Performance analysis of FSO communications under LOS blockage

José M. Garrido-Balsells; F. Javier Lopez-Martinez; Miguel Castillo-Vázquez; Antonio Jurado-Navas; Antonio Puerta-Notario

doi:10.1364/OE.25.025278

1. Introduction

Free-space optical (FSO) communications are currently a solid alternative to radio communications in many applications including last mile broadband access, cellular backhaul, inter-building connections and high-speed links of next generation all-optical networks [1]. The main advantage of FSO systems lies in the possibility of exploiting the huge unregulated bandwidth available in optical frequencies that allows data rates similar to those of fiber optic systems [2]. However, in FSO links the transmitted signal is affected by various factors before arriving at the receiver that degrade the link capacity and limit the transmission rate. These include atmospheric loss, atmospheric turbulence induced fading, and blockage. First, the high attenuation introduced by fog, rain or pollution in FSO communications causes the practical distance between transmitter and receiver to be limited up to several kilometers. A detailed study of atmospheric loss can be found in [3]. Nevertheless, the most impairing atmospheric effect is produced by random changes of the medium refractive index along the propagation path. This effect known as atmospheric turbulence leads to the appearance of random fading intervals in the received optical irradiance. Extensive research has been performed by the scientific community over the years in the search of a statistical distribution capable to model these irradiance fluctuations under any type of turbulence conditions. As a result of this research, different mathematical models for the probability density function (PDF) of the received optical irradiance have been proposed so far [4–10]. Such models adapt to a particular range of turbulence intensities, i.e. the Log-normal [5] distribution was proposed to model weak turbulence conditions, whereas the Gamma-Gamma [6] model was proposed to describe the irradiance behavior under moderate to strong turbulence conditions. Due to their mathematical tractability, these two distributions have been widely used to model the turbulence induced fading of the FSO links. More recently, an alternative statistical distribution, called ℳ or Malaga, was proposed in [10]. This distribution unifies in a closed-form expression most of the statistical models for the irradiance fluctuations proposed in the literature, including the popular Gamma-Gamma distribution. The ℳ distribution is applicable to unbounded optical wavefront transmission under the full range of atmospheric turbulence intensities, as detailed in [10–12]. The main characteristic of this new model is the definition of three different optical components in the signal traveling through the atmosphere: a line-of-sight (LOS) component and two scattered components. The first scatter component is coupled to the LOS component whilst the second one is the classical independent non-LOS (NLOS) scattering component. In [13], a reformulation of the ℳ distribution was presented and new and simpler analytical expressions for the model were proposed based on a mixture of the known Generalized-K and discrete Binomial and Negative Binomial distributions. Note that the Generalized-K distribution was introduced by Barakat [14] and Jakerman [15] to describe atmospheric laser transmissions under weak scattering conditions, and also applied in radar applications [16, 17]. In addition to the aforementioned atmospheric factors, FSO links are also sensitive to blockage due to moving objects intersecting the laser beam propagation path [18]. These interruptions can be quite frequent [19], specially in urban environments where the probability of appearance of obstacles in the near-to-ground air is noticeable higher. In this sense, although natural obstructions such as those caused by small birds are more common, currently, with the advent of unmanned aerial vehicles (UAVs), artificial obstructions caused by drones for civilian and military applications are also possible. Note that drones could potentially not only block the laser beam, but also take advantage of this situation to eavesdropping the signal, which also impacts the performance from a physical-layer security perspective [20,21]. Moreover, these blockages are particularly harmful in the current and future communication services based on real-time transmissions and high data rates. Therefore, their effect cannot be neglected in practical systems. For this reason, the performance analysis of FSO communication systems suffering a complete link blockage was recently addressed in [22]. Here, assuming a Gamma-Gamma fading model for the irradiance fluctuations, it was concluded that the link blockage causes the appearance of an outage floor in the high-SNR regime which is coincident with the blockage probability. This means that the OP in the presence of blockage cannot be improved by increasing the transmission power. However, the Gamma-Gamma model used in [22] is not the most appropriate to analyze the effect of blockages since it does not allow to accurately consider real scenarios in which only a partial obstruction of the laser beam occurs.

To cover these cases, in this paper, we extend the analysis presented in [22] and examine the effect of partial blockage in the performance of FSO links affected by turbulence fading. Unlike in [22], the ℳ distribution is assumed here to model the atmospheric turbulence induced fading. Thus, by using this distribution, we can explicitly consider the partial blockage of the laser beam. In particular, we analyze a type of partial blockage in which the coherent component of the laser beam is obstructed. This case corresponds to the possibility of a small object temporarily obstructing the propagating path within the transverse coherence diameter [4], considering that those objects are small compared to the divergence of the laser beam which can block the transverse coherence diameter but not the received power through NLOS scattering. Hereinafter, we refer to this type of partial blockage as LOS blockage. As a result of this analysis, a novel closed-form expression for the outage probability in terms of the turbulence model parameters and the LOS blockage probability is derived. Further, when LOS blockage occurs, the OP does not reach an irreducible error in the high-SNR regime. Thus, we can calculate the power offset required to overcome the effect of blockage in a very simple form. Note that, since the Gamma-Gamma distribution is a particular case of the ℳ distribution, the results derived here reduce to those presented in [22] when the coupling parameter of the ℳ distribution is one.

The remainder of this paper is organized as follows. Section 2 describes the system model and the different types of link blockage. In section 3, the PDF and MGF of the fading model under LOS blockage is derived. Section 4 focuses on the link outage analysis. Finally, results and most important conclusions are presented in section 5.

