-channel index is greater with an asymptotic trend to Ω′/ξ_g. Note that this growing is more noticeable for higher values of ρ. From Fig. 4(b), it is observed a similar behavior for the variance results. This means that in a scenario with finite number of β possible discrete sub-channels, under worse turbulence conditions, the optical power is concentrated only in the lowest order Generalized-K sub-channels, whereas under better turbulence conditions, the number of higher-order sub-channels involved is increased. Note that for the special case of ρ = 1 (in cyan colored lines), as expected, the variance is zero and the mean is linearly dependent on β. This fact is in coherence with the Fig. 3(a), where the optical signal is concentrated in a single Generalized-K channel with index k = β.

 

Fig. 4 Binomial (a–b) and Negative Binomial (c–d) mean and variance characteristics for the coefficients m_k and $m_k^{(G)}$, respectively, varying the small-scale shape parameter, β, and the coupling factor, ρ, considering normalized power Ω = 0.5 and ξ = 0.5.

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Likewise, Figs. 4(c) and 4(d) depict the values of E[k] and Var[k] with continuous curves associated to the real values of β. Here, the different discrete distribution that governs the sub-channel index k leads to a different behavior. In particular, the expected values in Fig. 4(c) are constant, regardless the β parameter, whereas the variance in Fig. 4(d) is a decreasing function with β. It can be deduced for this situation, where an infinite number of sub-channels may contribute, that for a fixed ρ and higher turbulence intensities (lower β), the number of sub-channels involved around the expected k is also higher. However, when increasing β, the number of relevant sub-channels tends to be concentrated with an asymptotic value of Ω′/ξ_g. As previously commented, the case of ρ = 1 has not been displayed since for β ∈ ℝ the amount of independent scattered component ξ_g is 0 and, consequently, the parameter p is1. This means that in the Bernoulli experiments related to the Negative Binomial description, the probability of success is 1 and then, every coefficient $m_k^{(G)}$ are forced to be zero since the probability of having β failures is zero. However, this inconsistency may be solved if the case of β ∈ ℕ is again observed. Note that in Fig. 3(a), when ρ = 1 the only active sub-channel is the corresponding to the highest limit (k = β). Then, it is reasonable to think that for the infinite sum associated to β ∈ ℝ, there is only one active channel with natural index k = ∞. However, as will be verified, the single Generalized-K sub-channel involved can be easily obtained when instead of using a natural index k, a real value β is employed.

Finally, it must be noted that, as expected, the optical irradiance distribution among subchannels follows the same behavior as the detailed in Fig. 3. Consequently, as explained before, it is desirable that most of the irradiance is allocated on higher order optical paths, which depends on the probability coefficients m_k and $m_k^{(G)}$ for β ∈ ℕ and β ∈ ℝ, respectively.

4.2. M distribution pdf (β ∈ ℕ)

Here, the proposed analytical expression for the ℳ model with β ∈ ℕ together with its physical interpretation are verified by comparing the analytical results of Eq. (14) to the numerical simulation results obtained by applying the histogram estimator described in [28] to random sequences generated according to the ℳ statistical distribution. In this sense, analytical and numerical results for the ℳ distribution pdf are depicted in Fig. 5 for different turbulence conditions (weak, moderate and strong) as a function of the normalized irradiance. Note that in this figure, the black solid line corresponds to the analytical results of the M distribution whereas the circles represent the numerical simulation results. As can be observed, both results (analytical and numerical) match perfectly in all cases. Moreover, to give a deeper explanation of these results, the individual Generalized-K pdfs that forms the sum of Eq. (14) are also included in Figs. 5(a)–5(c) with dash color lines. In particular, in Fig. 5(a), which corresponds to strong turbulence conditions with α = 2.1, β = 2 [7], and ρ = 0, both individual sub-channel pdfs are depicted. Note that, of the two, the lowest sub-channel (k = 1) is the dominant one, which makes sense being strong fluctuations. In Fig. 5(b), where moderate turbulence conditions are assumed, with α = 15, β = 10 [7] and ρ = 0.5, the number of sub-channels is increased. Note that, in this case, the 5 sub-channels shift the total pdf slightly to the right, which corresponds to the excitation of higher-order Generalized-K sub-channels with less severe fading. Results for weak turbulence conditions with α = 50, β = 14 [7] and ρ = 0.9 are shown in Fig. 5(c). Here, 14 optical paths form the analytical pdf presented with solid line. Of all of them, the sub-channels ranging from 7th to 10th are the most relevant, as can be also verified from Fig. 3(a). Finally, in Fig. 5(d), a comparison between the analytical results and numerical pdfs for the all the turbulence conditions with ρ = 1 is performed. Note that, assuming this coupling factor then ξ_g = 0 and the pdf of the M distribution is simplified as f_I(I) = K_G(I;α, β, Ω ′), where Ω′ is now the total power received.

