Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors

Antonio García-Zambrana; Beatriz Castillo-Vázquez; Carmen Castillo-Vázquez

doi:10.1364/OE.20.002096

1. Introduction

Atmospheric free-space optical (FSO) transmission using intensity modulation and direct detection (IM/DD) can provide high-speed links for a variety of applications, being an interesting alternative to consider for next generation broadband in order to support large bandwidth, unlicensed spectrum, excellent security, and quick and inexpensive setup [1]. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as atmospheric scintillation, severely degrading the link performance [2]. Additionally, since FSO systems are usually installed on high buildings, building sway causes vibrations in the transmitted beam, leading to an unsuitable alignment between transmitter and receiver and, hence, a greater deterioration in performance. Error control coding as well as diversity techniques can be used over FSO links to mitigate turbulence-induced fading [3–7]. In [8–10], selection transmit diversity is proposed for FSO links over strong turbulence channels, where the transmit diversity technique based on the selection of the optical path with a greater value of irradiance has shown to be able to extract full diversity as well as providing better performance compared to general FSO space-time codes (STCs) designs, such as conventional orthogonal space-time block codes (OSTBCs) and repetition codes (RCs). Recently, the optimality of selection transmit diversity as an optimal power allocation strategy for shot noise limited FSO systems has been proved in [11], proposing an extension of this scheme to systems with limited feedback. In [12], a novel approximate closed-form bit error-rate (BER) expression is derived for FSO links with transmit laser selection over K-distributed atmospheric turbulence channels. The combined effect of atmospheric and misalignment fading is analyzed in the case of single-input/single-output (SISO) FSO channels in [13]. In [14], the effects of atmospheric turbulence and misalignment considering aperture average effect were considered to study the outage capacity for SISO links. In [15] the error rate performance for coded FSO links over strong turbulence and misalignment fading channels is studied. The capacity calculation and the analysis of error rate performance for FSO links over log-normal and gamma-gamma turbulence and misalignment fading channels is presented in [16] using a wave optics based approach. In [17, 18], a wide range of turbulence conditions with gamma-gamma atmospheric turbulence and pointing errors is also considered on terrestrial FSO links, deriving closed-form expressions for the error-rate performance in terms of Meijer’s G-functions. In [19], average capacity optimization is considered in this same context by numerically solving the derivative of the corresponding capacity expression, also mathematically treated as a Meijer’s G-function. However, to the best of our knowledge, the performance of multiple-input/multiple-output (MIMO) FSO channels under the effects of atmospheric and misalignment fading has only been studied in [20, 21], being evaluated in terms of outage performance. In [20], the study of the outage probability and diversity gain has been considered for MIMO FSO communication systems impaired by log-normal atmospheric turbulence and misalignment fading, assuming repetition coding on the transmitter side and equal gain combining on the receiver side and showing that the diversity gain is conditioned only by misalignment parameters. In [21], comparing different diversity techniques, a significant improvement in outage performance is demonstrated when MIMO FSO links based on transmit laser selection are adopted in a strong turbulence scenario, showing that the diversity order is independent of the pointing error effects when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, and the fact that better performance is achieved when increasing the number of transmit apertures instead of the number of receive apertures in order to guarantee a same diversity order.

In this paper, asymptotic BER performance for FSO communication systems using IM/DD over atmospheric turbulence channels with pointing errors is analyzed when transmit laser selection on the transmitter side is assumed. Novel closed-form asymptotic expressions are derived when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters α and β, or pointing errors, following a misalignment fading model, as in [14, 15, 18, 20], where the effect of beam width, detector size and jitter variance is considered. Obtained results provide significant insight into the impact of various system and channel parameters, showing that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least 2(min{α,β})^1/2 times the value of the pointing error displacement standard deviation at the receiver, showing the same slope of the BER performance versus average signal-to-noise ratio (SNR) as in a similar FSO scenario where misalignment fading is not considered. However, different coding gain, i.e. the horizontal shift in the BER performance in the limit of large SNR, is achieved as a consequence of the severity of the pointing error effects and turbulence conditions. Here, not only rectangular pulses are considered but also on-off keying (OOK) formats with any pulse shape, corroborating the advantage of using pulses with high peak-to-average optical power ratio (PAOPR). Moreover, since proper FSO transmission requires transmitters with accurate control of their beamwidth, asymptotic expressions are used to find the optimum beamwidth that minimizes the BER at different turbulence conditions. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results, showing that asymptotic expressions here obtained lead to simple bounds on the bit error probability that get tighter over a wider range of SNR as the turbulence strength increases.