2. System model and LOS blockage

In this work, we assume an on-off keying (OOK) intensity modulation with a direct detection (IM/DD) scheme, so that the received optical irradiance, I_rx, is the variable to be considered. Thus, the instant photocurrent at the detector output is given by

y (t) = R \cdot I_{r x} (t) + n (t)

where R is the detector responsivity, which here is assumed equal to 1, and n(t) is the noise at the receiver, which is modeled as additive white Gaussian noise (AWGN) with zero mean and variance

σ_{n}^{2}

[23]. For a point-to-point optical communication link, due to the multiplicative effect of the random fading induced by the atmospheric turbulence [4], the value of I_rx can be obtained from the product of the received irradiance in absence of atmospheric turbulence, I₀, and the normalized optical irradiance, I, i.e. I_rx = I₀I. Note that here, I₀ is a deterministic value which includes the atmospheric loss while I is a random variable (RV) with statistical average E[I] = 1 which includes the refractive and diffractive effects of the atmospheric turbulence cells. Then, the electrical signal-to-noise ratio (SNR) at the receiver can be expressed as

γ = \frac{{(R I_{0} I)}^{2}}{σ_{n}^{2}} = γ_{0} I^{2}

where the parameter γ₀ represents the received electrical SNR in the absence of atmospheric turbulence.

2.1. Turbulence fading model

In order to model the optical irradiance fluctuations caused by the atmospheric turbulence, the statistical ℳ distribution is here assumed, considering an on-axis point receiver. This distribution relies on the use of doubly stochastic theory of scintillation in which the large- and small-scale turbulence eddies are supposed to induce refractive and diffractive effects on the light beam [4]. Thus, the normalized received irradiance I is considered as the product of two independent random variables X and Y, which represent the irradiance fluctuations arising from large- and small-scale fluctuations, respectively, as follows

I = Y \cdot X = {| U_{L} + U_{S}^{C} + U_{S}^{G} |}^{2} exp (2 χ),

where X = exp(2χ) and

Y = {| U_{L} + U_{S}^{C} + U_{S}^{G} |}^{2}

. As detailed in [10, 13], the RV X follows a Gamma distribution, while Y has the form of a Rician-Shadowed distribution. In this scheme, the small-scale fading characteristic of the atmospheric channel is modeled by three different signal components: a main LOS component, denoted as U_L, and two additional scattering components

U_{S}^{C}

, and

U_{S}^{G}

, as shown in Fig. 1. The first scattering component,

U_{S}^{C}

, is coupled to the LOS component and it is generated by the on-axis cells, i.e. by the quasi-forward eddies on the propagation axis. This fact makes this scattered optical field to coherently contribute to the LOS term. The second scattering component,

U_{S}^{G}

, is the classical NLOS optical signal due to the energy scattered by the off-axis eddies which is statistically independent of the other two components. The ℳ distribution is characterized by 3 parameters: α, β and ρ. The parameter α is related to the effective number of large-scale eddies, the parameter β is related to the amount of fading (AF) introduced by the diffraction effects associated to the small-scale eddies, and the coupling parameter ρ represents the amount of scattering power coupled to the LOS component and ranges from 0 to 1, i.e. 0 ≤ ρ ≤ 1. Note that for ρ → 1 the ℳ distribution reduces to the Gamma-Gamma distribution with parameters α and β. Following the notation of [13], the average optical power of the LOS component is given by Ω = E [|U_L|²], while the average power contained in both scattering components is

ξ = E [{| U_{S}^{C} |}^{2} + {| U_{S}^{G} |}^{2}]

. In turn, the power of the coupled-to-LOS and classic scattering components can be expressed as a function of the coupling parameter, ρ, as ξ_c = ρξ and ξ_g = (1 − ρ)ξ, respectively, being ξ = ξ_c + ξ_g. Finally, the average optical power associated to the coherent contribution of the LOS and the coupled-to-LOS components is denoted as Ω′. It is worth noting that although the received optical field is described here as the summation of three signals, it can also be physically interpreted as a summation of two effective optical components: the coherent component, composed of LOS and coupled-to-LOS terms, with average power Ω′, and the incoherent component, with average power ξ_g. This alternative description will be used in the next section. Note that since we assume that the total average optical irradiance received is normalized, then E [I] = Ω′ + ξ_g = 1.

Fig. 1 ℳ-model laser beam propagation scheme. The three components received are: the line-of-sight (LOS) term, U_L, the coupled-to-LOS scattering term, $U_{S}^{C}$ and the classic scattering term related to the off-axis eddies, $U_{S}^{G}$ .

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According to the reformulation of the ℳ distribution [13], the PDF of the normalized irradiance I can be written in a compact form as a mixture of the Generalized-K distribution with a Binomial or Negative binomial discrete distribution as follows

f_{I} (I) = \sum_{k = 1}^{\tilde{k}} {\tilde{m}}_{k} K_{G} (I; α, k, {\tilde{μ}}_{k}),

where the set of parameters {k̃, m̃_k μ̃_k} depends on the ℳ distribution parameters {α, β, ρ}. As detailed in [13], Eq. (4) can be seen as the superposition of k̃ independent Generalized-K sub-channels, K_G(I; α, k, μ̃_k), corresponding each one to a different physical optical path. Every sub-channel is weighted by a Binomial distribution m̃_k which depends on the inherent parameters of the ℳ distribution and is given by

{\tilde{m}}_{k} = {\begin{array}{l} (\begin{matrix} β - 1 \\ k - 1 \end{matrix}) p^{k - 1} {(1 - p)}^{β - k} & β \in ℕ \\ \frac{Γ (k - 1 + β)}{Γ (k) Γ (β)} p^{k - 1} {(1 - p)}^{β} & β \in ℝ \end{array},

where the parameter p can be interpreted, as introduced in [13], as the success probability in a Bernoulli experiment in which a success event is considered when the signal corresponding to any Generalized-K sub-channel travels through the on-axis coherent components, and a failure event is defined when that signal takes part of the off-axis independent term. This parameter is obtained as

p = {[1 + {(\frac{1}{β} \frac{Ω^{'}}{ξ_{g}})}^{- 1}]}^{- 1} .

