Abstract

The average bit error rate (BER) for binary phase-shift keying (BPSK) modulation in free-space optical (FSO) links over turbulence atmosphere modeled by the exponentiated Weibull (EW) distribution is investigated in detail. The effects of aperture averaging on the average BERs for BPSK modulation under weak-to-strong turbulence conditions are studied. The average BERs of EW distribution are compared with Lognormal (LN) and Gamma-Gamma (GG) distributions in weak and strong turbulence atmosphere, respectively. The outage probability is also obtained for different turbulence strengths and receiver aperture sizes. The analytical results deduced by the generalized Gauss-Laguerre quadrature rule are verified by the Monte Carlo simulation. This work is helpful for the design of receivers for FSO communication systems.

© 2014 Optical Society of America

1. Introduction

In recent years, free-space optical (FSO) communication is gaining increasing attention to overcome bandwidth shortage [13]. It is widely accepted as a complementary technology to near-ground radio-frequency wireless short-range connections. It has been used in the systems that have platforms with limited weight and space, require high data links, or must operate in an environment where fiber optic links are not practical, such as communications between buildings across cities or military tactical operations [4]. Actually, FSO communication can also be used over longer link distances, such as satellite communications, etc [5]. Despite the significant advantages, the performance of FSO communication will be degraded by several factors, such as weather conditions, building-sway, and the atmosphere turbulence [6]. Particularly, the atmospheric turbulence induces intensity fluctuations at the received signal, which is called scintillation in optical communication terminology. For shorter ranges, scintillation can be compensated by increasing the transmission power. However, if the link distance is longer than several kilometers, transmission power cannot reach the required level due to laser safety regulations and technical limitations [5]. Fading mitigation techniques are required in this case. Among variety of fading mitigation techniques, diversity and aperture averaging have been extensively studied [6, 7]. Of the three types of diversity, time diversity needs to perform coding and interleaving, and it also imposes long delay latencies and requires large memories for storing long data frames [8]. Wavelength diversity requires composite transmitter, in which the signal is transmitted towards a number of receivers at different wavelengths and each receiver detects the signal at a specific wavelength [9]. Not only the number of wavelengths but also the separation among the wavelengths can affect the performance of the FSO link [10]. In general, wavelength diversity is efficient, but at the cost of increased complexity of the optical system design [11]. For the case of spatial diversity, it is an attractive technique to mitigate fading of the received signal. However, it normally requires multiple transmitters and receivers [12]. Aperture averaging is another important technique to improve the FSO system performance in the presence of turbulence. As known, intensity fluctuations at a receiver will lead to the variance of the received power which depends on the size of the receiver aperture. However, increasing the size of the receiver aperture can reduce the power variance because the receiver will average the fluctuations over the aperture [13, 14]. In fact, it can also be regarded as a simple form of spatial diversity when the receiver aperture is larger than the fading correlation length [8]. However, not like the spatial diversity, extra hardware load is not required for aperture averaging technique. Currently, aperture averaging has been widely used in commercial FSO systems for its low cost and simplicity [1517].

Traditionally, the Lognormal (LN) and the Gamma-Gamma (GG) distribution are the most widely used models to analyze the performance of FSO systems through turbulence atmosphere [4]. In particular, the GG model has been also used to study the effects of aperture averaging [15]. However, very recently, a new probability density function (PDF) model called exponentiated Weibull (EW) has been first proposed in [18, 19] to describe the distribution of the received irradiance in FSO links. The studies of [18] showed that the proposed EW distribution offers excellent fit to PDF of irradiance for both simulation and experiment data under weak-to-moderate turbulence regime, under all aperture averaging conditions, as well as for point-like apertures [18]. It greatly outperforms the GG model when aperture averaging takes place. In [19], comparisons have been carried out among the PDF of the LN, GG and EW models in moderate-to-strong turbulence atmosphere with different receiver aperture sizes. It showed that the EW distribution presents perfect fit to not only the right tail of the PDF, but also the left tail unlike the LN and GG distribution. However, for the error performance of FSO systems based on the EW distribution, only a few works [20, 21] have been published up to now. The authors in [20] developed a closed-form approximation for the average bit error rate (BER) of on-off keying (OOK) modulation by using the generalized Gauss-Laguerre quadrature rule. Aperture averaging effects in weak and moderate turbulence regimes have been analyzed. In [21], a new closed-form expression for average BER of OOK modulation under the EW turbulence has been derived by using Meijer’s G-function. The authors analyzed the probability of fade and average BER of the EW model in moderate turbulence atmosphere, for two different aperture sizes, i.e., D=25mm and D=60mm. The GG and LN models were also presented for comparison.

Although OOK intensity modulation is extensively used in many existing works related to commercial FSO applications, it actually needs adaptive threshold to perform optimally in the presence atmospheric turbulence [22, 23]. And it also brings more synchronization complexity due to the possible long sequences of zeros and ones [24]. Subcarrier intensity modulation (SIM) is another attractive technology that could avoid the need for an adaptive threshold and be used to increase capacity by modulating multiple information sources onto different subcarriers [23]. Another advantage of this system is that a variety of modulation schemes could be employed at the electrical modulator [25]. The error rates for binary phase-shift keying (BPSK) of SIM-FSO systems over GG and negative exponential (NE) turbulence channels [22] and LN model [26] have been studied respectively. However, neither [22] nor [26] provided closed-form error rate expressions. In [27], the error rate performance of BPSK-based SIM-FSO communication systems over K-distributed atmospheric turbulence channels has been investigated and approximate closed-form expressions have been obtained for the system employing selection combining (SC) diversity schemes. Closed-form error rates of BPSK, MPSK and DPSK modulations for SIM-FSO systems over GG fading channels have been provided in [28]. The authors also investigated the error performance of subcarrier BPSK links over GG turbulence channels with pointing error in [29] as well as the error performance of MIMO FSO systems in [30]. However, there are no published works on the error performance of PSK based SIM-FSO systems over the EW fading channels up to now, to the best of our knowledge.

In this paper, the PSK pre-modulated subcarrier intensity modulations in the EW distribution based FSO systems are studied. Highly accurate average BER approximations for BPSK and MPSK modulations have been obtained by using the generalized Gauss-Laguerre quadrature rule. After that, the aperture averaging effects under different atmosphere turbulence conditions are investigated. Then, the average BERs of EW model are compared with LN and GG models in weak and strong turbulence condition, respectively. And finally, the outage probability of the EW turbulence model is derived.

