Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels

Ping Wang; Lu Zhang; Lixin Guo; Feng Huang; Tao Shang; Ranran Wang; Yintang Yang

doi:10.1364/OE.22.020828

1. Introduction

In recent years, free-space optical (FSO) communication is gaining increasing attention to overcome bandwidth shortage [1–3]. 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 [15–17].

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., $D = 25 m m$ and $D = 60 m m$ . 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 $P_{t}$ is given in [29] as

P_{t} = P [1 + ξ m (t)],

where

P

is the average power.

ξ

is the modulation index,

| ξ m (t) | \leq 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

i_{r} (t) = R P I (t) [1 + ξ m (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

σ_{n}^{2}

, 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

γ_{b} = {\bar{γ}}_{b} I^{2} = \frac{R^{2} {\bar{P}}^{2} ξ^{2} I^{2}}{σ_{n}^{2}},

where

{\bar{γ}}_{b} = \frac{R^{2} {\bar{P}}^{2} ξ^{2}}{σ_{n}^{2}}

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)]

f_{I} (I) = \frac{α β}{η} {(\frac{I}{η})}^{β - 1} \exp [- {(\frac{I}{η})}^{β}] \times {1 - \exp [- {(\frac{I}{η})}^{β}]}^{α - 1}, I \geq 0

and the corresponding cumulative distribution function (CDF) is given by [18, Eq. (8)]

F_{I} (I) = {1 - \exp [- {(\frac{I}{η})}^{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 ${\bar{P}}_{b}$ can be calculated by the conditional error rate $P_{b} (I)$ , i.e., the BER under AWGN [20], that is

{\bar{P}}_{b} = \int_{0}^{\infty} P_{b} (I) f_{I} (I) d I,

where

f_{I} (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

{\bar{P}}_{b} = - \int_{0}^{\infty} {P^{'}}_{b} (I) F_{I} (I) d I,

where

F_{I} (I)

is the CDF of the variable

I

and

{P^{'}}_{b} (I)

is the first order derivative of

P_{b} (I)

with respect to

I

.

3.1. BER of BPSK

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

P_{b, B P S K} (I) = \frac{1}{2} e r f c (\sqrt{γ_{b}}),

where

e r f c (\cdot)

is the complementary error function. The first order derivative of the conditional BER

P_{b, B P S K} (I)

can be obtained with the help of [32] as follows:

{P^{'}}_{b, B P S K} (I) = - \sqrt{\frac{{\bar{γ}}_{b}}{π}} \exp (- {\bar{γ}}_{b} I^{2}),

the detailed derivation process of

{P^{'}}_{b, B P S K} (I)

is provided in the appendix. Then, substituting Eqs. (5) and (9) into Eq. (7), the average BER for BPSK

{\bar{P}}_{b, B P S K}

can be written as

{\bar{P}}_{b, B P S K} = \int_{0}^{\infty} \sqrt{\frac{{\bar{γ}}_{b}}{π}} \exp (- {\bar{γ}}_{b} I^{2}) \times {1 - \exp [- {(\frac{I}{η})}^{β}]}^{α} d I,

if we perform the variable change

x = {\bar{γ}}_{b} I^{2}

, Eq. (10) takes the form

{\bar{P}}_{b, B P S K} = \frac{1}{2 \sqrt{π}} \int_{0}^{\infty} x^{- 1 / 2} \exp (- x) \times {1 - \exp [- {(\frac{\sqrt{x}}{η \sqrt{{\bar{γ}}_{b}}})}^{β}]}^{α} d x .

Equation (11) can be efficiently and accurately approximated by using the generalized Gauss- Laguerre quadrature function as [33] mentioned, $\int_{0}^{\infty} x^{β} e^{- x} f (x) d x = \sum_{i = 1}^{n} H_{i} f (x_{i})$ . Thus Eq. (11) can be expressed by a truncated series

{\bar{P}}_{b, B P S K} \approx \frac{1}{2 \sqrt{π}} \sum_{k = 1}^{n} H_{k} {1 - \exp [- {(\frac{\sqrt{x_{k}}}{η \sqrt{{\bar{γ}}_{b}}})}^{β}]}^{α},

where

x_{k}

is the

k

-th root of the generalized Laguerre polynomial

L_{n}^{(- 1 / 2)} (x)

and the weight

H_{k}

can be calculated by [34]

H_{k} = \frac{Γ (n + 1 / 2) x_{k}}{n! {(n + 1)}^{2} {[L_{n + 1}^{(- \frac{1}{2})} (x_{k})]}^{2}} .