The value of k̃ = β when β ∈ ℕ and k̃ → ∞ when β ∈ ℝ. However, as detailed in [13], it is possible to find a k̃= k_max which assures that the difference between the truncated and the exact formulation is bounded by an specified error tolerance parameter ∊, i.e. for k ≥ k_max the cumulative distribution function (CDF) of the corresponding m̃_k distribution is higher than (1−∊). Note also that, as explained in [13], the lower sub-channel orders are associated to more adverse turbulence conditions. The average optical irradiance for each Generalized-K sub-channel, μ̃_k, is expressed as

{\tilde{μ}}_{k} = {\begin{array}{l} \frac{k}{β} (ξ_{g} β + Ω^{'}) & β \in ℕ \\ k ξ_{g} & β \in ℝ \end{array},

and, finally, the PDF of the irradiance traveling through the k-th sub-channel is given by the generalized-K distribution as

K_{G} (I; α, k, ℐ_{k}) = \frac{2 B^{(α + k) / 2}}{Γ (α) Γ (k)} I^{(α + k) / 2 - 1} K_{α - k} (2 \sqrt{B \cdot I})

where B = αk/ℐ_k, with ℐ_k = E[I] being the average optical irradiance through the k-th sub-channel, and K_ν(·) is the modified Bessel function of the second kind. It must be noted that the generalized-K distribution is characterized by two shape parameters, i.e. α and k, that can be modified to describe different fading and shadowing scenarios [16]. As verified in [13] the higher the sub-channel order k is, the lower the amount of fading parameter (AF), thus implying that those sub-channels with a higher order are better for transmission. Conversely, a lower order is related to those sub-channels with worse atmospheric conditions and thus, with higher AF values.

2.2. LOS blockage

In an optical link, the width of the light beam produced by a transmitter laser expands as the propagation distance increases, as shown schematically in Fig. 2(a) in blue color. For horizontal FSO transmissions, a good approximation is to consider a Gaussian profile for the beam intensity [24], so that in the absence of atmospheric turbulence, the radius of this beam, at a distance z from the laser source, is given by [4]

W (z) = W_{0} \sqrt{{(1 - \frac{z}{F_{0}})}^{2} + {(\frac{2 z}{k W_{0}^{2}})}^{2}},

where W₀ is the beam radius at the transmitter, i.e. at z = 0, F₀ is the phase front radius of curvature at the transmitter output, k = 2π/λ is the wave number and λ is the wavelength. From Eq. (9), it follows that, during the designing of a FSO link, it is possible to control the beamwidth produced at a certain distance by adjusting properly the laser parameters W₀ and F₀. For example, assuming a collimated beam (i.e. F₀ → ∞), laser divergence can be reduced increasing the value of the initial radius W₀ by means of a beam expander.

Fig. 2 (a) Propagation model of a partially coherent Gaussian laser beam: W₀ is the initial beam radius, D_b = 2W_e is the beam diameter and D_c = 2ρ₀ is the transverse coherence diameter, both at z = L. (b) Laser beam radius, W_e, and transverse coherence radius, ρ₀, as a function of the propagation length, L, for moderate and strong turbulence conditions.

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However, in the presence of atmospheric turbulence, Eq. (9) is no longer valid and the beam spreading increases even more. In this case, a new effective beam radius W_e, greater than W, is defined [4] as

W_{e} (z) = W (z) \sqrt{1 + 1.625 σ_{R}^{12 / 5} Λ},

where W(z) is the beam radius in the absence of turbulence, given by Eq. (9),

σ_{R}^{2}

is the Rytov variance and Λ = 2z/kW². Note that the value of the Rytov variance is obtained with the expression

σ_{R}^{2} = 1.23 C_{n}^{2} k^{7 / 6} z^{11 / 6}

, being

C_{n}^{2}

the refraction index structure parameter, which is directly related to the turbulence strength [4].

In addition to the increase in the laser beam width, the atmospheric turbulence also destroys the spatial coherence of a laser beam as it propagates through the atmosphere. In particular, the transversal coherence radius ρ₀, which defines the coherent component of the beam, is reduced as the distance and the turbulence strength increase. This reduction is schematically depicted in Fig. 2(a) where the coherent component of the beam is represented in red. As reported in [4], assuming a plane wave (i.e. far from the transmitter), the spatial coherence radius at a distance z is given by

ρ_{0} (z) = {(1.46 C_{n}^{2} k^{2} z)}^{- 3 / 5}

where l₀ << ρ₀ << L₀, and l₀ and L₀ are the inner and outer scales of the turbulence, respectively.

Consider now an obstacle of diameter D between the transmitter and the receiver. Depending on the size of this obstacle compared to the laser beam diameter D_b = 2W_e and to the coherent beam component diameter D_c = 2ρ₀, two types of blockages can be occur. On the one hand, when D ≥ D_b, a total link blockage is produced and no signal is detected at the receiver. The effect of this type of total blockage on the FSO link performance is examined in [22]. On the other hand, when D < D_b, a partial link blockage is produced and, unlike in the previous case, one part of the signal still arrives at the receiver through the non-coherent component. In the case of partial blockage in which D = D_c, only the coherent component of the laser beam is blocked and, as mentioned above, we refer to this type of partial blockage to as LOS blockage. Note that this is the statistically more relevant case since the rest of partial blockage situations can be statistically analyzed as particular cases of the ℳ distribution or the ℳ distribution under LOS blockage considering an additional attenuation factor.