2. System and channel models

2.1 System model

In this paper, an optics intensity-modulated direct detection (IM/DD) system based on SIM technique is considered. The block diagram of generic SIM-FSO systems is shown in Fig. 1. At the transmitter, the subcarrier signal m(t) is pre-modulated by the source data. Before modulating a continuous wave laser beam, the subcarrier signal should be properly biased to ensure that the bias current is always equal to or greater than the threshold current [22]. The transmitted power of the modulated laser beam Pt is given in [29] as

Pt=P[1+ξm(t)],
where P is the average power. ξ is the modulation index, |ξm(t)|1 must be fulfilled in case over-modulation.

 

Fig. 1 Generic complex form of SIM-FSO system.

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At the receiver, the incident optical field is focused onto a photodetector by a lens. The received optical power is converted into electrical signal. The instantaneous photocurrent at the output of the photodetector is given in [29] as

ir(t)=RPI(t)[1+ξm(t)]+n(t),
where R is the photodetector responsivity and I(t) is the instantaneous channel gain, n(t) is the zero-mean additive white Gaussian noise (AWGN) with variance σn2, which is used to model the thermal noise and the background radiation shot noise. Usually, the Gaussian assumption is used for thermal and background noise. In case of shot noise, the noise power will depend on the received signal intensity, for the Gaussian approximation to be accurate, the intensity of the received signal has to be relatively high and the background radiation has to be signiðcant [8, 22]. The electrical signal-to-noise ratio (SNR) at the receiver is defined in [29] as
γb=γ¯bI2=R2P¯2ξ2I2σn2,
where γ¯b=R2P¯2ξ2σn2 is the average electrical SNR.

2.2. Atmospheric turbulence model

The PDF of received irradiance fluctuation for the EW model is given by [18, Eq. (7)]

fI(I)=αβη(Iη)β1exp[(Iη)β]×{1exp[(Iη)β]}α1,I0
and the corresponding cumulative distribution function (CDF) is given by [18, Eq. (8)]
FI(I)={1exp[(Iη)β]}α,
where β>0 and α>0 are shape parameters, and η>0 is a scale parameter. A group of expressions for calculation of the parameters has been developed as [18, Eqs. (10)-(12)]. In a recent work, new expressions for the parameters have been given as [19, Eqs. (20)-(22)].

3. Error rate analysis of subcarrier coherent modulations

For a FSO system over a turbulence channel, the average error rate P¯b can be calculated by the conditional error ratePb(I), i.e., the BER under AWGN [20], that is

P¯b=0Pb(I)fI(I)dI,
where fI(I) is the PDF of the received irradiance fluctuations. Equation (6) can be simplified with the help of the CDF of the variable I and the method of integration by parts, yielding
P¯b=0Pb(I)FI(I)dI,
where FI(I) is the CDF of the variable I and Pb(I) is the first order derivative of Pb(I) with respect to I.

3.1. BER of BPSK

For BPSK modulation, the conditional BER over an AWGN channel is given by [31]

Pb,BPSK(I)=12erfc(γb),
where erfc() is the complementary error function. The first order derivative of the conditional BER Pb,BPSK(I) can be obtained with the help of [32] as follows:
Pb,BPSK(I)=γ¯bπexp(γ¯bI2),
the detailed derivation process of Pb,BPSK(I) is provided in the appendix. Then, substituting Eqs. (5) and (9) into Eq. (7), the average BER for BPSK P¯b,BPSK can be written as
P¯b,BPSK=0γ¯bπexp(γ¯bI2)×{1exp[(Iη)β]}αdI,
if we perform the variable change x=γ¯bI2, Eq. (10) takes the form

P¯b,BPSK=12π0x1/2exp(x)×{1exp[(xηγ¯b)β]}αdx.

Equation (11) can be efficiently and accurately approximated by using the generalized Gauss- Laguerre quadrature function as [33] mentioned, 0xβexf(x)dx=i=1nHif(xi). Thus Eq. (11) can be expressed by a truncated series

P¯b,BPSK12πk=1nHk{1exp[(xkηγ¯b)β]}α,
where xk is the k-th root of the generalized Laguerre polynomial Ln(1/2)(x) and the weight Hk can be calculated by [34]

Hk=Γ(n+1/2)xkn!(n+1)2[Ln+1(12)(xk)]2.

3.2. BER of MPSK

For M>4, the BER of MPSK modulation over AWGN channel is given by [23]

Pb,MPSK=2log2MQ(2γblog2MsinπM),
where Q() is the Gaussian Q function defined as Q()=0.5erfc(z/2) [35]. The first order derivative of the conditional BER is
Pb,MPSK=2sin(πM)γ¯bπlog2Mexp(γ¯bI2sin2πMlog2M).
the detailed derivation process of Pb,MPSK(I) is provided in the appendix. Substituting Eqs. (5) and (15) into Eq. (7), the average BER of MPSK modulation can be obtained as
P¯b,MPSK=2sin(πM)γ¯bπlog2M0exp(γ¯bI2sin2πMlog2M)×{1exp[(Iη)β]}αdI,
performing the variable change x=γ¯blog2M(IsinπM)2, we get
P¯b,MPSK=1πlog2M0x1/2exp(x)×{1exp[(xηsin(πM)γ¯blog2M)β]}αdx,
using the generalized Gauss-Laguerre quadrature rule, we obtain
P¯b,MPSK1πlog2Mk=1nHk{1exp[(xkηsin(πM)γ¯blog2M)β]}α,
where xk is the k-th root of the generalized Laguerre polynomial Ln(1/2)(x) and Hk is given in Eq. (13).

4. Outage probability

The outage probability is defined as the probability that the instantaneous SNR falls below a threshold. It is another metric for quantifying communication systems in fading channels. It is given by [36]

Pout=Pr(γbγth)=Pr(Iγthγ¯b),
Pout=0γthγ¯bfI(I)dI.

In this paper, fI(I) is the PDF of the EW distribution. The outage probability can be obtained by using the CDF of I Eq. (5) as follows:

Pout==FI(γthγ¯b)={1exp[(1ηγn)β]}α,
where γn=γ¯bγth is the normalized electrical SNR following the definition in [36]. The outage probability in different turbulence and aperture averaging conditions are shown in Fig. 8.