3.2. BER of MPSK

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

P_{b, M P S K} = \frac{2}{\log_{2} M} Q (\sqrt{2 γ_{b} \log_{2} M} \sin \frac{π}{M}),

where

Q (\cdot)

is the Gaussian

Q

function defined as

Q (\cdot) = 0.5 e r f c (z / \sqrt{2})

[35]. The first order derivative of the conditional BER is

{P^{'}}_{b, M P S K} = - \frac{2 \sin (\frac{π}{M}) \sqrt{{\bar{γ}}_{b}}}{\sqrt{π \log_{2} M}} \exp (- {\bar{γ}}_{b} I^{2} \sin^{2} \frac{π}{M} \log_{2} M) .

the detailed derivation process of

{P^{'}}_{b, M P S K} (I)

is provided in the appendix. Substituting Eqs. (5) and (15) into Eq. (7), the average BER of MPSK modulation can be obtained as

{\bar{P}}_{b, M P S K} = \frac{2 \sin (\frac{π}{M}) \sqrt{{\bar{γ}}_{b}}}{\sqrt{π \log_{2} M}} \int_{0}^{\infty} \exp (- {\bar{γ}}_{b} I^{2} \sin^{2} \frac{π}{M} \log_{2} M) \times {1 - \exp [- {(\frac{I}{η})}^{β}]}^{α} d I,

performing the variable change

x = {\bar{γ}}_{b} \log_{2} M {(I \sin \frac{π}{M})}^{2}

, we get

{\bar{P}}_{b, M P S K} = \frac{1}{\sqrt{π} \log_{2} M} \int_{0}^{\infty} x^{- 1 / 2} \exp (- x) \times {1 - \exp [- {(\frac{\sqrt{x}}{η \sin (\frac{π}{M}) \sqrt{{\bar{γ}}_{b} \log_{2} M}})}^{β}]}^{α} d x,

using the generalized Gauss-Laguerre quadrature rule, we obtain

{\bar{P}}_{b, M P S K} \approx \frac{1}{\sqrt{π} \log_{2} M} \sum_{k = 1}^{n} H_{k} {1 - \exp [- {(\frac{\sqrt{x_{k}}}{η \sin (\frac{π}{M}) \sqrt{{\bar{γ}}_{b} \log_{2} M}})}^{β}]}^{α},

where

x_{k}

is the

k

-th root of the generalized Laguerre polynomial

L_{n}^{(- 1 / 2)} (x)

and

H_{k}

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]

P_{o u t} = P_{r} (γ_{b} \leq γ_{t h}) = P_{r} (I \leq \sqrt{\frac{γ_{t h}}{{\bar{γ}}_{b}}}),

P_{o u t} = \int_{0}^{\sqrt{\frac{γ_{t h}}{{\bar{γ}}_{b}}}} f_{I} (I) d I .

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

P_{o u t} = = F_{I} (\sqrt{\frac{γ_{t h}}{{\bar{γ}}_{b}}}) = {1 - \exp [- {(\frac{1}{η \sqrt{γ_{n}}})}^{β}]}^{α},

where

γ_{n} = \frac{{\bar{γ}}_{b}}{γ_{t h}}

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 $σ_{I}^{2}$ , but also by the ratio of aperture size ( $D$ ) to the atmospheric coherence radius ( $ρ_{0}$ ) as ( $D / ρ_{0}$ ), and/or the Rytov variance $σ_{R}^{2}$ [19]. The Rytov variance is defined as $σ_{R}^{2} = 1.23 C_{n}^{2} k^{7 / 6} L^{11 / 6}$ , where $C_{n}^{2}$ is the refractive-index structure constant and $L$ is the distance between the transmitter and receiver planes. $k = 2 π / λ$ 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 $10^{8}$ random numbers were generated in each case.