In order to illustrate the evolution of W_e and ρ₀ with the propagating length and to obtain realistic values of D_b and D_c, we have depicted in Fig. 2(b) the values of Eqs. (10) and (11) for moderate ( $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ ) and strong ( $C_{n}^{2} = 0.5 \cdot 10^{- 13} m^{- 2 / 3}$ ) turbulence intensities, assuming a wavelength λ = 1550 nm. In addition, two different transmitter beam radius have been applied (W₀ = 1 cm and W₀ = 2 cm). From this figure, it is observed that for moderate turbulence conditions and a laser with W₀ = 1 cm, at a distance L = 1600 m, an obstacle with D = 16 cm will produce a total blockage. Likewise, a smaller obstacle with D = 6 cm will cause a LOS blockage. For strong turbulence, at a half distance (L = 800 m), the obstacle diameters would be of D = 9 cm and D = 3 cm for total and LOS blockage cases, respectively. From these results, it should be noted that both total and LOS blockage are feasible considering real obstruction sizes such as the ones produced by small birds.

Finally, since the spatial coherence diameter, D_c decreases as distance increases, the probability of occurrence of LOS blockage, P_b, grows for large propagation lengths. This probability is defined as the obstruction probability of the transverse coherence area along the propagation path. The value of P_b depends on different factors such as the laser parameters, birds density and size, vegetation, etc. Some practical values of the probability of total blockage based on measurements are presented in [19]. Throughout this paper, we will analyze in depth the effect of LOS blockage in FSO links for a given blockage probability, P_b.

3. PDF and MGF of the turbulence fading model under LOS blockage

In this section, we derive de PDF and the MGF of the ℳ distribution considering the effect of the LOS blockage. These functions will be used in next section to perform an outage analysis.

3.1. PDF of the ℳ distribution under LOS blockage

Here, we modify the turbulence fading model described in section 2 in order to include the effect of the LOS blockage. Then, the PDF of the combined model with both effects the turbulence and LOS blockage is derived. It is worth noting that the inclusion of the LOS blockage on the ℳ distribution is possible due to its inherent physical interpretation, reported in [10]. According to this interpretation, we can assume that LOS (U_L) and coupled-to-LOS $U_{S}^{C}$ ) optical terms contain the energy coming from the coherent component of the laser beam and, similarly, we can also assume that the scattered term $U_{S}^{G}$ ) contains the optical energy coming from the area outside the coherent component of the laser beam. Thus, under these assumptions, the LOS blockage effect can be easily included in the ℳ distribution by adding a discrete RV b in Eq. (3) as follows:

I = {| b (U_{L} + U_{S}^{C}) + U_{S}^{G} |}^{2} exp (2 χ) .

Note that the RV b can take two possible values, b = 0 or b = 1, depending on whether a blockage of LOS components is considered or not, respectively. As previously discussed, the blockage probability (i.e. the probability of b = 0) depends on the specific configuration of the optical link.

With this change in Eq. (3), when a LOS blockage is produced, the combined power of the LOS and couple-to-LOS terms is forced to zero, i.e. Ω′ = 0, and then the normalized received irradiance, I, will only contain the power of the independent scattered term ξ_g. As a consequence, the Eqs. (5), (6), and (7) which define the weight coefficients m̃_k the p parameter and the average optical powers μ̃_k, respectively, are now simplified. More precisely, the parameter p, which is related to the probability of the optical signal to travel coupled to the LOS term within the coherence transverse disc, is now equal to zero, since

p = {[1 + {(\frac{1}{β} \frac{Ω^{'}}{ξ_{g}})}^{- 1}]}^{- 1} = \frac{Ω^{'}}{Ω^{'} + β ξ_{g}} = 0 .

Further, after substituting p = 0 into Eq. (5), the mixture weight coefficients given by m̃_k take now the values

{\tilde{m}}_{k} = {\begin{array}{l} 1 & k = 1 \\ 0 & k \neq 1 \end{array},

what indicates that, regardless of the value of β, the only active Generalized-K sub-channel is the first-order one, in which the transmitted optical power is μ̃₁ = ξ_g. Note that this result is in perfect agreement with the blockage scenario here described, and with the Generalized-K distribution behavior, which models a non-LOS Rayleigh fading channel when k = 1 [13, 16]. Thus, considering all these simplifications, when a LOS blockage is produced, then f_I(I) = K_G(I; α, 1, ξ_g). Therefore, if a non-zero P_b is assumed, the modified PDF of the normalized received optical irradiance I can be expressed as

f_{I, b} (I) = P_{b} f_{I} (I | b = 0) + (1 - P_{b}) f_{I} (I | b = 1),

where f_I(I|b = 0) stands for the PDF of the received irradiance under blockage and f_I(I|b = 1) stands for the general case of no blockage, given by the PDF of Eq. (4). Substituting the values of both functions, we obtain the PDF of the ℳ distribution under LOS blockage as

f_{I, b} (I) = P_{b} K_{G} (I; α, 1, ξ_{g}) + (1 - P_{b}) \sum_{k = 1}^{\tilde{k}} {\tilde{m}}_{k} K_{G} (I; α, k, {\tilde{μ}}_{k}) .

From Eq. (16) it follows that the independent scattered component power, ξ_g, is transmitted through the first-order sub-channel when LOS blockage is considered, whereas under a non-blockage scenario this power is split among all the active Generalized-K sub-channels, with probabilities controlled by the discrete mixture coefficients m̃_k.

It should be pointed out here that possible contributions to the independent scattered signal term produced by the refractive and diffractive phenomena induced by the obstacle are not considered in Eq. (16) since, as described in [25–27], those contributions to the propagating path oriented in the receiver direction can be considered negligible. As a consequence, we assume that the independent scattered term $U_{S}^{G}$ is not increased due to the blockage. Note that this can be regarded as a worst case scenario, since any additional contribution to $U_{S}^{G}$ component due to the previous phenomena would provide higher received power and better link performances. Finally, we also should remark that for the sake of clarity, we have not either considered misalignment effects between the transmitter and the receiver. However, these effects could be included by using the ℳ distribution with pointing errors reported in [28,29].

