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Performance analysis of a laser satellite-communication system with a three-layer altitude spectrum over weak-to-strong turbulence
Optik 148 (2017) 283–292
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Original research article
Performance analysis of a laser satellite-communication system with a three-layer altitude spectrum over weak-to-strong turbulence Peng Yue, Le Wu ∗ , Xiang Yi, Zhangjie Fu State Key Laboratory on Integrated Services Networks, Xidian University, Xi’an, China
a r t i c l e
i n f o
Article history: Received 29 January 2016 Received in revised form 9 August 2017 Accepted 11 August 2017 Keywords: Atmospheric turbulence Laser-satellite communication Scintillation index Three-layer altitude spectrum Gamma–Gamma distribution
a b s t r a c t In this article, the downlink bit-error-rate (BER) performance of scintillation models for a laser satellite communication system influenced by weak-to-strong turbulence on a slant path is investigated, which take optical wavelength, zenith angle, terrestrial turbulence strength and modulation schemes into account. To describe the properties of atmospheric turbulence which vary with the altitude between a satellite station and a ground station, a three-layer altitude spectrum is used. By adopting the extended Rytov theory valid in weak-to-strong turbulence, scintillation models of an unbounded plane wave for downlink path and spherical wave for uplink path are derived, respectively. The optical fading channel influenced by weak-to-strong turbulence is modeled by Gamma–Gamma distribution. The numeric results show that when the system is under strong terrestrial turbulence conditions or at large zenith angles, disparities in BER performance among different kinds of level of pulse position modulation (PPM) schemes are quite small. Under real circumstances, a longer optical wavelength and a suitable zenith angle range can be selected to reduce turbulence effects for the satellite communication system through weak-to-strong turbulence. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction A great many studies of impacts of atmospheric turbulence on horizontal or slant propagation paths for Free Space Optical (FSO) communication system have been conducted [1–5]. Statistics of atmospheric turbulence is often described by Kolmogorov spectrum [6–8]. However, recent studies and experiments show that Kolmogorov spectrum cannot properly describe the turbulence statistical characteristics in a vertical or slant path in the laser satellite communication system, and turbulence of stratosphere and upper troposphere along the propagation path between a satellite station and a ground station exhibits non-Kolmogorov statistical properties which change with the altitude [9–12]. Therefore, authors in Ref. [12] developed a two-layer turbulence spectrum model, in which numerical values 11/3 and 5, respectively, correspond to the spectral exponents of the troposphere and stratosphere in two-layer spectrum. However, Arkadi Zilberman proposed a three-layer altitude spectrum which has been proved matching the experiment data better [8,11]. In Ref. [8], with a three-layer altitude spectrum, the bit-error rate (BER) performance analysis of the optical communication links under weak turbulence conditions is developed. Meanwhile, a log-normal (LN) model is extensively used to
∗ Corresponding author. Tel.: +86 15291915621. E-mail addresses: [email protected] (P. Yue), [email protected] (L. Wu). http://dx.doi.org/10.1016/j.ijleo.2017.08.074 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
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Fig. 1. The power law exponent ˛(h) as a function of the altitude h.