In Fig. 2, the average BERs of BPSK modulation are plotted against the SNR ${\bar{γ}}_{b}$ in the weak turbulence regime with two aperture sizes, i.e., $D = 3 m m$ and $D = 25 m m$ . In our study, a Gaussian beam with $λ = 780 n m$ propagating along a horizontal path with $L = 375 m$ was considered and $C_{n}^{2} = 2.1 \times 10^{- 14} m^{2 / 3}$ was used. In addition, the initial beam radius is $W_{0} = 1.13 c m$ . $σ_{R}^{2}$ is equal to 0.15 and the atmospheric coherence radius is $ρ_{0} = 18.89 m m$ . Thus, the aperture of $3 m m$ can behave as a point receiver since it is smaller than $ρ_{0}$ . It is easily seen from the figure that the analytical results (solid curve for $D = 3 m m$ and dashed curve for $D = 25 m m$ , 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 ( $D = 3 m m$ ) requires larger SNR compared with the aperture-averaged receiver ( $D = 25 m m$ ) to reach the same BER. For example, to achieve the average BER of $10^{- 9}$ , about 25.5 dB of SNR is needed for the point receiver ( $D = 3 m m$ ) while only 15.5 dB is needed for the aperture-averaged receiver ( $D = 25 m m$ ). 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 condition $σ_{R}^{2} = 0.15$ . Link distance $L = 375 m$ and coherence radius $ρ_{0} = 18.89 m m$ .

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Figure 3 presents the average BER results for moderate turbulence condition ( $σ_{R}^{2} = 1.35$ ). Four aperture sizes ( $3 m m, 25 m m, 60 m m, 80 m m$ ) are chosen and the parameters are obtained from the best PDF fitting in [18] for $L = 1225 m$ . 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 ( $D = 25 m m$ ) offers a performance gain of about 28 dB compared with the point receiver ( $D = 3 m m$ ) in terms of the SNR to reach the average BER of $10^{- 9}$ , 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 condition $σ_{R}^{2} = 1.35$ . Link distance $L = 1225 m$ and coherence radius $ρ_{0} = 9.27 m m$ .

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Figure 4 shows the average BER as a function of SNR under strong turbulence condition ( $σ_{R}^{2} = 19.2$ ). 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 radius $W_{0} = 1.59 c m$ . A refractive-index structure constant $C_{n}^{2} = 4.58 \times 10^{- 13} m^{2 / 3}$ and an atmospheric coherence radius $ρ_{0} = 2.94 m m$ were used. In this paper, three apertures of $1.8 m m$ , $5 m m$ and $13 m m$ 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 $10^{- 2}$ , $10^{- 4}$ and $10^{- 6}$ for $1.8 m m$ , $5 m m$ and $13 m m$ , 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 condition $σ_{R}^{2} = 19.2$ . Link distance $L = 1500 m$ and coherence radius $ρ_{0} = 2.94 m m$ .

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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 $D = 3 m m$ and $25 m m$ in weak turbulence ( $σ_{R}^{2} = 0.15$ ). Due to the long Monte Carlo simulation time, the average BER of about $10^{- 5}$ is 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 $3 m m$ to $25 m m$ , a performance gain of about 5 dB in SNR at the BER of $10^{- 5}$ 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 condition $σ_{R}^{2} = 0.15$ .

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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 ( $σ_{R}^{2} = 0.15$ ) with two aperture sizes $D = 3 m m$ and $25 m m$ have been presented. The parameters used in this figure are given in Table 1. It can be seen from the figure that for $D = 3 m m$ , the average BER of EW is a little bit larger than that of LN. For $D = 25 m m$ , 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 $0.3 \times 10^{- 4}$ . It can also be seen that when the receiver size increases from $3 m m$ to $25 m m$ , a smaller SNR is needed to achieve the same BER for these two models. For instance, to achieve the BER of $10^{- 6}$ , about 18 dB is needed for $D = 3 m m$ while 12 dB is needed for $D = 25 m m$ of the EW distribution; 15.5 dB of SNR is needed for $D = 3 m m$ while 11.5 dB is needed for $D = 25 m m$ of the LN distribution. The aperture-averaged receiver ( $D = 25 m m$ ) offers a performance gain of about 6 dB and 4 dB compared with the point receiver ( $D = 3 m m$ ) 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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Table 1. Parameters for the LN and EW distributions used in Fig. 6.