5. Results and discussion

In this section, the analytical and Monte Carlo simulation results are illustrated in the figures below. It is believed that the shape parameter α is somehow affected not only by the scintillation index σI2, but also by the ratio of aperture size (D) to the atmospheric coherence radius (ρ0) as (D/ρ0), and/or the Rytov variance σR2 [19]. The Rytov variance is defined as σR2=1.23Cn2k7/6L11/6, where Cn2 is the refractive-index structure constant and L is the distance between the transmitter and receiver planes. k=2π/λ is the wavenumber, λ is the optical wavelength [18]. So the parameters used to perform the analytical calculation and Monte Carlo simulation are all extracted from the best PDF fitting in [18, 19]. In computing the generalized Gauss-Lagurre approximations, we choose n to be 30. During Monte Carlo simulation, the inverse CDF function method is used to generate random fading channel with the EW distribution. To reduce the statistical uncertainties in the numerical simulations of the irradiance, a total number of 108 random numbers were generated in each case.

In Fig. 2, the average BERs of BPSK modulation are plotted against the SNR γ¯b in the weak turbulence regime with two aperture sizes, i.e.,D=3mm and D=25mm. In our study, a Gaussian beam with λ=780nm propagating along a horizontal path with L=375m was considered and Cn2=2.1×1014m2/3 was used. In addition, the initial beam radius is W0=1.13cm. σR2 is equal to 0.15 and the atmospheric coherence radius is ρ0=18.89mm. Thus, the aperture of 3mm can behave as a point receiver since it is smaller than ρ0. It is easily seen from the figure that the analytical results (solid curve for D=3mm and dashed curve forD=25mm, respectively) of both aperture sizes have excellent agreements with the corresponding simulation results (circles). This confirms that our average BER model is accurate and the aperture averaging technology can improve the BER performance of FSO system. It is also seen that the point receiver (D=3mm) requires larger SNR compared with the aperture-averaged receiver (D=25mm) to reach the same BER. For example, to achieve the average BER of 109, about 25.5 dB of SNR is needed for the point receiver (D=3mm) while only 15.5 dB is needed for the aperture-averaged receiver (D=25mm). This is because that the aperture-averaged receiver will average the fluctuations over the aperture and the scintillation will be less compared to that of a point receiver when the aperture is larger than a spatial scale size that produces the irradiance fluctuations [13].

 

Fig. 2 Analytical and simulation results for average BER of subcarrier BPSK modulated FSO system over the EW turbulence channels against average electrical SNR under weak turbulence conditionσR2=0.15. Link distanceL=375mand coherence radiusρ0=18.89mm.

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Figure 3 presents the average BER results for moderate turbulence condition (σR2=1.35). Four aperture sizes (3mm,25mm,60mm,80mm) are chosen and the parameters are obtained from the best PDF fitting in [18] for L=1225m. It can be seen that the analytical results for all aperture sizes match well with the simulation results. As the aperture size increases, the average BER decreases. The effect of aperture averaging on average BER under the moderate turbulence condition is stronger than that under weak turbulence condition. For instance, the aperture-averaged receiver (D=25mm) offers a performance gain of about 28 dB compared with the point receiver (D=3mm) in terms of the SNR to reach the average BER of 109, which is larger than that in weak turbulence environment. It is because that as turbulence becomes more severe, aperture averaging will become more effective to inhibit variances of the power and intensity [14].

 

Fig. 3 Analytical and simulation results for average BER of subcarrier BPSK modulated FSO system over the EW turbulence channels against average electrical SNR under moderate turbulence conditionσR2=1.35. Link distanceL=1225mand coherence radiusρ0=9.27mm.

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Figure 4 shows the average BER as a function of SNR under strong turbulence condition (σR2=19.2). The parameters are selected from [19]. The simulated beam propagating along a 1500 m horizontal path was a 0.46 mrad full-angle diverging Gaussian beam with a transmitter beam radiusW0=1.59cm. A refractive-index structure constant Cn2=4.58×1013m2/3 and an atmospheric coherence radius ρ0=2.94mm were used. In this paper, three apertures of 1.8mm,5mm and 13mm are studied. It can be seen from the figure that the average BER for BPSK modulation in strong turbulence is also significantly improved by increasing the aperture size. For example, when the SNR is equal to 40 dB, the average BERs are 102, 104and 106 for 1.8mm, 5mm and 13mm, respectively.

 

Fig. 4 Analytical and simulation results for average BERs of subcarrier BPSK modulated FSO systems over the EW turbulence channels against average electrical SNR under strong turbulence conditionσR2=19.2. Link distanceL=1500mand coherence radiusρ0=2.94mm.

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Figure 5 shows the average BERs of FSO system over the EW turbulence channels using BPSK and MPSK modulation. The analytical results are following Eqs. (12) and (18) respectively. All the parameters are chosen from [18] with receiver sizes D=3mm and 25mm in weak turbulence (σR2=0.15). Due to the long Monte Carlo simulation time, the average BER of about 105is included. It is clearly seen that all the analytical results match well with the corresponding Monte Carlo simulation results. The BER increases with the increase of M and the BER performance with subcarrier BPSK intensity modulation is better for both aperture sizes. It is also found that the BER performance can be improved by increasing the receiver aperture size for both modulations. For example, by increasing D from 3mm to 25mm, a performance gain of about 5 dB in SNR at the BER of 105 is achieved for each modulation.

 

Fig. 5 Analytical and simulation results for average BERs of subcarrier BPSK and MPSK modulated FSO systems over the EW turbulence channels against average electrical SNR under weak turbulence conditionσR2=0.15.

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Traditionally, the analysis of BER in the FSO community has been done by modeling the atmospheric turbulence as LN, in the weak turbulence regime, and GG, in the moderate-to-strong turbulence regime [21]. In this work, EW and LN models are compared for the weak turbulence regime, and then, EW and GG models are compared for the strong turbulence regime. The BER results for BPSK modulation in GG and LN fading channels here are based on the Monte Carlo simulation. The parameters with the subscript ‘fit’ mean that the values are obtained from the best PDF fitting [18, 19]. In Fig. 6, the average BERs for BPSK modulation of the LN and EW models under weak turbulence atmosphere (σR2=0.15) with two aperture sizes D=3mm and 25mm have been presented. The parameters used in this figure are given in Table 1. It can be seen from the figure that for D=3mm, the average BER of EW is a little bit larger than that of LN. For D=25mm, the average BERs of the EW and LN are almost equal. For example, when SNR is equal to 10 dB, the average BERs of the EW and the LN are about 0.3×104. It can also be seen that when the receiver size increases from 3mmto 25mm, a smaller SNR is needed to achieve the same BER for these two models. For instance, to achieve the BER of 106, about 18 dB is needed for D=3mm while 12 dB is needed for D=25mmof the EW distribution; 15.5 dB of SNR is needed for D=3mm while 11.5 dB is needed for D=25mmof the LN distribution. The aperture-averaged receiver (D=25mm) offers a performance gain of about 6 dB and 4 dB compared with the point receiver (D=3mm) in terms of the SNR for EW and LN, respectively. Thus, the effect of aperture averaging is larger on the PSK-based average BER of EW than that of LN.