describe the weak turbulent fluctuation channel [1,13]. In effect, when zenith angles exceed 60◦ , optical links will experience moderate-to-strong refractive fluctuations [14,15]. In this situation, Gamma–Gamma distribution is being adopted as it is applicable to weak-to-strong turbulence. In this article, based on a three-layer altitude spectrum and an extended Rytov theory, theoretical expressions governing the scintillation index are derived, respectively, when the unbounded plane wave and spherical wave propagates through weak-to-strong turbulence along a slant path. Ignoring inner-scale and outer-scale effects, we investigate the BER performance of a laser satellite communication system with some restrictions of parameters such as the turbulent strength, zenith angles and wavelengths. 2. The three-layer altitude spectrum The three-layer altitude spectrum model ˚n (, ˛, h) [11,16] is defined by
k ˛/2−11/6
˚n (, ˛, h) = 0.033
L
Cn2 (h) −˛ ,
(1)
where is the spatial frequency, ˛ is the spectral exponent varying with the altitude h, k is the electromagnetic wave number which can be written as k = 2/ at the wavelength of , Cn2 (h) is the refractive index structure parameter, and L is the propagation distance expressed as L = (H3 − H0 ) sec () in meters(m). In this formula, H3 is the altitude of the satellite or the space station, H0 is the height of a ground station. is the zenith angle. Cn2 (h) varies with h and gets the unit of m−2/3 , and the H − V5/7 model [1,11] is used to describe Cn2 (h)
w 2
Cn2 (h) = 0.00594 −16
+2.7 × 10
27
10
10−5 h
h exp − 1500
exp −
h 1000
h + A0 exp − 100
,
(2)
where w denotes the wind speed and has the units of m/s (generally w = 21m/s). A0 is the terrestrial refractive index structure parameter in m−2/3 . ˛ (h) takes the form [11]
˛ (h) =
˛1
1 + h/H1
b1 +
b1
b1 ·
˛2 · h/H1 1 + h/H1
1
1 + h/H2
b2 +
b2
b2 ,
˛3 · h/H2 1 + h/H2
(3)
where ˛1 = 11/3, ˛2 = 10/3, ˛3 = 5 represent the numerical values of the spectral exponents of the surface layer, troposphere and upper stratosphere above them, respectively. H1 and H2 are boundary layer heights. b1 and b2 are parameters of flatness between layers. There are no exact values for H1 , H2 , b1 and b2 . In the following parts, we set those values as [11], H1 = 2km, H2 = 8km, b1 = 8 and b2 = 10. The curve of ˛ (h) is showed in Fig. 1. It is observed in Fig. 1 that for the atmosphere under the altitude of 2 km, the spectral exponent of the three-layer altitude spectrum is 11/3, which is equal to the Kolmogorov spectrum. This means that the three-layer altitude spectrum presents the Kolmogorov properties for near-ground troposphere. For the upper troposphere with the altitude between 2 and 8 km, the spectral exponent of the three-layer altitude spectrum falls to 10/3, representing the non-Kolmogorov characteristics. Then the spectral exponent goes to 5 for the stratosphere with the altitude higher than 8 km, also showing the non-Kolmogorov properties in range. In Fig. 2 we also see that the flatness factors b1 and b2 make the spectral exponent curve transit smoothly between layers.
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Fig. 2. BER versus the zenith angle with different terrestrial turbulent strengths and modulation schemes for the wavelength of 1550 nm.
3. Scintillation index models in a weak-to-strong turbulence region Under weak turbulence circumstances, an unbounded plane wave is applied to approximating a downlink propagation beam, and for an uplink path, we consider it a spherical beam model. In this situation, expressions of the scintillation index for both downlink and uplink path are given by Ref. [16]:
⎧ 7 5 11 ⎪ ⎪ 1.303 ⎪ ⎪ d2 = − d k 6 (H3 − H0 ) 6 sec 6 ⎨ 2 ˛/2 cos ˛/4 ,
H3 ⎪ ⎪ ⎪ 2 ˛/2−1 ⎪ Cn (h) dh ⎩ d = Re
(4)
H0
⎧ 7 5 11 ⎪ ⎪ 1.303 ⎪ 2 =− 6 6 6 ⎪ k − H ) (H sec u 3 0 u ⎨ 2 ˛/2 cos ˛/4 ,
H3 ⎪ ⎪ ˛/2−1 ⎪ ⎪ Cn2 (h) ˛/2−1 (1 − ) dh ⎩ u = Re
(5)
H0
where d2 and u2 denote the downlink and uplink scintillation index, respectively. is the normalized variable [1] written as = (h − H0 ) / (H3 − H0 ). is the zenith angle. Re ( ) takes the real part of the expressions. ( ) is the Gamma function [17]. Cn2 (h) is the refractive index structure parameter provided by Eq. (2). ˛ (h) is the spectral exponent described by Eq. (3). In a strong turbulence regime, the asymptotic theory predicts that scintillation I2 (L) for an unbounded plane wave or a spherical wave can be expressed in the form [18,19]
H3
∞
I2 (L) = 1 + 322 k2 sec
× exp
−
1
DS 0
H0
L k
w ,
0
˚n (, ˛, h) sin2
d
L2 2k
(6)
ddh,
where is the normalized distance variable, and w , is defined by
1 − , < , w , =
(7)
1 − , >
¯ = 1 − . In Eq. (6), the exponential function is a low pass space filter defined where is the optical beam parameter, and by the phase structure function DS ( , ˛), and is the distance of two observation sites. Owing to L2 /k «1, a geometrical optics approximation can be used [1]. At the same time, the phase structure function can be reduced to the wave structure function Dpl ( , ˛) Ds ( , ˛)
∼ = Dpl ( , ˛)
H3
= 82 k2 sec
∞
˚n (, ˛, h) [1 − J0 ( )] ddh H0
0
.
(8)
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Replacing ˚n (, ˛, h) with a three-layer altitude spectrum (Eq. (1)), and using Lw , /k instead of , we get
DS
L
L w , , ˛ ∼ w , , ˛ = Dpl k k
H3
= 2.606k1/6 L11/6 sec H0
k (2−˛)/2 −˛ ˛−2 Cn2 (h) ( ) 2 ˛ . L
(9)
˛−2 −˛/2 dh × w , ˛/2
By substituting Eq. (8) into Eq. (6), the scintillation index for an unbounded plane wave can be written as
⎧ H3 ∞ k ˛/2−11/6 ⎪ 2 ⎪ = 1 + 2.606 sec Cn2 (h) L2 2 5−˛ ⎪ (L) I ⎪ L ⎪ ⎪ H0 0 ⎪ ⎪
1−˛/2 ⎪ ⎪ k ⎪ 2 ˛−2 ˛−2 ⎪ × exp −E (˛) d ddh − +1 ⎪ ⎪ ˛−1 L ⎪ ⎪ ⎪ ⎪ 2(˛−4)/(˛−2) ⎪ ⎪ ⎪ = 1 + E (˛) × d2 ⎪ ⎪ ⎨ , 4 (6 − ˛) / (˛ − 2) ˛/2 cos ˛/4 [E (˛)](˛−6)/(˛−2) ⎪ ⎪ E − = (˛) ⎪ ⎪ (˛ − 2) d ⎪ ⎪ ⎪ ⎪ (˛−6)/(˛−2) H3 ⎪ ⎪ ⎪ ⎪ 2 ˛−4 ⎪ × Cn (h) dh − +1 ⎪ ⎪ ˛−1 ⎪ H0 ⎪ ⎪ H ⎪ ⎪ 3 ⎪ 22−˛ ˛ −˛/2 cos ˛/4 ⎪ ⎪ − Cn2 (h) dh E = ⎩ (˛) d
(10)
H0
∼ L2 4 2 /4k2 and exp − L2 2 /k = ∼ 1 are used. where the geometrical optics approximation sin2 L2 /2k = A conventional Rytov approximation is limited to weak turbulent fluctuations, which cannot properly characterize the coherence loss caused by the increase of propagation distances and the turbulence strength. Adopting an extended Rytov theory, we can deduce the scintillation index models with large-scale and small-scale turbulence effects considered through weak-to-strong turbulence. With inner-scale and outer-scale effects neglected, the space filter functions of large-scale GX () and small-scale GY () are expressed as [1]
GX () = exp
GY () =
−
2
,
X2
(11)
11/3 2 + Y2
11/6 ,
(12)
where X and Y , respectively, represent the large-scale and small-scale spatial cut-off frequencies, both of which are used to cut off the effects caused by turbulence whose scales are between a spatial coherence radius 0 and a scatter disk L/ (k 0 ). 2 Along a downlink propagation path, scintillation index models of the log-irradiance for large-scale ln and small-scale X 2 ln Y take the form as follows [1]
2 ln X
H3
= 8 k sec
˚n (, ˛, h) GX () 1 − cos H0
2 ln Y
∞
2 2
H3
0
2 2
= 8 k sec
∞
˚n (, ˛, h) GY () 1 − cos
H0
0
2 h sec k 2 h sec k
ddh,
(13)
ddh.
