We derived the average bit error rate (BER) of coherent freespace optical (FSO) systems with digital binary phase shift keying (BPSK) modulations over atmospheric turbulence channels with a gamma–gamma distribution. To obtain a generalized derivation in a closedform expression, we used special integrals and transformations of the Meijer G function. Furthermore, we numerically analyzed and simulated the average BER behavior according to the average SNR for different turbulence strengths. Simulation results are demonstrated to confirm the analytical results.
I. INTRODUCTION
Freespace optical (FSO) communication systems
[1

5]
are commonly used to provide an attractive and costeffective link for highdatarate wireless transmission. FSO systems support diverse applications ranging from highly directive pointtopoint links for terrestrial lastmile and longhaul intersatellite solutions to quick and efficient deployment in densely populated urban areas or in unstructured environments such as disasterprone areas.
FSO systems utilize a freespace medium for transmission, but they are inherently affected by atmospheric conditions, among which turbulence has the most significant effect, especially for highdatarate pointtopoint links. As such, it is an interesting problem to analyze the degradation of signal strength due to scintillation of the optical signal as well as link performance against atmospheric turbulence channels. In atmospheric turbulence channels, the coherence time of the channels is on the order of milliseconds, which is typically much larger than the onebit time interval of gigabitpersecond (Gbps) FSO signals
[6

11]
. Hence, for a onebit time interval, the FSO channels are modeled as a constant and random variable that is governed by a lognormal, K, or gammagamma distribution. The gammagamma distribution is a tractable mathematical model with a multiplication of two parameters of smallscale and largescale irradiance fluctuations, the probability density functions (PDFs) of which are independent gamma distributions, and it provides excellent agreement between theoretical and simulation results
[6