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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 ( $σ_{R}^{2} = 19.2$ ) with three aperture sizes, i.e., $D = 1.8 m m$ , $D = 5 m m$ and $D = 13 m m$ . 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 $0.3 \times 10^{- 2}$ . For the EW distribution, when SNR is equal to 40 dB, the average BERs are $10^{- 2}$ , $10^{- 4}$ and $10^{- 6}$ for $1.8 m m$ , $5 m m$ and $13 m m$ , 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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Table 2. Parameters for the GG and EW distributions used in Fig. 7.

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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 $D = 3 m m$ (solid curve) and $25 m m$ (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 $σ_{R}^{2}$ 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 $10^{- 3}$ in weak turbulence ( $σ_{R}^{2} = 0.15$ ) while it is larger than $10^{- 1}$ in moderate turbulence ( $σ_{R}^{2} = 1.30, 1.35$ ) for $D = 3 m m$ . 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 ( $D = 25 m m$ ) compared with the point receiver ( $D = 3 m m$ ) in terms of the SNR to achieve the outage probability of $10^{- 5}$ for weak turbulence ( $σ_{R}^{2} = 0.15$ ).

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 $P_{b, B P S K} (I)$ with respect to $I$

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

P_{b, B P S K} (I) = \frac{1}{2} e r f c (\sqrt{γ_{b}}),

where

γ_{b} = {\bar{γ}}_{b} I^{2}

[29].

The first order derivative of $P_{b} (I)$ with respect to $I$ can be obtained as follows

{P^{'}}_{b, B P S K} (I) = \frac{d P_{b, B P S K} (I)}{d I} = \frac{1}{2} \frac{d}{d I} e r f c (\sqrt{{\bar{γ}}_{b}} I),

according to the method of composite function derivation provided in [32], Eq. (23) can be expressed as

{P^{'}}_{b, B P S K} (I) = \frac{1}{2} \frac{d [e r f c (\sqrt{{\bar{γ}}_{b}} I)]}{d (\sqrt{{\bar{γ}}_{b}} I)} \cdot \frac{d (\sqrt{{\bar{γ}}_{b}} I)}{d I},

using

\frac{d}{d z} e r f c (z) = - \frac{2}{\sqrt{π}} e^{- z^{2}}

[32] and

\frac{d (\sqrt{{\bar{γ}}_{b}} I)}{d I} = \sqrt{{\bar{γ}}_{b}}

, Eq. (24) can be written as

{P^{'}}_{b, B P S K} (I) = - \frac{\sqrt{{\bar{γ}}_{b}}}{\sqrt{π}} \exp (- {\bar{γ}}_{b} I^{2}) .

Thus, the first order derivate of $P_{b, B P S K} (I)$ with respect to $I$ is ${P^{'}}_{b, B P S K} (I) = - \frac{\sqrt{{\bar{γ}}_{b}}}{\sqrt{π}} \exp (- {\bar{γ}}_{b} I^{2})$ .

2. The first order derivative of $P_{b, M P S K} (I)$ with respect to $I$

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

P_{b, M P S K} = \frac{2}{\log_{2} M} Q (\sqrt{2 γ_{b} \log_{2} M} \sin \frac{π}{M}),

where

Q (\cdot)

is the Gaussian

Q

function defined as

Q (\cdot) = 0.5 e r f c (z / \sqrt{2})

[35] and

γ_{b} = {\bar{γ}}_{b} I^{2}

[29]. Thus

P_{b, M P S K} = \frac{1}{\log_{2} M} e r f c (\sqrt{{\bar{γ}}_{b} \log_{2} M} I \sin \frac{π}{M}) .