(14)
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By substituting the space filter function Eqs. (11) and (12) into Eqs. (13) and (14), respectively, and replacing ˚n (, ˛, h) with a three-layer altitude spectrum, scintillation index models of the log-irradiance for large-scale and small-scale turbulence become
⎧ 2 L ∞ 2 2 z ⎪ ˛/2−11/6 1−˛ 2 2 2 ⎪ Cn (h)(k/L) exp − 2 ddz ⎪ ⎪ ln X = 1.303k k ⎪ X ⎪ 0 0 ⎪ ⎪ H ⎪ 3 ⎪ 3−˛/2 3 − ˛/2 ˛/2 cos ˛/4 ⎪ ⎪ ⎨ Cn2 (h) 2 dh d2 LX2 /k =− d
⎪ ⎪ = EX (˛) × d2 × X 3−˛/2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 − ˛/2 ˛/2 cos ˛/4 ⎪ ⎪ ⎩ EX (˛) = − d
(15)
,
H0
H3
Cn2 (h) 2 dh
H0
⎧ L ∞ 11/3 ⎪ ˛/2−11/6 1−˛ 2 2 ⎪ = 2.606k Cn2 (h)(k/L) ⎪ ln Y 11/6 ddz ⎪ 2 + 2 ⎪ 0 0 ⎪ Y ⎪ ⎪ ⎪ ⎪ 2 ˛/2 cos ˛/4 17/6 − ˛/2 ˛/2 − 1 ⎪ ⎪ =− ⎨ (11/6) d
⎪ 1−˛/2 ⎪ = EY (˛) × d2 × Y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ˛/2 17/6 − ˛/2 ˛/2 − 1 cos ˛/4 ⎪ ⎪ ⎩ EY (˛) = − d (11/6)
H3
Cn2 (h)dh
d2 (LY2 /k)
1−˛/2
H0
(16)
,
H3
Cn2 (h)dh
H0
where X = LX2 /k and Y = LY2 /k are the non-dimension spatial cut-off frequencies of large-scale turbulence and small-
scale turbulence, respectively. Geometrical optics approximations are used as 1 − cos 2 z/k ∼ = 0.5 2 z/k 2 ∼ 1, > Y >> k/L. 1 − cos z/k =
2
, « X , and
To determine X and Y , we should utilize the scintillation index in the weak turbulence and saturated region. The expression of the scintillation index in a weak and saturated strong turbulence region is deduced from Eq. (10)
I2
∼ =
2 2 , + ln ln X Y
2 2(˛−4)/(˛−2)
2 = 1 + E (˛) × d 1 + 2ln X
d2 << 1 ,
.