 

Fig. 6 The average BERs for subcarrier BPSK modulated FSO system over the EW and LN turbulence channels against average SNR under weak turbulence condition.

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Tables Icon

Table 1. Parameters for the LN and EW distributions used in Fig. 6.

In Fig. 7, the average BERs for BPSK modulation in GG and EW fading channels are plotted against the SNR in the strong turbulence regime (σR2=19.2) with three aperture sizes, i.e.,D=1.8mm, D=5mm and D=13mm. Parameters used are listed in Table 2. It can be seen from the figure that as the receiver sizes increase from 1.8 to 13, the average BERs do not change too much for the GG. For example, when SNR is equal to 40 dB, the average BERs of the GG for three aperture sizes are all about 0.3×102. For the EW distribution, when SNR is equal to 40 dB, the average BERs are 102, 104and 106 for 1.8mm, 5mm and 13mm, respectively. It means that under strong turbulence atmosphere condition, there is almost no improvement for the performance of the GG model with the increase of aperture size. However, the EW model is more efficient for aperture averaging in strong turbulence atmosphere.

 

Fig. 7 The average BER for subcarrier BPSK modulated FSO system over the EW and GG turbulence channels against average SNR under strong turbulence condition.

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Tables Icon

Table 2. Parameters for the GG and EW distributions used in Fig. 7.

Figure 8 plots the outage probability against the normalized electrical SNR. The outage probability in weak-to-moderate turbulence channels following the EW distribution with D=3mm (solid curve) and 25mm (dashed curve) has been shown. It can be seen from the figure that the outage probability is affected not only by the normalized electrical SNR, but also by the turbulence strength and the receiver size. Along with the increase of SNR, the outage probability decreases. However, as the Rytov variance σR2 increases from 0.15 to 1.35, the outage probability increases. It is also seen that the outage probability in the weak turbulence atmosphere is much smaller than that in the moderate turbulence atmosphere. For example, when SNR is equal to 10 dB, the outage probability is about 103in weak turbulence (σR2=0.15) while it is larger than 101 in moderate turbulence (σR2=1.30,1.35) for D=3mm. This indicates that the performance of FSO is deteriorated with the increase of atmosphere turbulence, which has been also confirmed in [36, 37]. In addition, the effect of aperture averaging for the outage probability can also be seen in this figure. It is found that as the receiver aperture size increases, the outage probability evidently decreases. For example, a performance gain of about 8 dB is obtained by the aperture-averaged receiver (D=25mm) compared with the point receiver (D=3mm) in terms of the SNR to achieve the outage probability of 105 for weak turbulence (σR2=0.15).

 

Fig. 8 The outage probability against the normalized electrical SNR.

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6. Conclusions

In summary, the performance of the SIM-FSO systems with BPSK modulation in turbulent atmosphere with the EW distribution was studied. Highly-accurate closed-form approximations of the average BERs for BPSK and MPSK modulations were derived by using the generalized Gauss-Laguerre quadrature rule. The effects of aperture averaging under weak-to-strong turbulence conditions were investigated in detail. The comparisons of the BER performance between the LN and EW models in weak turbulence atmosphere, GG and EW models in strong turbulence atmosphere were carried out respectively. The results showed that the aperture averaging technique can significantly improve the system performance in turbulence atmosphere and the subcarrier BPSK modulation system has superior BER performance. The effect of turbulence strength on the outage probability of the EW fading channels was also studied in this paper, which can be restrained by aperture averaging. This work is applicable to the receiver design for PSK-SIM based FSO communication systems.

Appendix

1. The first order derivative of Pb,BPSK(I) with respect to I

For BPSK modulation, the conditional BER over an AWGN channel can be written as

Pb,BPSK(I)=12erfc(γb),
where γb=γ¯bI2 [29].

The first order derivative of Pb(I) with respect to I can be obtained as follows

Pb,BPSK(I)=dPb,BPSK(I)dI=12ddIerfc(γ¯bI),
according to the method of composite function derivation provided in [32], Eq. (23) can be expressed as
Pb,BPSK(I)=12d[erfc(γ¯bI)]d(γ¯bI)d(γ¯bI)dI,
using ddzerfc(z)=2πez2 [32] and d(γ¯bI)dI=γ¯b, Eq. (24) can be written as

Pb,BPSK(I)=γ¯bπexp(γ¯bI2).

Thus, the first order derivate of Pb,BPSK(I) with respect to I is Pb,BPSK(I)=γ¯bπexp(γ¯bI2).

2. The first order derivative of Pb,MPSK(I) with respect to I

For M>4, the BER of MPSK modulation over AWGN channel is given as

Pb,MPSK=2log2MQ(2γblog2MsinπM),
where Q() is the Gaussian Q function defined as Q()=0.5erfc(z/2) [35] and γb=γ¯bI2 [29]. Thus

Pb,MPSK=1log2Merfc(γ¯blog2MIsinπM).

The first order derivative of Pb,MPSK(I) with respect to I can be obtained as follows

Pb,MPSK=dPb,MPSKdI=1log2MddIerfc(γ¯blog2MIsinπM),
according to the method of composite function derivation provided in [32], and performing the variable change z=γ¯blog2MIsinπM, Eq. (28) can be written as
Pb,MPSK=1log2Md[erfc(z)]dzdzdI,
using ddzerfc(z)=2πez2 [32] and dzdI=γ¯blog2MsinπM, Eq. (29) can be written as

Pb,MPSK=2γ¯bsinπMπlog2Mexp(γ¯bI2sin2πMlog2M).

Thus, the first order derivate of Pb,MPSK(I) with respect to I is Pb,MPSK=2γ¯bsinπMπlog2Mexp(γ¯bI2sin2πMlog2M).

Acknowledgments

Project supported by Nature Science Basic Research Plan in Shaanxi Province of China (Grant No. 2014JM8340), the China Postdoctoral Science Special Foundation (Grant No. 201104659), the China Post-doctoral Science Foundation (Grant No. 20100481322), the Fundamental Research Funds for the Central Universities (Grant No. NSIY041404) and this work is also partly supported by 111 Project of China (Grant No. B08038).