2. System and channel model

We adopt a multiple-input/single-output (MISO) array based on M laser sources, assumed to be intensity-modulated only and all pointed towards a distant photodetector, assumed to be ideal noncoherent (direct-detection) receiver, as shown in Fig. 1. The sources and the detector are physically situated so that all transmitters are simultaneously observed by the receiver. The use of infrared technologies based on IM/DD links is considered. Here, not more than one laser is simultaneously operating, following the transmit laser selection (TLS) scheme based on the selection of the optical path with a greater value of fading gain (irradiance) [8]. It is assumed that channel state information (CSI) is known not only at the receiver but also at the transmitter (CSIT). The knowledge of CSIT is feasible for FSO channels given that scintillation is a slow time varying process relative to the large symbol rate. This transmit diversity technique is well known for MIMO radio-frequency (RF) systems [22], also presenting in our FSO scenario as a promising approach for reducing complexity since one can employ a lower number of optical chains. So, whilst full diversity is achieved, hardware complexity and cost are reduced. In this context, it must be noted that non-ideal hardware aspects related to the selection switch, such as attenuation and incapability of switching instantaneously, may decrease theoretical performance. The instantaneous current y_m(t) in the receiving photodetector corresponding to the information signal transmitted from the mth laser, when this one is operating, can be written as

y_{m} (t) = η i_{m} (t) x (t) + z (t)

where η is the detector responsivity, assumed hereinafter to be the unity, X ≜ x(t) represents the optical power supplied by the mth source and I_m ≜ i_m(t) the equivalent real-valued fading gain (irradiance) through the optical channel between the mth laser and the receive aperture. Additionally, the fading experienced between source-detector pairs I_m is assumed to be statistically independent. Z ≜ z(t) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at the detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance σ² = N₀/2, i.e. Z ∼ N(0,N₀/2), independent of the on/off state of the received bit. It must be emphasized that z(t) is related to the only one photodetector, not being associated with the transmit laser operating. Since the transmitted signal is an intensity, X must satisfy ∀t x(t) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of X is limited. The received electrical signal Y_m ≜ y_m(t), however, can assume negative amplitude values. We use Y_m, X, I_m and Z to denote random variables and y_m(t), x(t), i_m(t) and z(t) their corresponding realizations. Additionally, we consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [8,9,21].

Fig. 1 FSO system with transmit laser selection.

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The irradiance is susceptible to either atmospheric turbulence conditions and pointing error effects. In this case, it is considered to be a product of two independent random variables, i.e. $I_{m} = I_{m}^{(a)} I_{m}^{(p)}$ , representing $I_{m}^{(a)}$ and $I_{m}^{(p)}$ the attenuation due to atmospheric turbulence and the attenuation due to geometric spread and pointing errors, respectively, between mth transmitter and receiver. Although the effects of turbulence and pointing are not strictly independent, for smaller jitter values they can be approximated as independent [16]. To consider a wide range of turbulence conditions (weak to strong), the gamma-gamma turbulence model proposed in [2,23] is here assumed, whose probability density function (PDF) is given by

f_{I_{m}^{(a)}} (i) = \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} i^{((α + β) / 2) - 1} K_{α - β} (2 \sqrt{α β_{i}}), i \geq 0

where Γ(·) is the well-known Gamma function and K_ν(·) is the νth-order modified Bessel function of the second kind [24, eqn. (8.43)]. The parameters α and β can be selected to achieve a good agreement between Eq. (2) and measurement data [23]. Alternatively, assuming spherical wave propagation, α and β can be directly linked to physical parameters through the following expresions [6,23,25]:

α = {[exp (\frac{0.49 χ^{2}}{{(1 + 0.18 d^{2} + 0.56 χ^{12 / 5})}^{7 / 6}}) - 1]}^{- 1}

β = {[exp (\frac{0.51 χ^{2} {(1 + 0.69 χ^{12 / 5})}^{- 5 / 6}}{{(1 + 0.9 d^{2} + 0.62 d^{2} χ^{12 / 5})}^{7 / 6}}) - 1]}^{- 1}

where

χ^{2} = 0.5 C_{n}^{2} k^{7 / 6} L^{11 / 6}

and d = (kD²/4L)^1/2. Here, k = 2π/λ is the optical wave number, λ is the wavelength, D is the diameter of the receiver collecting lens aperture and L is the link distance in meters.

C_{n}^{2}

stands for the altitude-dependent index of the refractive structure parameter and varies from 10⁻¹³ m^−2/3 for strong turbulence to 10⁻¹⁷ m^−2/3 for weak turbulence [2, 25]. Since the mean value of this turbulence model here considered is normalized and the second moment is given by E[I²] = (1 + 1/α)(1 + 1/β), the scintillation index (SI), a parameter of interest used to describe the strength of atmospheric fading, is defined as

S I = \frac{E {[I]}^{2}}{{(E [I])}^{2}} - 1 = \frac{1}{α} + \frac{1}{β} + \frac{1}{α β} .

It must be noted that PDF in Eq. (2) contains other turbulence models adopted in strong turbulence FSO scenarios such as the K-distribution (β = 1 and α > 0) and the negative exponential distribution (β = 1 and α → ∞) [5, 10, 15]. From the point of view of scintillation index, it is easy to deduce the fact that the strength of atmospheric fading represented by the gamma-gamma distributed turbulence model with channel parameters β = 1 and increasing α tends to be closer and closer to 1, SI corresponding to the negative exponential distributed turbulence model. Regarding to the impact of pointing errors, we use the general model of misalignment fading given in [14] by Farid and Hranilovic, wherein the effect of beam width, detector size and jitter variance is considered. Assuming a Gaussian spatial intensity profile of beam waist radius, ω_z, on the receiver plane at distance z from the transmitter and a circular receive aperture of radius r, the PDF of

I_{m}^{(p)}

is given by

f_{I_{m}^{(p)}} (i) = \frac{φ^{2}}{A_{0}^{φ^{2}}} i^{φ^{2} - 1}, 0 \leq i \leq A_{0}

where φ= ωz_eq/2σ_s is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver,

ω_{z_{e q}}^{2} = ω_{z}^{2} \sqrt{π} erf (v) / 2 v exp (- v^{2})

,

v = \sqrt{π} r / \sqrt{2} ω_{z}

, A₀ = [erf(v)]² and erf(·) is the error function [24, eqn. (8.250)]. Here, independent identical Gaussian distributions for the elevation and the horizontal displacement (sway) are considered, being

σ_{s}^{2}

the jitter variance at the receiver. It must be commented that the pointing error model in Eq. (6) is not general enough to be directly applied to MISO FSO systems even when the lasers are pointed at the same detector since pointing displacements will not be exactly on the same transverse plane for all lasers, not being strictly applicable in that case. Nonetheless, this can be considered negligible under the assumption that the link distance L between transmit lasers and photodetector is several orders of magnitude the spacing between transmitters. Knowing that the coherence length of the optical beams is on the order of centimeters, this can be easily justified if the transmitters are placed a few centimeters apart so that the fading is approximately independent of one another [26]. Using the previous PDFs for turbulence and misalignment fading, a closed-form expression of the combined PDF of I_m was derived in [17] as