(17)
d2 >> 1
And the non-dimension spatial cut-off frequencies of large-scale turbulence and small-scale turbulence X and Y are, respectively, described as
X =
2 L kl
L 2 = k X k
X
L
=
1 c1 + c2 L/k 02
∼ =
⎧ 1 ⎪ ⎪ ⎨ c1 , k 2 ⎪ ⎪ ⎩ c L0 , 2
L 1 2
L Y = Y2 = k k
lY
⎧ ⎪ ⎨ c3 ,
c4 L ∼ = c3 + = c L k 02 ⎪ ⎩ 42, k 0
L k 02 L k 02
L k 02 L k 02
<< 1 ,
(18)
>> 1
<< 1 ,
(19)
>> 1
where lX denotes the lower bound of the largest scales, and lY represents the upperbound of the smallest scales. c1 , c2 , L/k, lY ∼ L/k. In a strong turbulence regime, c3 , c4 are constants to be determined. In a weak turbulence regime, lX ∼ = = lX ∼ = L/ (k 0 ), lY ∼ = 0 . According to Eq. (17), in a weak turbulence regime, scintillation index models of the log-irradiance for large-scale and small-scale turbulence is set as
2 = 0.49d2 ln X 2 ln = 0.51d2 Y
.
(20)
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Since the second of Eq. (17) shows the limit of the small-scale intensity scintillation in a saturated region, 2expression namely Y2 = exp ln − 1 → 1, for a moderate-to-strong turbulence regime, we have Y
2(˛−4)/(˛−2)
2 ln = 0.5E (˛) × d2 X
.
(21)
2 ln → ln 2 Y
Combining Eqs. (15) and (16) with Eqs. (20) and (21), constants c1 , c2 , c3 , c4 in Eqs. (18) and (19) can be determined. Hence, X and Y are derived X =
1 c1 + c2 L/k 02
Y = c3 +
c4 L k 02
=
=
0.49 EX (˛)
1/(˛/2−3)
0.51 1/(1−˛/2) EY (˛)
+
+ 0.5
2(˛−4)/(˛−2)−1 E (˛) × d2 EX (˛)
1/(˛/2−3) −1
,
(22)
1/(1−˛/2)
ln 2
(23)
.
EY (˛) × d2
If we insert Eqs. (22) and (23) into Eqs. (15) and (16), respectively, the large-scale and small-scale scintillation index models of the log-irradiance for a downlink propagation path become 2 ln = EX (˛) × d2 × X 3−˛/2 X
= EX (˛) × d2 ×
0.49 EX (˛)
1/(˛/2−3)
+
E (˛)
2(˛−4)/(˛−2)−1 1/(˛/2−3)
˛/2−3 ,
(24)
× d2
2EX (˛)
1−˛/2
2 ln = EY (˛) × d2 × Y Y
=
EY (˛) × d2
×
0.51 1/(1−˛/2) EY (˛)
+
ln 2
1/(1−˛/2) 1−˛/2 .
(25)
EY (˛) × d2
Merging Eqs. (24) and (25), the scintillation index model of a downlink path through weak-to-strong turbulence is represented as
2 2 2 I,pl + ln −1 (L) = exp ln X Y
1−˛/2
= exp EX (˛) × d2 × X 3−˛/2 + EY (˛) × d2 × Y
−1
(26)
,
where ˛ is the spectral exponent described by Eq. (3). d2 denotes the downlink scintillation index presented by Eq. (4). X is the non-dimension spatial cut-off frequencies of large-scale written as Eq. (22). In a similar way, the scintillation index model of an uplink path through weak-to-strong turbulence is given by
1−˛/2
2 I,sp (L) = exp EX (˛) × u2 × X 3−˛/2 + EY (˛) × u2 × Y
− 1,
(27)
where u2 denotes the uplink scintillation index presented by Eq. (5). Y is the non-dimension spatial cut-off frequencies of small-scale written as Eq. (23). 4. BER analysis In a wireless laser communication system with the Intensity Modulation/Direct Detection (IM/DD) technology, the received optical beam is converted to an electronic signal through a photo diode. Shot noise and thermal noise are introduced by the receiver, and the noise obeys a Gaussian distribution with the mean zero and the variance. A statistical channel model is defined as follows [20,21] y = RIx + n,
(28)
where x denotes the modulated signal. y denotes the received signal. R represents the photoelectric responsivity of the positive intrinsic-negative (PIN) photodetector. I is the channel fading. n denotes additive Gaussian white noise. The on-off keying (OOK) scheme has been extensively used to represent the weak turbulence channel. The conditional BER of OOK scheme can be written as BEROOK (I) =
1 erfc 2
SNR
√ I , 2 2
where SNR is the signal-to-noise ratio, and erfc ( ) is complementary error function [17].