3.2. MGF of the ℳ distribution under LOS blockage

According to the definition of the moment generating function given by [17], the MGF of the ℳ distribution can be written as

M_{I} (s) ≜ \int_{0}^{\infty} exp (- s I) f_{I} (I) d I,

where f_I(I) is the PDF given by Eq. (4). This function can be also expressed by substituting Eq. (4) into Eq. (17) as follows

M_{I} (s) = \sum_{k = 1}^{\tilde{k}} {\tilde{m}}_{k} M_{I, Kg}^{(k)} (s),

where

M_{I, Kg}^{(k)} (s)

represents the MGF of the k-th sub-channel Generalized-K distribution which can be calculated in the same way as in Eq. (4) in [17], as

M_{I, Kg}^{(k)} (s) = {(\frac{α k}{\tilde{μ_{k}} s})}^{\frac{α + k - 1}{2}} exp (\frac{α k}{2 \tilde{μ_{k}} s}) W_{- \frac{α + k - 1}{2}, \frac{α - k}{2}} (\frac{α k}{\tilde{μ_{k}} s}),

being W_v,μ (·) the Whittaker function, Eq. (9.220) in [30], which is defined through the Tricomi confluent hypergeometric function, Eq. (07.45.02.0001.01) in [31], U(a, b, z), as

W_{v, μ} (z) = z^{μ + \frac{1}{2}} exp (- \frac{z}{2}) U (μ - v + \frac{1}{2}, 2 μ + 1, z) .

Then, after some analytical manipulation, the MGF of the Generalized-K distribution can be written as

M_{I, Kg}^{(k)} (s) = {(\frac{α k}{\tilde{μ_{k}} s})}^{α} U (α, α - k + 1, \frac{α k}{\tilde{μ_{k}} s}) .

Note that, by definition, the use of the Tricomi confluent hypergeometric function forces the second parameter not to be integer, which means that the large-scale parameter α must be noninteger, forcing the more general requirement of real for α. This assumption is applied here to follow a convenient analytical study but the results obtained are also valid and applicable to scenarios with α ∈ ℕ. In this sense, applying the primary definition of U(a, b, z) given in Eq. (07.33.02.0001.01) in [31],

\begin{array}{l} M_{I, Kg}^{(k)} (s) = & \frac{Γ (k - α)}{Γ (k)} {(\frac{α k}{\tilde{μ_{k}} s})}^{α} {{}_{1}F}_{1} (α, α - k + 1, \frac{α k}{\tilde{μ_{k}} s}) + \\ + \frac{Γ (α - k)}{Γ (α)} {(\frac{α k}{\tilde{μ_{k}} s})}^{k} {{}_{1}F}_{1} (k, k - α + 1, \frac{α k}{\tilde{μ_{k}} s}) . \end{array}

where ₁F₁() is the Kummer confluent hypergeometric function of the first kind.

Finally, the MGF of the ℳ distribution with blockage, given in Eq. (16), can be calculated as

M_{I, b} (s) = P_{b} M_{I} (s | b = 0) + (1 - P_{b}) M_{I} (s | b = 1),

where the ℳ distribution MGF under LOS blockage is

M_{I} (s | b = 0) = M_{I, Kg}^{(1)} (s)

and the term corresponding to the generalized no blockage case is

M_{I, b} (s | b = 1) = \sum_{k = 1}^{\tilde{k}} {\tilde{m}}_{k} M_{I, Kg}^{(k)} (s) .

4. Outage performance

Due to the slow-fading characteristic of the free-space optical channel, an appropriate metric for the link performance is the outage probability. Thus, in this section, we derive a novel closed-form general expression for the outage probability in FSO ℳ-turbulence links under LOS blockage. In addition, a further asymptotic analysis in the high-SNR regime allows to deduce a simpler approximate expression for the outage probability.

4.1. General analysis

The outage probability, denoted as P_out, is defined as the probability that the instantaneous SNR, given by γ, falls below a specified threshold, denoted as γ_th, which represents a protection value of the SNR above which the quality of the channel is satisfactory. Thus, P_out can be expressed as P_out = Pr {γ < γ_th}, where Pr{·} denotes probability, or also in terms of the optical irradiance, using the Eq. (2), as

P_{out} = Pr {I < \sqrt{\frac{γ_{t h}}{γ_{0}}}} = Pr {I < \frac{1}{\sqrt{γ_{n}}}},

where, as defined in Section 2 (Eq. 2), γ₀ is the received electrical SNR in absence of turbulence and γ_n = γ₀/γ_th is the normalized received electrical SNR in absence of atmospheric turbulence.

According to the above definition, P_out can be obtained from the CDF of the normalized received irradiance under LOS blockage, F_I,b(I), since

P_{out} = \int_{0}^{\frac{1}{\sqrt{γ_{n}}}} f_{I, b} (I) d I = F_{I, b} (\frac{1}{\sqrt{γ_{n}}}),

and, in turn, the ℳ distribution CDF can be defined from Eq. (16), as

F_{I, b} (I) = P_{b} F_{I, Kg} (I; α, 1, ξ_{g}) + (1 - P_{b}) \sum_{k = 1}^{\tilde{k}} {\tilde{m}}_{k} F_{I, Kg} (I; α, k, {\tilde{μ}}_{k}),

where F_I,Kg(I; α, k, μ̃_k) is the CDF associated to the k-th Generalized-K sub-channel. This function can be calculated by applying the equivalence between the Meijer-G and the Bessel K functions, Eq. (03.04.26.0009.01) in [31], to the integral of the Generalized-K PDF defined in Eq. (8), leading to the following expression

F_{I, Kg} (I; α, k, ℐ_{k}) = \frac{1}{Γ (α) Γ (k)} {(B \cdot I)}^{\frac{α + k}{2}} G_{1, 3}^{2, 1} (\begin{array}{l} B \cdot I | & \begin{matrix} 1 - \frac{α + k}{2} \\ \frac{α - k}{2}, \frac{α - k}{2}, \frac{α + k}{2} \end{matrix} \end{array}),

where again B = αk/ℐ_k, Γ(·) is the Gamma function and

G_{p, q}^{m, n} (\cdot)

is the Meijer-G function. Note that this expression is very similar to the Eq. (11) obtained in [8] for the Gamma-Gamma distribution.