f_{I_{l m}} (i) = \frac{α β φ^{2}}{A_{0} Γ (α) Γ (β)} G_{1, 3}^{3, 0} (\frac{α β}{A_{0}} i | \begin{matrix} φ^{2} \\ φ^{2} - 1, α - 1, β - 1 \end{matrix}), i \geq 0

where

G_{p, q}^{m, n} [\cdot]

is the Meijer’s G-function [24, eqn. (9.301)]. Even though Meijer’s G-function can be expressed in terms of more familiar generalized hypergeometric functions, this PDF appears to be cumbersome to use in order to obtain simple closed-form expressions in the analysis of MIMO FSO systems, leading to numerical solutions that obscure the impact of the basic system and channel parameters on performance. To overcome this inconvenience, the PDF in Eq. (7) is approximated by a single polynomial term as f_{I_m} (i) ≈ ai^b, based on the fact that the asymptotic behavior of the system performance is dominated by the behavior of the PDF near the origin, i.e. f_{I_m} (i) at i → 0 determines high SNR performance [27]. Hence, using the series expansion corresponding to the Meijer’s G-function [28, eqn. (07.34.06.0006.01)] and considering the fact that the two parameters α and β related to the atmospheric conditions are greater than each other, different asympotic expressions for Eq. (7), depending on the relation between the values of φ² and min{α,β}, can be written as

f_{I_{m}} (i) \approx \frac{φ^{2} {(α β)}^{min {α, β}} Γ (| α - β |)}{A_{0}^{min {α, β}} Γ (α) Γ (β) (φ^{2} - min {α, β})} i^{min {α, β} - 1}, φ^{2} > min {α, β}

f_{I_{m}} (i) \approx \frac{φ^{2} {(α β)}^{φ^{2}} Γ (α - φ^{2}) Γ (β - φ^{2})}{A_{0}^{φ^{2}} Γ (α) Γ (β)} i^{φ^{2} - 1}, φ^{2} < min {α, β}

It must be noted that the assumption α ≠ β can be assumed under a wide variety of simulated turbulence conditions [23].

3. Asymptotic error-rate performance analysis

In this section, using the previous asymptotic expressions for the combined PDF, we reveal some insights into the BER performance of FSO links with transmit laser selection over atmospheric turbulence and misalignment fading channels. We can take advantage of these simpler expressions in order to quantify the bit error-rate probability at high SNR, showing that the asymptotic performance of this metric as a function of the average SNR is characterized by two parameters: the diversity and coding gains. Following the TLS scheme based on the selection of the optical path with a greater value of irradiance, our MISO system model can be considered as an equivalent SISO system model where the channel irradiance corresponding to the TLS scheme, I_max, can be written as I_max = max_m_=1,2,···_M I_m. In this way, according to Eq. (1) and the OOK signaling [21, appendix], a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of d, the statistical channel model corresponding to this MISO configuration can be written as

Y = X I_{\max} + Z, X \in {0, d}, Z \sim N (0, N_{0} / 2)

Assuming channel side information at the receiver, the conditional BER is given by

P_{b} (E | I_{\max}) = Q (\sqrt{d^{2} i^{2} / 2 N_{0}})

where Q(·) is the Gaussian-Q function defined as

Q (x) = \frac{1}{\sqrt{2 π}} \int_{x}^{\infty} e^{- \frac{t^{2}}{2}} d t

. Substituting the value of the Euclidean distance d gives

P_{b} (E | I_{\max}) = Q (\sqrt{2 γ ξ} i)

where

γ = P_{opt}^{2} T_{b} / N_{0}

represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats, T_b parameter is the bit period, P_opt is the average optical power transmitted and ξ represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, as explained in a greater detail in [21, appendix]. Hence, the average BER, P_b(E), can be obtained by averaging P_b (E|I_max) over the PDF as follows

P_{b} (E) = \int_{0}^{\infty} Q (\sqrt{2 γ ξ} i) f_{I_{\max}} (i) d i .