(29)
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Table 1 Parameters for the numerical analysis. Symbol
Description
Value
H3 H2 H1 H0 A0
Height of the satellite station Boundary layer heights between the surface layer and the troposphere Boundary layer heights between the troposphere and the stratosphere Height of the ground station Wavelength Zenith angle Terrestrial refractive index structure parameter for weak turbulence Terrestrial refractive index structure parameter for moderate turbulence Terrestrial refractive index structure parameter for strong turbulence
35800 km 8 km 2 km 0 km 1550 nm, 1310 nm, 850 nm 0◦ –80◦ 1.7 × 10 − 14 m−2/3 3.0 × 10 − 13 m−2/3 1.7 × 10 − 12 m−2/3
Compared with OOK, M-level pulse position modulation (M-PPM) requires a lower average transmitted power. In a MPPM scheme, there is no thresholds at the receiver. Moreover, to a certain degree, M-PPM can resist background noise. In the absence of turbulence, the conditional BER of an M-PPM system can be written as [22]
SNR
M BERM−PPM (I) ≈ erfc 4
Mlog2 M 4
I
(30)
.
In the study of atmospheric turbulence, many density functions are adopted to describe the scintillation, such as lognormal (LN) distribution [1] which is used in a weak turbulence region, and K-distribution [23] is commonly used in a strong turbulence region. However, Gamma–Gamma distribution [2,24] is a model that can describe the probability density function (PDF) of the irradiance over weak-to-strong turbulence. The PDF of Gamma–Gamma distribution takes the form
⎧ (a+b)/2 2(ab) (a+b)/2−1 ⎪ ⎪ Ka−b 2 abI , I > 0 ⎨ fI (I, a, b) = (a) (b) I ⎪ 1 ⎪ ⎩a = 2 = X
1
2 exp ln −1 X
,
b=
1 Y2
=
1
,
(31)
2 exp ln −1 Y
where Kv ( ) is the second kind of modified Bessel function in v-order [17], and a and b denote the large-scale and small-scale turbulence cells. Hence, the average BER for both OOK and M-PPM schemes can be written as
⎧ ∞ SNR ⎪ 1 ⎪ ⎪ BER = erfc √ I × fI (I)dI ⎪ ⎪ OOK 2 2 2 ⎨ 0 , ∞ ⎪ ⎪ SNR Mlog2 M M ⎪ ⎪ BER I × fI (I)dI = erfc ⎪ M−PPM ⎩ 4 4
(32)
0
where fI (I) denotes the PDF of Gamma–Gamma distribution, and M is the level of a PPM modulation scheme. 5. Numeric results In this section, using the previously derived formula (Eq. (32)), we estimate the BER performance of a laser satellite communication system on a downlink propagation path employing the M-PPM schemes over a Gamma–Gamma distribution channel. In a subsequent section, we make a numerical analysis of downlink BER performance under various conditions. The parameters used in the numerical analyses are listed in Table 1. In Figs. 2–4, downlink BER performance for OOK, 2-PPM, 4-PPM, and 8-PPM schemes are presented as a function of the zenith angle. Here, we make a comparison among three kinds of terrestrial turbulent strength at different wavelengths ( = 850nm, = 1310nm, = 1550nm). SNR is set as 20 dB. Figs. 2–4 show that the BER performance for OOK scheme is equal to 2-PPM in value because they have the same BER formula. As it is shown in Figs. 2–4, BER rises with the increase of the zenith angle. We show that the BER increases slowly until the zenith angle approach 60◦ , and then increases rapidly as the zenith angle continue to increase. This appears for various wavelengths. For zenith angle larger than 60◦ , the effect of atmospheric turbulence reinforces, the BER performance severely degraded. In an actual system, we can select a suitable zenith angle range to help improve the system reliability. It is also observed from Figs. 2–4 that for weak, moderate and strong terrestrial turbulent strength, when zenith angle exceeds 60◦ , the disparities in these three kinds of level of PPM schemes decrease gradually with the increase of the zenith angle. But when the zenith angle is under 60◦ , a higher modulation level M of PPM scheme shows a better BER performance. However, for strong terrestrial turbulent strength, differences of BER performance among those PPM schemes are quite small. It indicates that at large zenith angles or under the condition of strong terrestrial turbulence, the improvement of the BER performance is not obvious simply by increasing the modulation
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Fig. 3. BER versus the zenith angle with different terrestrial turbulent strengths and modulation schemes for the wavelength of 1310 nm.