8]
.
Many authors have intensively researched and analyzed several implementation techniques for FSO systems under turbulence. In Ref.
[8]
, a performance analysis for intensity modulationdirect detection (IMDD) FSO systems over gammagamma turbulence channels was presented. For coherent FSO systems, Refs.
[12]
and
[13]
proposed alternative implementations enabling a higher receiver sensitivity than that of IMDD, especially when the power of the local oscillator laser is sufficiently high; Refs.
[6]
and
[14]
presented analyses of coherent heterodyne DPSK systems over K and gammagamma turbulence channels, respectively, considering thermal noise caused by the high operating temperature of FSO systems. Coherent PSK requires the proper control of laser coherence, which is challenging in FSO systems because of difficulties associated with the phaselocking of the local oscillators. However, coherent PSK is expected to provide performance benefits over DPSK. Furthermore, there are known results indicating that the performance of coherent FSO systems is limited by the shot noise of the receiver, which heterodynes with a local oscillator laser having sufficiently high power; this needs to be analyzed further under turbulence channels.
In this study, we derived a generalized closedform expression for the average BER performance of coherent FSO systems by using binary phase shift keying (BPSK) over atmospheric turbulence channels, in which the turbulenceinduced fading of the signal intensity is described by a gammagamma distribution. Moreover, theoretical results are provided to understand the degradation of performance as a function of scintillation depth. Analytical results are further confirmed through Monte Carlo simulations and VPI transmission Maker results.
II. SYSTEM AND CHANNEL MODEL
The overall architecture of coherent FSO systems is shown in
Fig. 1 (a)
. The optical modulator processes data by using a laser at a transmitter. The output signal of the optical modulator is transmitted via atmospheric turbulence channels between telescopes. In
Fig. 1 (b)
, the received signal is combined with a local oscillator laser through a halfmirror at a receiver. Then, a photodetector detects the compound optical signals to generate the photocurrent. Finally, a decision module extracts data. In this section, we explain the coherent PSK receiver in detail because it is the basic model for coherent receivers and can be adapted to other modulation receivers. In
Fig. 1 (b)
, the received optical signal and the local oscillator laser in scalar form can be expressed, respectively, as
[12
,
13]
(a) Overall architecture of coherent FSO systems. (b) Structure of a dualphotodiode balanced receiver of coherent FSO systems.
where
E
_{S}
is the electrical field of the received signal,
I
is the intensityfading coefficient,
a
= ± 1 is the information,
f
_{c}
is the optical carrier frequency, and
E
_{LO}
is the electric field of the local oscillator laser. The total power of the received signal and local oscillator laser is
where
P
_{S}
is the power of the received signal and
P
_{LO}
is the power of the local oscillator laser. Additionally, the output current of the photodetector is expressed as
where
R
is the responsivity of the receiver; the short noise (
i_{SH}
(
t
)) and thermal noise (
i_{TH}
(
t
)) have power spectral densities (PSDs) of
G
_{SH}
=
RP
_{LOq}
and
, respectively;
q
is the electron charge;
k
is the Boltzmann constant;
B
is the noiseequivalent bandwidth of the filter; and
R
_{L}
is the load resistance. When
i_{PD}
(
t
) is lowpass filtered to limit the noise power, the instantaneous signaltonoise ratio (SNR) is given as
In coherent FSO systems, if the power of the local oscillator laser were sufficiently high, the second term in the denominator of Eq. (4) would vanish. Thus, Eq. (4) is reduced to
[5]
As in
[7]
and
[8]
, the PDF of a gammagamma distribution is represented by the product of smallscale and largescale irradiance fluctuations, both of which have gamma distributions. The gammagamma distribution is
where I >0,
is the average irradiance of the channel,
α
and
β
are the scintillation parameters,
K
_{s}
(ㆍ) is the modified Bessel function of the second kind of order ε , and Γ(ㆍ) is the Gamma function. Here,
α
and
β
are defined based on the atmospheric conditions, as in
[8]
.
III. DERIVATION OF THE AVERAGE BER
In this section, we derive the average BER of coherent FSO systems according to BPSK modulation. We first calculate the average SNR (µ) using
[15
, Eq. 07.34.21.0009.01] as follows:
The conditional BER,
P
_{b}
(
I
), for coherent systems is represented as
where
erfc
is the complementary error function. To obtain a closedform expression, we used the following Meijer G functions that were reported in
[15
, Eq. 07.34.03.0619.01 and Eq. 07.34.03.0605.01], which are, respectively, expressed as
and
Thus, the conditional BER (Eq. 8) and gammagamma distribution (Eq. 6) can be represented by the Meijier G function using Eqs. 9 and 10. Thus, the average BER (
) can be obtained using the following integral:
By substituting Eqs. (6) and (8) into Eq. (11), we can obtain the following equation:
Finally, by using the classic Meijer integral of the two G functions
[15
, Eq. 07.34.21.0011.01], Eq. 12 is simplified as follows:
IV. NUMERICAL RESULTS
Figure 2
represents the results for a closedform expression of the average BER,
, as a function of the average SNR at different turbulence strengths. For instance, we considered the following turbulence strengths: (
α
:
β
)∈ {(4,4),(6,6),(8,8),(10,10)}. Monte Carlo simulation results are included as a reference to validate our theoretical analysis. For creating gammagamma turbulence channels in the simulation, we used the multiplication of two random variables with a gamma distribution
[3]
. Then, we confirm that the PDF from a gammagamma random variable using the histogram method in MATLAB is the same as Eq. (6). Owing to the long simulation time, only simulation results up to BER = 10
^{−5}
are included. The simulation results demonstrate an excellent agreement with the results of theoretical analysis. Considering that BER = 10
^{−9}
is a practical performance target for an FSO system, our analytical results can serve as a simple and reliable method to estimate BER performance without resorting to lengthy simulations. In addition, to clarify our analysis further, we confirm its accuracy through systemlevel simulations using VPI transmission Maker. Because the validity of the FSO toolboxes of VPI transmission Maker has been established by verified FSO experimental data, we consider the results from VPI transmission Maker as an alternative experimental approach. The outputs of VPI transmission Maker are completely consistent with that of our analysis, as well as that of our MATLAB simulation.
Comparison of the average BER performance as a function of the average SNR for (α ,β )∈{(4,4), (6,6), (8,8), (10,10)}.
Figure 2
plots the average BER as a function of the average SNR for different turbulence channel strengths. Recall that the channel strength depends on the scintillation parameters,
α
and
β
, such that the turbulence effects become stronger as
α
and
β
decrease. We observe that the BER of the coherent FSO system under a strong turbulence effect of (α,β) = (4,4) is significantly higher than that under a weak turbulence effect of (α,β) = (10,10). More specifically, in the weak turbulence case, the average BER with an SNR of 20 dB is less than 10
^{−8}
. In the strong turbulence case, the average BER severely increases to 10
^{−3}
at 20dB SNR.
V. CONCLUSION
We obtained a closedform expression for the average BER of coherent FSO systems over atmospheric turbulence channels with a gammagamma distribution by using special integrals and transformations of the Meijer G function. Furthermore, we simulated the average BER performance using the Monte Carlo method to confirm the theoretical results. Simulation results show an excellent agreement with the analytical results. Therefore, we can more easily predict BER performance by using a simple closedform expression without any complicated calculation. In practical terms, when we establish coherent FSO systems, we can create an engineering table by using the derived BER.
Acknowledgements
This paper was supported by Research Fund, Kumoh National Institute of Technology.
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