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13. F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46(11), 2099–2108 (2007). [CrossRef]   [PubMed]  

14. H. Yuksel, S. Milner, and C. C. Davis, “Aperture averaging for optimizing receiver design and system performance on free-space optical communication links,” J. Opt. Netw. 4(8), 462–475 (2005). [CrossRef]  

15. F. S. Vetelino, C. Young, and L. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. 46(18), 3780–3789 (2007). [CrossRef]   [PubMed]  

16. N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE 5338, 233–242 (2004). [CrossRef]  

17. I. E. Lee, Z. Ghassemlooy, W. P. Ng, and M. A. Khalighi, “Joint optimization of a partially coherent Gaussian beam for free-space optical communication over turbulent channels with pointing errors,” Opt. Lett. 38(3), 350–352 (2013). [CrossRef]   [PubMed]  

18. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012). [CrossRef]   [PubMed]  

19. R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013). [CrossRef]  

20. X. Yi, Z. J. Liu, and P. Yue, “Average BER of free-space optical systems in turbulent atmosphere with exponentiated Weibull distribution,” Opt. Lett. 37(24), 5142–5144 (2012). [CrossRef]   [PubMed]  

21. R. Barrios and F. Dios, “Probability of fade and BER performance of FSO links over the exponentiated Weibull fading channel under aperture averaging,” Proc. SPIE 8540, 85400D (2012). [CrossRef]  

22. W. O. Popoola and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol. 27(8), 967–973 (2009). [CrossRef]  

23. W. O. Popoola, “Subcarrier intensity modulated free-space optical communication systems,” Diss. Northumbria Univ. (2009).

24. A. E. Morra, H. S. Khallaf, H. M. H. Shalaby, and Z. Kawasaki, “Performance analysis of both shot- and thermal-noise limited multipulse PPM receivers in gamma-gamma atmospheric channels,” J. Lightwave Technol. 31(19), 3142–3150 (2013). [CrossRef]  

25. X. G. Song and J. L. Cheng, “Subcarrier intensity modulated optical wireless communications using noncoherent and differentially coherent modulations,” J. Lightwave Technol. 31(12), 1906–1913 (2013).

26. J. Li, J. Q. Lin, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun. 55(8), 1598–1606 (2007). [CrossRef]  

27. H. Samimi and P. Azmi, “Subcarrier intensity modulated free-space optical communications in K-distributed turbulence channels,” J. Opt. Commun. Netw. 2(8), 625–632 (2010). [CrossRef]  

28. X. G. Song, M. B. Niu, and J. L. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett. 16(4), 540–543 (2012). [CrossRef]  

29. X. G. Song, F. Yang, and J. L. Cheng, “Subcarrier intensity modulated optical wireless communications in atmospheric turbulence with pointing error,” J. Opt. Commun. Netw. 5(4), 349–358 (2013).

30. X. G. Song and J. L. Cheng, “Subcarrier intensity modulated MIMO optical communications in atmospheric turbulence,” J. Opt. Commun. Netw. 5(9), 1001–1009 (2013).

31. M. K. Simon and M. S. Alouini, in Digital Communication Over Fading Channels: A Unified Approach to Perform Analysis (John Wiley, 2000).

32. Wolfram, http://mathworld.wolfram.com.

33. P. Conçus, D. Cassatt, G. Jaehnig, and E. Melby, “Tables for the evaluation of 0xβexf(x)dx by Gauss-Laguerre quadrature,” Math. Comput. 17(83), 245–256 (1963).

34. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

35. G. K. Karagiannidis and S. Lioumpas, “An improved approximation for the Gaussian Q-function,” IEEE Commun. Lett. 11(8), 644–646 (2007). [CrossRef]  

36. J. M. Garrido-Balsells, A. Jurado-Navas, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “On the capacity of M-distributed atmospheric optical channels,” Opt. Lett. 38(20), 3984–3987 (2013). [CrossRef]   [PubMed]  

37. L. Q. Han, Q. Wang, and K. Shida, “Outage probability of free space optical communication over atmosphere turbulence,” in Proceeding of WASE International Conference on Information Engineering (2010), pp. 127–130.