Now, substituting $I = γ_{n}^{- 1 / 2}$ into Eq. (27), the outage probability under LOS blockage is derived as

P_{out} = P_{b} F_{I, Kg} (γ_{n}^{- 1 / 2}; α, 1, ξ_{g}) + (1 - P_{b}) \sum_{k = 1}^{\tilde{k}} {\tilde{m}}_{k} F_{I, Kg} (γ_{n}^{- 1 / 2}; α, k, {\tilde{μ}}_{k}) .

Note that it is also possible to more compactly express P_out as the combination of the outage probabilities associated to the signal propagation through each k-th Generalized-K sub-channel,

P_{out, Kg}^{(k)}

, governed by the mixture coefficients of the Binomial or Negative Binomial distributions, m̃_k, as

P_{out} = P_{b} P_{out, Kg}^{(1)} + (1 - P_{b}) \sum_{k = 1}^{\tilde{k}} {\tilde{m}}_{k} P_{out, Kg}^{(k)} .

It is worth noting that the derived closed-form expressions given by Eq. (29) and Eq. (30) are general and hold for any value of the SNR. However, taking advantage of the asymptotic behavior of the outage probability in the high-SNR regime [32], a simpler expression to estimate the performance of a FSO link under LOS blockage can be achieved. To this end, a MGF-based approach, as in [32,33], is followed below.

4.2. Asymptotic analysis

Assuming that the channel irradiance PDF is suitable to be expanded through Taylor series at I = 0 (Maclaurin series), we can apply the procedure detailed in [32] to approximate the outage probability for large enough values of γ₀ by the following expression

P_{out} \approx \frac{a}{D_{M}} {(\frac{γ_{th}}{γ_{0}})}^{D_{M} / 2} = \frac{a}{D_{M}} γ_{n}^{- D_{M} / 2},

where D_M is the diversity order. Note that the dependece here on D_M/2 is due to the cuadratic relation between the electrical SNR γ and the RV I inherent to the IM/DD system considered, i.e. I = (γ/γ₀)^1/2, from Eq. (2). The parameter a is the diversity gain, obtained from the first non-zero derivative of the ℳ distribution PDF at I = 0 [32],

a = \frac{f_{I}^{(t)} (0)}{t!},

where t = D_M − 1. When assuming the considerations described in [32], there is a direct relation between the behavior of the irradiance PDF at I = 0 and the decaying order of the corresponding MGF at high values of s. In this sense, for s → ∞, the MGF can be expressed as M_I,b(s) = b_M|s|^−D_M + o (|s|^−D_M), with b_M = a · Γ(D_M) and o(x) being a polynomial function whose limit verifies that lim_x→0o(x)/x = 0. Then, the MGF verifies also that

lim_{s \to \infty} s^{D_{M}} M_{I, b} (s) = b_{M} = a \cdot Γ (D_{M}) .

In order to obtain the diversity order of the ℳ distribution, let us first calculate the diversity order of the k-th Generalized-K sub-channel, denoted as D_k, that must verify that ${lim}_{s \to \infty} s^{D_{k}} M_{I, Kg}^{(k)} (s) = b_{k}$ . From Eq. (22), and applying that ₁F₁(·, ·, 0) = 1,

\begin{array}{l} lim_{s \to \infty} s^{D_{k}} M_{I, Kg}^{(k)} (s) & = \frac{Γ (k - α)}{Γ (k)} {(\frac{α k}{\tilde{μ_{k}} s})}^{α} lim_{s \to \infty} \frac{s^{D_{k}}}{s^{α}} + \\ + \frac{Γ (α - k)}{Γ (α)} {(\frac{α k}{\tilde{μ_{k}} s})}^{k} lim_{s \to \infty} \frac{s^{D_{k}}}{s^{k}}, \end{array}

in which the first term leads to a finite non-zero value only for D_k = α, whereas the second term, for D_k = k. So, the diversity order for the k-th sub-channel is given by

D_{k} = min {α, k} .

The parameter b_k is then

b_{k} = {\begin{array}{l} \frac{Γ (k - α)}{Γ (k)} \frac{α k}{\tilde{μ_{k}}} & for D_{k} = α if α < k \\ \frac{Γ (α - k)}{Γ (α)} \frac{α k}{\tilde{μ_{k}}} & for D_{k} = k if α > k \end{array} .

With regard to the ℳ distribution, the diversity order D_M must verify Eq. (33) which, by introducing Eq. (18), leads to

b_{M} = \sum_{k = 1}^{\tilde{k}} {\tilde{m}}_{k} [lim_{s \to \infty} s^{D_{M}} M_{I, Kg}^{(k)} (s)] .

The lowest value of D_M that makes the previous equation be a constant is D_M = min{D_k} for k = 1 . . . k̃ governed again by the Binomial or Negative Binomial distribution coefficients m̃_k.

At this point, it must be noted that from the definition of the ℳ distribution parameters and the equivalences described in [13], any combination of {α, β, ρ} leads to a finite non-zero sequence of m̃_k for k = 1 . . . k̃, except in two particular cases: the first one corresponds to the case of having a blockage event, for which the only non-zero coefficient is m̃₁ = 1. The second one is the corresponding to the special scenario with ρ = 1, i.e. the whole scattered optical power is coupled to the LOS signal component. In this case, as detailed in [13], the coefficients m̃_k = 0 for every k ≠ β (if β ∈ ℕ) or k ≠ ∞ (if β ∈ ℝ), turning the ℳ distribution into a Gamma-Gamma distribution. In this situation, our analysis reduces to the case of having a full blockage of the link, which was the scenario analyzed in [22].