According to order statistics [29], the PDF corresponding to I_max is given by f_{I_max} (i) = Mf_{I_m} (i)[F_{I_m} (i)]^M^–1, where F_{I_m} (i) the cumulative density function (CDF). Using Eq. (8) the asymptotic expressions for f_{I_max} (i) can be written as

f_{I_{\max}} (i) \approx {(\frac{{(Ω_{\min} M)}^{1 / M} {(A_{0}^{- 1} α β)}^{Ω_{\min}} φ^{2} Γ (| α - β |)}{Γ (Ω_{\max}) (φ^{2} - Ω_{\min}) Γ (Ω_{\min} + 1)})}^{M} i^{M Ω_{\min} - 1}, φ^{2} > Ω_{\min}

f_{I_{\max}} (i) \approx {(\frac{{(φ^{2} M)}^{1 / M} {(A_{0}^{- 1} α β)}^{φ^{2}} Γ (α - φ^{2}) Γ (β - φ^{2})}{Γ (α) Γ (β)})}^{M} i^{M φ^{2} - 1}, φ^{2} < Ω_{\min}

where Ω_min = min{α,β} and Ω_max = max{α,β} are used for readability. To evaluate the integral in Eq. (11), we can use that the Q-function is related to the complementary error function erfc(·) by

erfc (x) = 2 Q (\sqrt{2} x)

[24, eqn. (6.287)] and the fact that

\int_{0}^{\infty} erfc (x) x^{a - 1} d x = Γ ((1 + a) / 2) / (π^{1 / 2} a)

[24, eqn. (6.281)], obtaining the corresponding closed-form asymptotic solutions for the BER as can be seen in

P_{b} (E) ≐ {({(\frac{φ^{2} {(A_{0}^{- 1} α β)}^{Ω_{\min}} Γ (| α - β |) Γ {(\frac{M Ω_{\min} + 1}{2})}^{1 / M}}{(φ^{2} - Ω_{\min}) {(2 \sqrt{π})}^{1 / M} Γ (Ω_{\max}) Γ (Ω_{\min} + 1)})}^{- \frac{2}{Ω_{\min}}} γ ξ)}^{- \frac{M Ω_{\min}}{2}}, φ^{2} > Ω_{\min}

P_{b} (E) ≐ {({(\frac{{(A_{0}^{- 1} α β)}^{φ^{2}} Γ (α - φ^{2}) Γ (β - φ^{2}) Γ {(\frac{M φ^{2} + 1}{2})}^{1 / M}}{{(2 \sqrt{π})}^{1 / M} Γ (α) Γ (β)})}^{- \frac{2}{φ^{2}}} γ ξ)}^{- \frac{M φ^{2}}{2}}, φ^{2} < Ω_{\min}

The results corresponding to this asymptotic analysis are illustrated in the Fig. 2, when different levels of turbulence strength of (α,β) = (6,3), (α,β) = (4,2) and (α,β) = (2,1) are assumed, corresponding to values of scintillation index of SI = 0.55, SI = 0.87 and SI = 2, respectively, and where rectangular pulse shapes with ξ = 1 are used for values of M = {1,2,4,6}. Monte Carlo simulation results are furthermore included as a reference, confirming the accuracy and usefulness of the derived results. Due to the long simulation time involved, simulation results only up to BER=10⁻⁸ are included. Simulation results corroborate that asymptotic expressions here obtained lead to simple bounds on the bit error probability that get tighter over a wider range of SNR as the turbulence strength increases. It is straightforward to show that the average BER behaves asymptotically as (G_cγξ)^−G_d, where G_d and G_c denote diversity order and coding gain, respectively [27, 30]. At high SNR, if asymptotically the error probability behaves as (G_cγξ)^−G_d, the diversity order G_d determines the slope of the BER versus average SNR curve in a log-log scale and the coding gain G_c (in decibels) determines the shift of the curve in SNR. Additionally, as previously reported by the authors [8,21], a relevant improvement in performance must be noted as a consequence of the pulse shape used, providing an increment in the average SNR of 10log₁₀ ξ decibels. At this point, it can be convenient to compare with the BER performance obtained in a similar context when misalignment fading is not present. Knowing that the impact of pointing errors in our analysis can be suppressed by assuming A₀ → 1 and φ² → ∞ [14], the corresponding asymptotic expression can be easily derived from Eq. (13a) as follows