References

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  1. P. Deng, M. Kavehrad, Z. W. Liu, Z. Zhou, and X. H. Yuan, “Capacity of MIMO free space optical communications using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence,” Opt. Express 21(13), 15213–15229 (2013).
    [Crossref] [PubMed]
  2. M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in FSO communications,” J. Lightwave Technol. 32(7), 1323–1328 (2014).
    [Crossref]
  3. C. Y. Chen, H. M. Yang, Z. Zhou, W. Z. Zhang, M. Kavehrad, S. F. Tong, and T. S. Wang, “Effects of source spatial partial coherence on temporal fade statistics of irradiance flux in free-space optical links through atmospheric turbulence,” Opt. Express 21(24), 29731–29743 (2013).
    [Crossref] [PubMed]
  4. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
    [Crossref]
  5. B. Epple, “Simplified channel model for simulation of free-space optical communications,” J. Opt. Commun. Netw. 2(5), 293–304 (2010).
    [Crossref]
  6. S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm. 6(8), 2813–2819 (2007).
    [Crossref]
  7. X. Tang, Z. Ghassemlooy, S. Rajbhandari, W. O. Popoola, and C. G. Lee, “Coherent heterodyne multilevel polarization shift keying with spatial diversity in a free-space optical turbulence channel,” J. Lightwave Technol. 30(16), 2689–2695 (2012).
    [Crossref]
  8. M. A. Khalighi, N. Schwartz, N. Aitamer, and S. Bourennane, “Fading reduction by aperture averaging and spatial diversity in optical wireless systems,” J. Opt. Commun. Netw. 1(6), 580–593 (2009).
    [Crossref]
  9. V. Xarcha, A. N. Stassinakis, H. E. Nistazakis, G. P. Latsas, M. P. Hanias, G. S. Tombras, and A. Tsigopoulos, “Wavelength diversity for Free Space Optical Systems: Performance evaluation over log normal turbulence channels,” in Proceeding of IEEE on Microwave Radar and Wireless Communications (IEEE 2012), pp. 678–683.
    [Crossref]
  10. K. Kiasaleh, “On the scintillation index of a multiwavelength Gaussian beam in a turbulent free-space optical communications channel,” J. Opt. Soc. Am. A 23(3), 557–566 (2006).
    [Crossref] [PubMed]
  11. H. E. Nistazakis and G. S. Tombras, “On the use of wavelength and time diversity in optical wireless communication systems over gamma-gamma turbulence channels,” Opt. Laser Technol. 44(7), 2088–2094 (2012).
    [Crossref]
  12. V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006).
    [Crossref]
  13. F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46(11), 2099–2108 (2007).
    [Crossref] [PubMed]
  14. H. Yuksel, S. Milner, and C. C. Davis, “Aperture averaging for optimizing receiver design and system performance on free-space optical communication links,” J. Opt. Netw. 4(8), 462–475 (2005).
    [Crossref]
  15. F. S. Vetelino, C. Young, and L. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. 46(18), 3780–3789 (2007).
    [Crossref] [PubMed]
  16. N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE 5338, 233–242 (2004).
    [Crossref]
  17. I. E. Lee, Z. Ghassemlooy, W. P. Ng, and M. A. Khalighi, “Joint optimization of a partially coherent Gaussian beam for free-space optical communication over turbulent channels with pointing errors,” Opt. Lett. 38(3), 350–352 (2013).
    [Crossref] [PubMed]
  18. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012).
    [Crossref] [PubMed]
  19. R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
    [Crossref]
  20. X. Yi, Z. J. Liu, and P. Yue, “Average BER of free-space optical systems in turbulent atmosphere with exponentiated Weibull distribution,” Opt. Lett. 37(24), 5142–5144 (2012).
    [Crossref] [PubMed]
  21. R. Barrios and F. Dios, “Probability of fade and BER performance of FSO links over the exponentiated Weibull fading channel under aperture averaging,” Proc. SPIE 8540, 85400D (2012).
    [Crossref]
  22. W. O. Popoola and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol. 27(8), 967–973 (2009).
    [Crossref]
  23. W. O. Popoola, “Subcarrier intensity modulated free-space optical communication systems,” Diss. Northumbria Univ. (2009).
  24. A. E. Morra, H. S. Khallaf, H. M. H. Shalaby, and Z. Kawasaki, “Performance analysis of both shot- and thermal-noise limited multipulse PPM receivers in gamma-gamma atmospheric channels,” J. Lightwave Technol. 31(19), 3142–3150 (2013).
    [Crossref]
  25. X. G. Song and J. L. Cheng, “Subcarrier intensity modulated optical wireless communications using noncoherent and differentially coherent modulations,” J. Lightwave Technol. 31(12), 1906–1913 (2013).
  26. J. Li, J. Q. Lin, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun. 55(8), 1598–1606 (2007).
    [Crossref]
  27. H. Samimi and P. Azmi, “Subcarrier intensity modulated free-space optical communications in K-distributed turbulence channels,” J. Opt. Commun. Netw. 2(8), 625–632 (2010).
    [Crossref]
  28. X. G. Song, M. B. Niu, and J. L. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett. 16(4), 540–543 (2012).
    [Crossref]
  29. X. G. Song, F. Yang, and J. L. Cheng, “Subcarrier intensity modulated optical wireless communications in atmospheric turbulence with pointing error,” J. Opt. Commun. Netw. 5(4), 349–358 (2013).
  30. X. G. Song and J. L. Cheng, “Subcarrier intensity modulated MIMO optical communications in atmospheric turbulence,” J. Opt. Commun. Netw. 5(9), 1001–1009 (2013).
  31. M. K. Simon and M. S. Alouini, in Digital Communication Over Fading Channels: A Unified Approach to Perform Analysis (John Wiley, 2000).
  32. Wolfram, http://mathworld.wolfram.com .
  33. P. Conçus, D. Cassatt, G. Jaehnig, and E. Melby, “Tables for the evaluation of ∫0∞xβe−xf(x)dx by Gauss-Laguerre quadrature,” Math. Comput. 17(83), 245–256 (1963).
  34. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).
  35. G. K. Karagiannidis and S. Lioumpas, “An improved approximation for the Gaussian Q-function,” IEEE Commun. Lett. 11(8), 644–646 (2007).
    [Crossref]
  36. J. M. Garrido-Balsells, A. Jurado-Navas, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “On the capacity of M-distributed atmospheric optical channels,” Opt. Lett. 38(20), 3984–3987 (2013).
    [Crossref] [PubMed]
  37. L. Q. Han, Q. Wang, and K. Shida, “Outage probability of free space optical communication over atmosphere turbulence,” in Proceeding of WASE International Conference on Information Engineering (2010), pp. 127–130.

2014 (1)

2013 (9)

C. Y. Chen, H. M. Yang, Z. Zhou, W. Z. Zhang, M. Kavehrad, S. F. Tong, and T. S. Wang, “Effects of source spatial partial coherence on temporal fade statistics of irradiance flux in free-space optical links through atmospheric turbulence,” Opt. Express 21(24), 29731–29743 (2013).
[Crossref] [PubMed]

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[Crossref]

I. E. Lee, Z. Ghassemlooy, W. P. Ng, and M. A. Khalighi, “Joint optimization of a partially coherent Gaussian beam for free-space optical communication over turbulent channels with pointing errors,” Opt. Lett. 38(3), 350–352 (2013).
[Crossref] [PubMed]

A. E. Morra, H. S. Khallaf, H. M. H. Shalaby, and Z. Kawasaki, “Performance analysis of both shot- and thermal-noise limited multipulse PPM receivers in gamma-gamma atmospheric channels,” J. Lightwave Technol. 31(19), 3142–3150 (2013).
[Crossref]

X. G. Song and J. L. Cheng, “Subcarrier intensity modulated optical wireless communications using noncoherent and differentially coherent modulations,” J. Lightwave Technol. 31(12), 1906–1913 (2013).

P. Deng, M. Kavehrad, Z. W. Liu, Z. Zhou, and X. H. Yuan, “Capacity of MIMO free space optical communications using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence,” Opt. Express 21(13), 15213–15229 (2013).
[Crossref] [PubMed]

X. G. Song, F. Yang, and J. L. Cheng, “Subcarrier intensity modulated optical wireless communications in atmospheric turbulence with pointing error,” J. Opt. Commun. Netw. 5(4), 349–358 (2013).

X. G. Song and J. L. Cheng, “Subcarrier intensity modulated MIMO optical communications in atmospheric turbulence,” J. Opt. Commun. Netw. 5(9), 1001–1009 (2013).