After this discussion and assuming ρ < 1, since the first-order Generalized-K sub-channel is always active, i.e. m̃₁ ≠ 0, the diversity order of the ℳ distribution is D_M = 1, independently on the turbulence parameters. Thus, Eq. (37) can be simplified to only the first term of the summation and, then, the parameter b_M is obtained from

b_{M} = {\tilde{m}}_{1} b_{1} = {\tilde{m}}_{1} \frac{Γ (α - 1)}{Γ (α)} \frac{α}{{\tilde{μ}}_{1}} = \frac{{\tilde{m}}_{1}}{{\tilde{μ}}_{1}} \frac{α}{α - 1},

which can be introduced in Eq. (31) in order to calculate the asymptotic outage probability expression at high SNR values in a ℳ-modeled atmospheric optical link, as

P_{out} \approx b_{M} γ_{n}^{- 1 / 2} .

Finally, when assuming the partial LOS component blockage with probability P_b, as previously described, the outage probability for high SNR values can be expressed as P_out = P_bP_out(b = 0) + (1 − P_b)P_out(b = 1), where P_out(b = 1) is given by the above equation and P_out(b = 0), i.e. the outage probability under LOS blockage, is obtained by applying m̃₁ = 1 μ̃₁ = ξ_g, as shown in Eq. (29). Hence, the complete outage probability asymptotic expression with LOS blockage is, after some analytical manipulation,

P_{out} \approx \frac{α}{α - 1} [P_{b} \frac{1}{ξ_{g}} + (1 - P_{b}) \frac{{\tilde{m}}_{1}}{{\tilde{μ}}_{1}}] γ_{n}^{- 1 / 2}

being governed by the first-order Generalized-K sub-channel behavior, in any case.

Inspecting this equation, it is evident that blockage causes the outage probability to be increased for a given value of γ_n. However, unlike the scenario considered in [22], the outage probability does not tend to an irreducible floor in the high-SNR regime. This result can be used to design the required power boost to overcome the SNR loss Δ due to blockage, for a given P_b and a target outage probability as

Δ (dB) ≜ 10 {log}_{10} [\frac{γ_{n}^{(P_{b})}}{γ_{n}^{(P_{b} = 0)}}] \approx 20 {log}_{10} [1 + P_{b} (\frac{{\tilde{m}}_{1}}{ξ_{g} {\tilde{μ}}_{1}} - 1)] .

5. Results and discussion

In this section, we present results of the effect of partial blockage in both the PDF of ℳ distribution and the outage probability. To obtain these results, we use the closed-form expressions deduced in the previous sections. In all cases, conditions of moderate to strong atmospheric turbulence are assumed with α = 4.2, β = 3 and ρ ranging from 0 to 1. Note that when ρ → 1, the amount of optical power coupled to the LOS component is increased, corresponding to a reduction of the turbulence intensity (in this case D_c ≈ D_b). On the contrary, when ρ → 0, the amount of optical power coupled to the LOS component is decreased, corresponding to an increase of the turbulence intensity (in this case D_c << D_b). Thus, by varying the value of ρ, turbulence conditions are modified.

First, in Fig. 3, the PDF of the ℳ distribution under LOS blockage, f_I,b(I) is shown. In particular, Fig. 3(a) displays the PDF for different values of the coupling parameter ρ (i.e. as a function of the turbulence conditions), while Fig. 3(b) presents the PDF for different values of the blockage probability P_b. In both figures, results were obtained using Eq. (16) and, so, assuming that an obstacle with diameter D = D_c temporary blocks the coherent component of the optical beam with a probability P_b. To cover all possible cases, in Fig. 3(a), two extreme blockage situations are considered. On the one hand, with solid line, the figure shows the PDF of the conventional ℳ distribution without blockage (P_b = 0) for different values of ρ (i.e. different turbulence conditions). Note that, in this case, when ρ → 1, the ℳ distribution becomes a Gamma-Gamma distribution, being not possible to consider partial blocking as described in Section 1; for this reason we have adopted ρ = 0.99 to approximate to this situation. On the other hand, with dotted line, the figure shows the PDF of the ℳ distribution with maximum LOS blockage probability (P_b = 1). Here, it is clearly observed that as ρ increases the PDF concentrates on smaller values of the normalized irradiance. This is due to a larger portion of the signal being obstructed when turbulence decreases. In fact, in the limit, when ρ → 1, the distribution degenerates in a deterministic distribution centered at zero, as expected in the case of a complete link blockage. The effect of P_b on the PDF of the ℳ distribution with LOS blockage is shown in Fig. 3(b). Here, we have chosen the worst case of the previous figure (i.e. ρ = 0.99). It is observed that the behavior of the PDF near I = 0 is clearly affected as P_b is increased, so that the probability of this range of low irradiances is also increased. Note that these results are in agreement with the scenario described in [22], since the modified ℳ distribution PDF with ρ = 1 is given by

f_{I, b} (I) = P_{b} δ (I) + (1 - P_{b}) f_{I, G G} (I; α, β)

where f_I,GG(I; α, β) is the PDF of the Gamma-Gamma distribution with parameters {α, β}. Thus, these results corroborate that the blockage probability P_b modulates the relevance among the statistical behavior of the LOS and coupled-to-LOS components, described through the different Generalized-K distributed sub-channels, and the Rayleigh distributed non-LOS component under LOS blockage, described exclusively through the first order Generalized-K sub-channel.

Fig. 3 (a) PDF of the ℳ distribution under LOS blockage for different values of ρ assuming P_b = 0 and P_b = 1. (b) PDF of the ℳ distribution under LOS blockage for different values of blockage probability, P_b, assuming a very high coupling factor ρ = 0.99.