P_{b} (E) ≐ {({(\frac{{(α β)}^{Ω_{\min}} Γ (| α - β |) Γ {(\frac{M Ω_{\min} + 1}{2})}^{1 / M}}{{(2 \sqrt{π})}^{1 / M} Γ (Ω_{\max}) Γ (Ω_{\min} + 1)})}^{- \frac{2}{Ω_{\min}}} γ ξ)}^{- \frac{M Ω_{\min}}{2}} .

In Fig. 3, BER performance in FSO links using transmit laser selection over gamma-gamma atmospheric with/without pointing errors, assuming M = {1,4}, (α,β) = (4,2) and values of normalized jitter of σ_s/r = {1,3} and normalized beamwidth of ω_z/r = {5,10}, are displayed. From this asymptotic analysis, it can be deduced that the main aspect to consider in order to optimize the error-rate performance is the relation between the values of φ² and min{α,β}. So, it is shown that the diversity order is independent of the pointing errors when the equivalent beam radius at the receiver is at least 2(min{α,β})^1/2 times the value of the pointing error displacement standard deviation at the receiver. Once this condition is satisfied, taking into account the coding gain in Eq. (13a), the impact of the pointing error effects translates into a coding gain disadvantage, D[dB], relative to gamma-gamma atmospheric turbulence without misalignment fading given by

D [d B] ≜ \frac{20}{Ω_{\min}} {log}_{10} (\frac{φ^{2}}{A_{0}^{Ω_{\min}} (φ^{2} - Ω_{\min})}) .

As in the outage performance analysis of MIMO FSO links in a strong turbulence scenario [21], it is also here corroborated that the impact of pointing errors is not related to the MISO nature of the FSO system, obtaining the same coding gain disadvantage regardless the number of transmit lasers. According to this expression, it can be observed in Fig. 3 that coding gain disadvantages of 23.9, 34.4 and 39.5 decibels are achieved for values of (ω_z/r,σ_s/r) = (5, 1), (ω_z/r,σ_s/r) = (10, 1) and (ω_z/r,σ_s/r) = (10, 3), respectively. Additionally, knowing that our turbulence model can be reduced to the simpler negative exponential turbulence model when β = 1 and α → ∞, it can be noted that the expression in Eq. (15) can be reduced to the equivalent coding gain disadvantage obtained in the outage performance analysis over strong turbulence and misalignment fading channels when Ω_min = 1 [21, eqn. (12)]. For the better understanding of the impact of pointing errors, the coding gain disadvantage D in Eq. (15) as a function of the distance L between transmitter and receiver is depicted in Fig. 4 assuming spherical wave propagation, a normalized jitter of σ_s/r = 1.5 and values of normalized beamwidth of ω_z/r = {4.75,5,5.25}. The parameters α and β are calculated from Eq. (3) and Eq. (4), and values of λ = 1550 nm,

C_{n}^{2} = 1.7 \times 10^{- 14} m^{- 2 / 3}

, and D/L → 0 are adopted [6, 25]. From this figure, it can be deduced that the value of ω_z/r that represents a lower coding gain disadvantage is depending on L, suggesting the fact that different values of normalized beamwidth might be used depending on the distance between transmitter and receiver in order to reduce the impact of pointing errors. Once the equivalent beam radius at the receiver is guaranteed to be at least 2(min{α,β})^1/2 times the value of the pointing error displacement standard deviation at the receiver, since proper FSO transmission requires transmitters with accurate control of their beamwidth, the optimization procedure is finished by finding the optimum beamwidth, ω_z/r, that gives the minimum BER performance in Eq. (13a) [14, 31–33]. As previously explained, it can be observed that this is equivalent to minimize the expression in Eq. (15), being independent of the number of transmit lasers. In this way, the optimum beamwidth can be achieved using numerical optimization methods for different values of normalized jitter, σ_s/r and turbulence conditions [34]. Numerical results for the optimum beamwidth are used in Fig. 4 in order to minimize D for a value of normalized jitter of σ_s/r = 1.5 and a range of distances from 3 km to 7 km in discrete steps of 250 m when the stochastic function minimizer simulated annealing is adopted. In Fig. 5, numerical results for the optimum beamwidth are displayed for distances between transmitter and receiver of L = {3,5,7} km and a range of values in the normalized jitter from σ_s/r = 1 to σ_s/r = 10 in discrete steps of 0.5 when λ = 1550 nm,