J. M. Garrido-Balsells, A. Jurado-Navas, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “On the capacity of M-distributed atmospheric optical channels,” Opt. Lett. 38(20), 3984–3987 (2013).
[Crossref] [PubMed]

2012 (6)

X. G. Song, M. B. Niu, and J. L. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett. 16(4), 540–543 (2012).
[Crossref]

R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012).
[Crossref] [PubMed]

X. Yi, Z. J. Liu, and P. Yue, “Average BER of free-space optical systems in turbulent atmosphere with exponentiated Weibull distribution,” Opt. Lett. 37(24), 5142–5144 (2012).
[Crossref] [PubMed]

R. Barrios and F. Dios, “Probability of fade and BER performance of FSO links over the exponentiated Weibull fading channel under aperture averaging,” Proc. SPIE 8540, 85400D (2012).
[Crossref]

X. Tang, Z. Ghassemlooy, S. Rajbhandari, W. O. Popoola, and C. G. Lee, “Coherent heterodyne multilevel polarization shift keying with spatial diversity in a free-space optical turbulence channel,” J. Lightwave Technol. 30(16), 2689–2695 (2012).
[Crossref]

H. E. Nistazakis and G. S. Tombras, “On the use of wavelength and time diversity in optical wireless communication systems over gamma-gamma turbulence channels,” Opt. Laser Technol. 44(7), 2088–2094 (2012).
[Crossref]

2010 (2)

2009 (2)

2007 (5)

F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46(11), 2099–2108 (2007).
[Crossref] [PubMed]

F. S. Vetelino, C. Young, and L. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. 46(18), 3780–3789 (2007).
[Crossref] [PubMed]

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm. 6(8), 2813–2819 (2007).
[Crossref]

J. Li, J. Q. Lin, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun. 55(8), 1598–1606 (2007).
[Crossref]

G. K. Karagiannidis and S. Lioumpas, “An improved approximation for the Gaussian Q-function,” IEEE Commun. Lett. 11(8), 644–646 (2007).
[Crossref]

2006 (2)

2005 (1)

2004 (1)

N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE 5338, 233–242 (2004).
[Crossref]

2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

1963 (1)

P. Conçus, D. Cassatt, G. Jaehnig, and E. Melby, “Tables for the evaluation of ∫0∞xβe−xf(x)dx by Gauss-Laguerre quadrature,” Math. Comput. 17(83), 245–256 (1963).

Aitamer, N.

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Andrews, L.

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Azmi, P.

Barrios, R.

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[Crossref]

R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012).
[Crossref] [PubMed]

R. Barrios and F. Dios, “Probability of fade and BER performance of FSO links over the exponentiated Weibull fading channel under aperture averaging,” Proc. SPIE 8540, 85400D (2012).
[Crossref]

Bourennane, S.

Cassatt, D.

P. Conçus, D. Cassatt, G. Jaehnig, and E. Melby, “Tables for the evaluation of ∫0∞xβe−xf(x)dx by Gauss-Laguerre quadrature,” Math. Comput. 17(83), 245–256 (1963).

Castillo-Vázquez, M.

Chan, V. W. S.

Chen, C. Y.

Cheng, J. L.

Conçus, P.

P. Conçus, D. Cassatt, G. Jaehnig, and E. Melby, “Tables for the evaluation of ∫0∞xβe−xf(x)dx by Gauss-Laguerre quadrature,” Math. Comput. 17(83), 245–256 (1963).

Davis, C. C.

Deng, P.

Dios, F.

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[Crossref]

R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012).
[Crossref] [PubMed]

R. Barrios and F. Dios, “Probability of fade and BER performance of FSO links over the exponentiated Weibull fading channel under aperture averaging,” Proc. SPIE 8540, 85400D (2012).
[Crossref]

Epple, B.

Fritzsche, D.

N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE 5338, 233–242 (2004).
[Crossref]

Garrido-Balsells, J. M.

Ghassemlooy, Z.

Hulea, M.

Jaehnig, G.

P. Conçus, D. Cassatt, G. Jaehnig, and E. Melby, “Tables for the evaluation of ∫0∞xβe−xf(x)dx by Gauss-Laguerre quadrature,” Math. Comput. 17(83), 245–256 (1963).

Jurado-Navas, A.

Karagiannidis, G. K.

G. K. Karagiannidis and S. Lioumpas, “An improved approximation for the Gaussian Q-function,” IEEE Commun. Lett. 11(8), 644–646 (2007).
[Crossref]

Kavehrad, M.

Kawasaki, Z.

Khalighi, M. A.

Khallaf, H. S.

Kiasaleh, K.

Lee, C. G.

Lee, I. E.

Li, J.

J. Li, J. Q. Lin, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun. 55(8), 1598–1606 (2007).
[Crossref]

Lin, J. Q.

J. Li, J. Q. Lin, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun. 55(8), 1598–1606 (2007).
[Crossref]

Lioumpas, S.

G. K. Karagiannidis and S. Lioumpas, “An improved approximation for the Gaussian Q-function,” IEEE Commun. Lett. 11(8), 644–646 (2007).
[Crossref]

Liu, Z. J.

Liu, Z. W.

Melby, E.

P. Conçus, D. Cassatt, G. Jaehnig, and E. Melby, “Tables for the evaluation of ∫0∞xβe−xf(x)dx by Gauss-Laguerre quadrature,” Math. Comput. 17(83), 245–256 (1963).

Milner, S.

Morra, A. E.

Navidpour, S. M.

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm. 6(8), 2813–2819 (2007).
[Crossref]

Ng, W. P.

Nistazakis, H. E.

H. E. Nistazakis and G. S. Tombras, “On the use of wavelength and time diversity in optical wireless communication systems over gamma-gamma turbulence channels,” Opt. Laser Technol. 44(7), 2088–2094 (2012).
[Crossref]

Niu, M. B.

X. G. Song, M. B. Niu, and J. L. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett. 16(4), 540–543 (2012).
[Crossref]

Paris, J. F.

Perlot, N.

N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE 5338, 233–242 (2004).
[Crossref]

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Popoola, W. O.

Puerta-Notario, A.

Rajbhandari, S.

Recolons, J.

Samimi, H.

Schwartz, N.

Shalaby, H. M. H.

Song, X. G.

Tang, X.

Taylor, D. P.

J. Li, J. Q. Lin, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun. 55(8), 1598–1606 (2007).
[Crossref]

Tombras, G. S.

H. E. Nistazakis and G. S. Tombras, “On the use of wavelength and time diversity in optical wireless communication systems over gamma-gamma turbulence channels,” Opt. Laser Technol. 44(7), 2088–2094 (2012).
[Crossref]

Tong, S. F.