 

Fig. 5 Analytical and numerical ℳ distribution pdfs for for β ∈ ℕ under strong (a), moderate (b) and weak (c) turbulence conditions, showing the associated Generalized-K sub-channel pdfs in dashed colored lines. The case of maximum coupling parameter ρ = 1 is considered in (d).

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From this figure, it follows that when α and β parameters increases, the irradiance probability is concentrated around the normalized mean value initially defined, i.e. I = 1. In the limit, when α → ∞ and β → ∞, the pdf tends to a Dirac delta function, corresponding to the ideal AWGN channel.

Note that the amount of circles in the numerical results increases for lower fading conditions, since the histogram estimator applies a fixed number of intervals on the normalized intensity random sequences generated, with decreasing variances.

4.3. Generalized ℳ distribution pdf (β ∈ ℝ)

Now, similarly to the previous subsection, the proposed generalized analytical expression for the ℳ model with β ∈ ℝ is verified. In this case, analytical results obtained from Eq. (24) are compared to numerical simulation results and shown in Fig. 6. In order to calculate the value of k_max, an error ε = 0.01 (i.e. an accuracy of the 99% in the ℳ pdf model) is assumed in all cases. Moreover, to show the accuracy of the proposed expression, real values for the large scale and small scale parameters (α and β) are employed. Fig. 6 depicts results for different turbulence conditions: strong, in Fig. 6(a), moderate, in Fig. 6(b) and weak in Fig. 6(c). Again, the black solid line represents the analytical results of the ℳ distribution, whereas the circles represent the numerical simulation results. As expected, from this figure, it follows that analytical and numerical results are virtually indistinguishable. Thus, this fact corroborates the validity of the proposed expression. Fig. 6 also shows that when the turbulence intensity is decreased, the number of contributing Generalized-K sub-channels is increased.

 

Fig. 6 Analytical and numerical ℳ distribution pdfs for β ∈ ℝ under strong (a), moderate (b) and weak (c) turbulence conditions, showing the associated Generalized-K sub-channel pdfs in dashed colored lines. The case of ρ = 0.9 is considered in (d) for the same previous turbulence scenarios, as well as two extremely weak turbulence intensity with ρ = 0.9 and ρ = 1 in blue and violet colors, respectively.

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In particular, for the strong turbulence conditions of Fig. 6(a), only k_max = 6 sub-channels contribute to the signal propagation. The pdfs of these sub-channels are displayed with colored dashed lines. When turbulence fluctuations are reduced from strong to moderate, in Fig. 6(b), or to weak scintillation conditions, in Fig. 6(c), the number of contributing sub-channels is increased to k_max = 10 and k_max = 38, respectively. In both cases, the lower-order sub-channels become less relevant. Finally, the comparison between analytical and numerical pdfs for different turbulence conditions is presented in Fig. 6(d). Here, due to the singularity for ρ = 1, the coupling parameter is adjusted to be 0.9 to apply Eq. (24). In this case, it is not obvious to obtain the pdf expression when ρ = 1 from the Eq. (20) but, for the sake of coherence with the case of β ∈ ℕ, it is reasonable to assume that now, the pdf can also be reduced to a single Generalized-K sub-channel, i.e. f_I(I) = K_G(I;α, β, Ω ′), with Ω 2032representing the whole received optical power. This fact is corroborated in magenta colored results in Fig. 6(d), where this analytical simplification is compared to simulation results for extremely low turbulence intensity parameters α = 100.5 and β = 50.6, with ρ = 1. As expected, the analytical an numerical pdf curves match perfectly. The behavior is also the same than for β ∈ ℕ, tending to the AWGN ideal non-fading channel when α → ∞ and β → ∞.

5. Concluding remarks

In this paper, new closed-form analytical expressions for the ℳ distribution which models the received optical irradiance in free space optical channels are proposed. These expressions are based on a superposition of Generalized-K distributions weighted by Binomial or Negative Binomial distributed coefficients. The validity of the proposed expressions has been corroborated by comparing them to numerical histogram simulation results, showing an optimum matching for every turbulence conditions. The proposed expressions offer two main advantages from previously reported expression. The first one is that the proposed new generalized pdf is simpler than the previous pdf, since it can be expressed as a finite sum, even in the case of β ∈ ℝ, thus leading to compact and easily computable closed-form analytical pdf expressions. The second one is that this new expressions allow to give a novel physical interpretation to the ℳ model in which the turbulence induced scintillation effects are decomposed in Generalized-K distributed sub-channels, each one characterized by the ℳ -model parameters, i.e. α, β and ρ. Moreover, in the analysis here performed, the probability for the optical signal traveling through a sub-channel has been calculated, as well as the optical power distribution among the contributing sub-channels. It is worth noting that the proposed new expressions together with their physical interpretation provide a very valuable tool for analyzing the effects of turbulence induced scintillation in free-space optical communication links under any turbulence conditions.