C_{n}^{2} = 1.7 \times 10^{- 14} m^{- 2 / 3}

, and D/L → 0 are considered. From this figure, it can be deduced that the BER optimization provides numerical results following a linear performance for each value of distance L, where its corresponding slope is subject to the turbulence conditions. This leads to easily obtain a first-degree polynomial given by

ω_{z} / r_{optimum} \approx (- 0.034 Ω_{\min}^{2} + 0.72 Ω_{\min} + 2.15) σ_{s} / r

where the slope follows a quadratic form in Ω_min. As can be seen in this figure, it is clearly depicted that the approximative analytical expression remains very accurate to numerical results. The use of this expression is also shown in Fig. 4, where results assuming the optimum beamwidth corresponding to a normalized jitter of σ_s/r = 1.5 are also included for a range of distances of 3 km to 7 km. Here, it is shown that the impact of the pointing errors is more severe as the distance between the transmitter and receiver is smaller, and, hence, the atmospheric turbulence strength is lower. In agreement with the analysis of error rate performance for FSO links over log-normal and gamma-gamma turbulence and misalignment fading channels presented in [16] using a wave optics based approach, obtained results corroborate that the stronger turbulence channels are more robust to pointing errors.

Fig. 2 BER performance in FSO IM/DD links using transmit laser selection over gamma-gamma atmospheric and misalignment fading channels, when different levels of turbulence (a) (α,β) = (6,3), (b) (α,β) = (4,2) and (c) (α,β) = (2,1) are assumed together with values of normalized jitter of σ_s/r = {1,2,3} and a normalized beamwidth of ω_z/r = 5. BER results corresponding to the same FSO scenario without pointing errors and the non-turbulence case are also included.

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Fig. 3 BER performance in FSO links using transmit laser selection over gamma-gamma atmospheric with/without pointing errors, assuming M = {1,4} and values of normalized jitter of σ_s/r = {1,3} and normalized beamwidth of ω_z/r = {5,10}. BER result corresponding to the non-turbulence case is also included as a reference.

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Fig. 4 Coding gain disadvantage D in Eq. (15) as a function of the distance L between transmitter and receiver in FSO links over gamma-gamma atmospheric when λ = 1550 nm, $C_{n}^{2} = 1.7 \times 10^{- 14} m^{- 2 / 3}$ , and D/L → 0, assuming a normalized jitter of σ_s/r = 1.5 and values of normalized beamwidth of ω_z/r = {4.75,5,5.25}.

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Fig. 5 Optimum normalized beamwidth versus normalized jitter, σ_s/r in FSO links over gamma-gamma atmospheric when λ = 1550 nm, $C_{n}^{2} = 1.7 \times 10^{- 14} m^{- 2 / 3}$ , D/L → 0 and distances between transmitter and receiver of L = {3,5,7} km.