The first order derivative of $P_{b, M P S K} (I)$ with respect to $I$ can be obtained as follows

{P^{'}}_{b, M P S K} = \frac{d P_{b, M P S K}}{d I} = \frac{1}{\log_{2} M} \frac{d}{d I} e r f c (\sqrt{{\bar{γ}}_{b} \log_{2} M} I \sin \frac{π}{M}),

according to the method of composite function derivation provided in [32], and performing the variable change

z = \sqrt{{\bar{γ}}_{b} \log_{2} M} I \sin \frac{π}{M}

, Eq. (28) can be written as

{P^{'}}_{b, M P S K} = \frac{1}{\log_{2} M} \frac{d [e r f c (z)]}{d z} \cdot \frac{d z}{d I},

using

\frac{d}{d z} e r f c (z) = - \frac{2}{\sqrt{π}} e^{- z^{2}}

[32] and

\frac{d z}{d I} = \sqrt{{\bar{γ}}_{b} \log_{2} M} \sin \frac{π}{M}

, Eq. (29) can be written as

{P^{'}}_{b, M P S K} = - \frac{2 \sqrt{{\bar{γ}}_{b}} \sin \frac{π}{M}}{\sqrt{π \log_{2} M}} \exp (- {\bar{γ}}_{b} I^{2} \sin^{2} \frac{π}{M} \log_{2} M) .

Thus, the first order derivate of $P_{b, M P S K} (I)$ with respect to $I$ is ${P^{'}}_{b, M P S K} = - \frac{2 \sqrt{{\bar{γ}}_{b}} \sin \frac{π}{M}}{\sqrt{π \log_{2} M}} \exp (- {\bar{γ}}_{b} I^{2} \sin^{2} \frac{π}{M} \log_{2} M)$ .

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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Aperture D(mm)	Conditions		LN	EW
Aperture D(mm)	$σ_{R}^{2}$	$σ_{I f i t}^{2}$	$σ_{\ln I}^{2}$	$α_{f i t}$	$β_{f i t}$	$η_{f i t}$
3	0.15	0.078	0.08	5.24	1.78	0.62
25	0.15	0.011	0.011	3.75	5.44	0.89

Aperture D(mm)	Conditions		GG		EW
Aperture D(mm)	$σ_{R}^{2}$	$σ_{I f i t}^{2}$	$α_{f i t}$	$β_{f i t}$	$α_{f i t}$	$β_{f i t}$	$η_{f i t}$
1.8	19.2	3.37	0.88	0.88	2.69	0.43	0.178
5		3.27	0.96	0.96	20.27	0.29	0.007
13		2.18	1.28	1.28	39.11	0.29	0.005

Aperture D(mm)	Conditions		LN	EW
Aperture D(mm)	$σ_{R}^{2}$	$σ_{I f i t}^{2}$	$σ_{\ln I}^{2}$	$α_{f i t}$	$β_{f i t}$	$η_{f i t}$
3	0.15	0.078	0.08	5.24	1.78	0.62
25	0.15	0.011	0.011	3.75	5.44	0.89

Aperture D(mm)	Conditions		GG		EW
Aperture D(mm)	$σ_{R}^{2}$	$σ_{I f i t}^{2}$	$α_{f i t}$	$β_{f i t}$	$α_{f i t}$	$β_{f i t}$	$η_{f i t}$
1.8	19.2	3.37	0.88	0.88	2.69	0.43	0.178
5		3.27	0.96	0.96	20.27	0.29	0.007
13		2.18	1.28	1.28	39.11	0.29	0.005

Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels

Abstract

1. Introduction

2. System and channel models

2.1 System model

2.2. Atmospheric turbulence model

3. Error rate analysis of subcarrier coherent modulations

3.1. BER of BPSK

3.2. BER of MPSK

4. Outage probability

5. Results and discussion

6. Conclusions

Appendix

1. The first order derivative of $P_{b, B P S K} (I)$ with respect to $I$

2. The first order derivative of $P_{b, M P S K} (I)$ with respect to $I$

Acknowledgments

References and links

Cited By

Figures (8)

Tables (2)

Equations (30)

Optics Express

Abstract

1. Introduction

2. System and channel models

2.1 System model

2.2. Atmospheric turbulence model

3. Error rate analysis of subcarrier coherent modulations

3.1. BER of BPSK

3.2. BER of MPSK

4. Outage probability

5. Results and discussion

6. Conclusions

Appendix

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

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

Acknowledgments

References and links

Cited By

Figures (8)

Tables (2)

Equations (30)

Optics Express

1. The first order derivative of $P_{b, B P S K} (I)$ with respect to $I$

2. The first order derivative of $P_{b, M P S K} (I)$ with respect to $I$