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Fig. 4 shows now the impact of LOS blockage on the outage probability for different values of turbulence conditions (i.e different ρ values). To obtain these results, we have used Eqs. (21), (40) and (41). Again, the extreme situations with P_b = 0 and P_b = 1 are first considered. Therefore, the results are grouped in two sets of curves which define the lower and upper bounds of the outage probability. Note that any blockage probability is between these two bounds. In particular, with solid line are depicted the general results of P_out obtained with Eq. (29), while with dotted line are shown the asymptotic results of P_out given by Eq. (40). Logically, the outage probability is worse in case of LOS blockage (P_b = 1) than when there is no blockage (P_b = 0). However, from Fig. 4 it is observed that the difference between both bounds decreases when turbulence increases (ρ → 0). In fact, this difference can be obtained from Eq. (41) as

Δ_{\max} (dB) = 10 {log}_{10} [\frac{γ_{n}^{(P_{b} = 1)}}{γ_{n}^{(P_{b} = 0)}}] \approx 20 {log}_{10} [\frac{{\tilde{μ}}_{1}}{ξ_{g} {\tilde{m}}_{1}}] .

Note that, for instance, when ρ = 0.8, Δ_max = 36.1 dB while for ρ = 0.2, Δ_max = 7.5 dB. Thus, these results indicate that the impact of LOS blockage on P_out decreases as atmospheric turbulence increases. The value of Δ (dB), i.e. the transmit power increment needed to maintain the desired outage probability when a LOS blockage is produced is depicted in Fig. 5(a) as a function of P_b for different turbulence conditions. A target P_out = 10⁻³ is here assumed. From this figure, it is clearly observed that, since LOS blockage is more harmful as turbulence is reduced (due to more signal being blocked), the value of Δ (dB) increases as ρ is reduced. So, for P_b = 10⁻¹ and ρ = 0.1, a Δ ≈ 1.4 dB is needed to maintain the P_out, while for the same P_b and ρ = 0.9, Δ ≈ 32 dB. Thus, for lower scattered energy coupled to the LOS component, the required power boost is lower and only noticeable for a high P_b. On the contrary, higher levels of power increments are necessary to maintain the performance for higher ρ values, even under negligible blockage probabilities. Fig. 5(b) presents the dependence of P_b on the outage probability. Again, we have chosen the worst case of the previous figure (i.e. ρ = 0.99). As expected, for fixed turbulence conditions, P_out decreases when P_b is reduced. Further, it is also observed that the asymptotic results of P_out (depicted with dotted lines) are a good approximation to P_out for values of γ_n > 50 dB and 0.2 ≤ ρ ≤ 0.8. Thus, Eq. (40) provides a very simple tool to predict the outage probability behavior of a ℳ-modeled FSO link, in the high received SNR regime, with or without LOS blockage.

Fig. 4 Outage probability of a FSO link over ℳ-turbulence channels and LOS blockage with different values of ρ and blockage probabilities P_b = 0 and P_b = 1.

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Fig. 5 (a) Power boost needed to achieve a P_out = 10⁻³ under LOS blockage as a function of the coupling factor, ρ, and the blockage probability P_b. (b) Outage probability of a FSO link over ℳ-turbulence channels with ρ = 0.99 and several values of the LOS blockage probabilities.

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Finally, Fig. 6 shows the impact of the turbulence conditions, through the coupling parameter ρ, on the outage probability considering different values of P_b and γ_n. We observe that for low values of ρ and moderately low values of P_b, the outage probability is barely altered. However, as ρ grows, the effect of the LOS blockage becomes prominent. In fact, when γ_n is high enough, the value of P_b can change the OP behavior with respect to the turbulence conditions. In particular, when P_b is low (P_b < 10⁻²), the OP increases as turbulence conditions worsen (ρ → 0). However, when P_b is higher (P_b > 10⁻²), then, the OP reduces when turbulence conditions worsen. Note that, in this case, an increase in the turbulence intensity can improve the link performance. In the limit case of ρ → 1, the outage probability tends to P_b; i.e. this scenario reduces to the one assumed in [22], due to the equivalence between the ℳ and the Gamma-Gamma distributions.

Fig. 6 Impact of the coupling factor, ρ, on the outage probability for several normalized SNR, γ_n, and blockage probabilities, P_b.

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6. Concluding remarks

In this paper, we have investigated the effect of partial link blockage on the performance of FSO communication links affected by turbulence fading. We derived new analytical expressions for the PDF and the MGF of the ℳ distribution considering a particular type of partial blockage, referred as LOS blockage, in which only the coherent component of the laser been is obstructed. From these functions, we deduced two closed-form expressions for the outage probability in FSO link under LOS blockage. The first expression is general and valid under any SNR, whereas the second one is an approximation applicable only in the high SNR regime. Since these new expressions can estimate the performance of FSO link affected by both turbulence and blockage, they are valuable tools for FSO system design. Obtained results demonstrate a strong dependence of the outage performance with the turbulence intensity and with the LOS blockage probability.

Funding

Universidad de Málaga (Plan Propio de Investigación); Andalucía Talent Hub Program co-funded by the Marie Curie actions (291780).

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Performance analysis of FSO communications under LOS blockage

Abstract

1. Introduction

2. System model and LOS blockage

2.1. Turbulence fading model

2.2. LOS blockage

3. PDF and MGF of the turbulence fading model under LOS blockage

3.1. PDF of the ℳ distribution under LOS blockage

3.2. MGF of the ℳ distribution under LOS blockage

4. Outage performance

4.1. General analysis

4.2. Asymptotic analysis

5. Results and discussion

6. Concluding remarks

Funding

References and links

Cited By

Figures (6)

Equations (43)

Optics Express