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Finally, to fully exploit the obtained results in this paper and knowing that the impact of pointing errors in our analysis can be suppressed by assuming A₀ → 1 and φ²→ ∞, we can use the expression in Eq. (14) to compare the BER performance between different combining techniques in the context of FSO links and gamma-gamma fading. In [6], closed-form expressions for the diversity gain and the combining gain of MIMO FSO with repetition coding (RC) across lasers at the transmitter and equal gain combining (EGC) or maximal ratio combining (MRC) at the receiver have been provided, concluding that in gamma-gamma fading MRC achieves only small to moderate performance gains compared to EGC [6, Fig. 4], which is more attractive in practice because of its considerably lower implementation complexity. Modifying the combining gain advantage corresponding to RC-EGC in [6, eqn. (27)] in order to consider that the sum of the receive aperture areas is the same as the aperture area of a system with no receive diversity, allowing the systems to be compared fairly [3], the combining gain advantage of TLS compared to RC-EGC can be written as

G_{TLS} [dB] ≜ \frac{20}{Ω_{\min}} {log}_{10} (\frac{M^{Ω_{\min}} Γ (Ω_{\min} + 1)}{Γ {(M Ω_{\min} + 1)}^{1 / M}})

where M is the number of trasmit lasers with the TLS scheme or the sum of transmit lasers and receive apertures with the RC-EGC scheme. In Fig. 6, results corresponding to G_TLS versus the distance L between transmitter and receiver in FSO links and gamma-gamma fading when λ = 1550 nm,

C_{n}^{2} = 1.7 \times 10^{- 14} m^{- 2 / 3}

, and D/L → 0 are displayed for values of M = {2,4,6,8}. It can be noted that relevant improvement in performance is achieved when transmit laser selection is assumed, being this superiority even more relevant when: firstly, the turbulence strength is greater because of the fact that the distance L is increased; and, secondly, the number of transmit lasers is greater.

Fig. 6 Combining gain advantage, G_TLS, of TLS over RC-EGC versus the distance L between transmitter and receiver in FSO links and gamma-gamma fading when λ = 1550 nm, $C_{n}^{2} = 1.7 \times 10^{- 14} m^{- 2 / 3}$ , and D/L → 0, assuming M trasmit lasers with the TLS scheme and the same diversity order as in RC-EGC scheme.

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At last, considering BER=10⁻⁹ is a practical performance target for a FSO link and the fact that asymptotic expressions here obtained have shown to be simple bounds on the bit error probability that get tighter over a wider range of SNR as the turbulence strength increases, we can use the expression in Eq. (14) as a simpler alternative to the aproximate closed-form BER expression recently derived in [12] for FSO links with transmit laser selection over K-distributed atmospheric turbulence channels without pointing errors. This is based on the approximation of K-distribution (β = 1 and α > 0) by a finite sum of weighted negative exponential functions, but not valid for α ≤ 3 [35]. In order to solve this inconvenience, bound in Eq. (14) can be proposed to use in this FSO scenario of more severe atmospheric turbulence conditions.

4. Conclusions

In this paper, asymptotic BER performance for FSO communication systems using IM/DD over atmospheric turbulence channels with pointing errors is analyzed when transmit laser selection on the transmitter side is assumed. Novel closed-form asymptotic expressions are derived when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters α and β, or pointing errors, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Obtained results provide significant insight into the impact of various system and channel parameters, showing that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least 2(min{α,β})^1/2 times the value of the pointing error displacement standard deviation at the receiver, showing the same slope of the BER performance versus average signal-to-noise ratio (SNR) as in a similar FSO scenario where misalignment fading is not considered. However, different coding gain, i.e. the horizontal shift in the BER performance in the limit of large SNR, is achieved as a consequence of the severity of the pointing error effects and turbulence conditions. Moreover, since proper FSO transmission requires transmitters with accurate control of their beamwidth, asymptotic expressions are used to find the optimum beamwidth that minimizes the BER at different turbulence conditions. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results, showing that asymptotic expressions here obtained lead to simple bounds on the bit error probability that get tighter over a wider range of SNR as the turbulence strength increases.

Acknowledgments

The authors would like to thank the anonymous reviewers for their useful comments that helped to improve the presentation of the paper. The authors are grateful for financial support from the Junta de Andalucía (research group “Communications Engineering (TIC-0102)”).

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Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors

Abstract

1. Introduction

2. System and channel model

3. Asymptotic error-rate performance analysis

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (20)

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