Uysal, M.

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm. 6(8), 2813–2819 (2007).
[Crossref]

Vetelino, F. S.

Wang, T. S.

Yang, F.

Yang, H. M.

Yi, X.

Young, C.

Yuan, X. H.

Yue, P.

Yuksel, H.

Zhang, W. Z.

Zhou, Z.

Appl. Opt. (2)

IEEE Commun. Lett. (2)

X. G. Song, M. B. Niu, and J. L. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett. 16(4), 540–543 (2012).
[Crossref]

G. K. Karagiannidis and S. Lioumpas, “An improved approximation for the Gaussian Q-function,” IEEE Commun. Lett. 11(8), 644–646 (2007).
[Crossref]

IEEE Trans. Commun. (1)

J. Li, J. Q. Lin, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun. 55(8), 1598–1606 (2007).
[Crossref]

IEEE Trans. Wirel. Comm. (1)

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm. 6(8), 2813–2819 (2007).
[Crossref]

J. Lightwave Technol. (6)

J. Opt. Commun. Netw. (5)

J. Opt. Netw. (1)

J. Opt. Soc. Am. A (1)

Math. Comput. (1)

P. Conçus, D. Cassatt, G. Jaehnig, and E. Melby, “Tables for the evaluation of ∫0∞xβe−xf(x)dx by Gauss-Laguerre quadrature,” Math. Comput. 17(83), 245–256 (1963).

Opt. Eng. (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (2)

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[Crossref]

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[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Generic complex form of SIM-FSO system.
Fig. 2
Fig. 2 Analytical and simulation results for average BER of subcarrier BPSK modulated FSO system over the EW turbulence channels against average electrical SNR under weak turbulence condition σ R 2 =0.15 . Link distance L=375m and coherence radius ρ 0 =18.89mm .
Fig. 3
Fig. 3 Analytical and simulation results for average BER of subcarrier BPSK modulated FSO system over the EW turbulence channels against average electrical SNR under moderate turbulence condition σ R 2 =1.35 . Link distance L=1225m and coherence radius ρ 0 =9.27mm .
Fig. 4
Fig. 4 Analytical and simulation results for average BERs of subcarrier BPSK modulated FSO systems over the EW turbulence channels against average electrical SNR under strong turbulence condition σ R 2 =19.2 . Link distance L=1500m and coherence radius ρ 0 =2.94mm .
Fig. 5
Fig. 5 Analytical and simulation results for average BERs of subcarrier BPSK and MPSK modulated FSO systems over the EW turbulence channels against average electrical SNR under weak turbulence condition σ R 2 =0.15 .
Fig. 6
Fig. 6 The average BERs for subcarrier BPSK modulated FSO system over the EW and LN turbulence channels against average SNR under weak turbulence condition.
Fig. 7
Fig. 7 The average BER for subcarrier BPSK modulated FSO system over the EW and GG turbulence channels against average SNR under strong turbulence condition.
Fig. 8
Fig. 8 The outage probability against the normalized electrical SNR.

Tables (2)

Tables Icon

Table 1 Parameters for the LN and EW distributions used in Fig. 6.

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Table 2 Parameters for the GG and EW distributions used in Fig. 7.

Equations (30)

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P t =P[1+ξm(t)],
i r (t)=RPI(t)[1+ξm(t)]+n(t),
γ b = γ ¯ b I 2 = R 2 P ¯ 2 ξ 2 I 2 σ n 2 ,
f I (I)= αβ η ( I η ) β1 exp[ ( I η ) β ]× {1exp[ ( I η ) β ]} α1 ,I0
F I (I)= {1exp[ ( I η ) β ]} α ,
P ¯ b = 0 P b (I) f I (I)dI ,
P ¯ b = 0 P b (I) F I (I)dI,
P b,BPSK (I)= 1 2 erfc( γ b ),
P b,BPSK (I)= γ ¯ b π exp( γ ¯ b I 2 ),
P ¯ b,BPSK = 0 γ ¯ b π exp( γ ¯ b I 2 )× {1exp[ ( I η ) β ]} α dI ,
P ¯ b,BPSK = 1 2 π 0 x 1/2 exp(x)× {1exp[ ( x η γ ¯ b ) β ]} α dx.
P ¯ b,BPSK 1 2 π k=1 n H k {1exp[ ( x k η γ ¯ b ) β ]} α ,
H k = Γ(n+1/2 ) x k n! (n+1) 2 [ L n+1 ( 1 2 ) ( x k )] 2 .
P b,MPSK = 2 log 2 M Q( 2 γ b log 2 M sin π M ),
P b,MPSK = 2sin( π M ) γ ¯ b π log 2 M exp( γ ¯ b I 2 sin 2 π M log 2 M).
P ¯ b,MPSK = 2sin( π M ) γ ¯ b π log 2 M 0 exp( γ ¯ b I 2 sin 2 π M log 2 M)× {1exp[ ( I η ) β ]} α dI ,
P ¯ b,MPSK = 1 π log 2 M 0 x 1/2 exp(x)× {1exp[ ( x ηsin( π M ) γ ¯ b log 2 M ) β ]} α dx,
P ¯ b,MPSK 1 π log 2 M k=1 n H k {1exp[ ( x k ηsin( π M ) γ ¯ b log 2 M ) β ]} α ,
P out = P r ( γ b γ th )= P r (I γ th γ ¯ b ),
P out = 0 γ th γ ¯ b f I (I)d I.
P out == F I ( γ th γ ¯ b )= {1exp[ ( 1 η γ n ) β ]} α ,
P b,BPSK (I)= 1 2 erfc( γ b ),
P b,BPSK (I)= d P b,BPSK (I) dI = 1 2 d dI erfc( γ ¯ b I),
P b,BPSK (I)= 1 2 d[erfc( γ ¯ b I)] d( γ ¯ b I) d( γ ¯ b I) dI ,
P b,BPSK (I)= γ ¯ b π exp( γ ¯ b I 2 ).
P b,MPSK = 2 log 2 M Q( 2 γ b log 2 M sin π M ),
P b,MPSK = 1 log 2 M erfc( γ ¯ b log 2 M Isin π M ).
P b,MPSK = d P b,MPSK dI = 1 log 2 M d dI erfc( γ ¯ b log 2 M Isin π M ),
P b,MPSK = 1 log 2 M d[erfc(z)] dz dz dI ,
P b,MPSK = 2 γ ¯ b sin π M π log 2 M exp( γ ¯ b I 2 sin 2 π M log 2 M).

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