Fig. 4. BER versus the zenith angle with different terrestrial turbulent strengths and modulation schemes for the wavelength of 850 nm.
Fig. 5. BER versus SNR with different terrestrial turbulent strengths and modulation schemes for the wavelength of 1550 nm.
level M. Furthermore, numeric results obtained from Figs. 2–4 show that BER rises with the decrease of the wavelength for different terrestrial turbulence strengths. Although BER performance of a long optic wavelength is better than that of a short one, considering the limitation of the power of a laser device, an appropriate wavelength should be selected in the actual laser communication system. Downlink BER performance for OOK, 2-PPM, 4-PPM and 8-PPM schemes under different terrestrial turbulent strength conditions are given in Figs. 5–7, as a function of SNR. Here, we make a comparison among three kinds of wavelengths ( = 850nm, = 1310nm, = 1550nm). The zenith angle takes = 0◦ . As it is shown in Figs. 5–7, no matter which kind of wavelength we choose, there is a rapid decline in the differences in BER among these three kinds of PPM schemes at a certain SNR, when the optical signal go through from weak-to-strong terrestrial turbulence. Furthermore, it is found that for a certain modulation level of PPM schemes, a longer wavelength helps us get a lower BER. With respect to the 8-PPM scheme, for example, when we set SNR as 12 dB for weak terrestrial turbulence, Fig. 5 shows that the value of BER can reach nearly 10−12 at 1550 nm, but in Fig. 7 for the wavelength at 850 nm, the value of BER only gets 10−8 . Moreover, for both weak and moderate terrestrial turbulent strength, 8-PPM gets the minimum of BER among these three kinds of level of PPM schemes. In other words, under weak and moderate terrestrial turbulent strength conditions, a higher modulation level M of PPM scheme can help improve the BER performance. However, when the optical beam propagates through strong terrestrial turbulence on a slant path, the improvement of BER performance is not notable just by increasing the modulation level M.
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Fig. 6. BER versus SNR with different terrestrial turbulent strengths and modulation schemes for the wavelength of 1310 nm.
Fig. 7. BER versus SNR with different terrestrial turbulent strengths and modulation schemes for the wavelength of 850 nm.
It indicates that the terrestrial turbulent strength is the key factor that affects the BER performance of this communication system. 6. Conclusion In this article, based on a three-layer altitude spectrum and an extended Rytov theory, the uplink and downlink scintillation index models in weak-to-strong turbulence are derived. Then, using Gamma–Gamma distribution, we theoretically investigate the BER performance of the different (OOK and M-PPM) schemes for a satellite communication system through weak-to-strong turbulence. In conclusion, M-PPM shows a better BER performance than OOK in the same condition. Under real circumstances, a longer optical wavelength and a suitable zenith angle range can be selected to reduce turbulence effects. What is more, when the zenith angle falls below 60◦ and the system is under the condition of weak and moderate terrestrial turbulent strength, PPM scheme of a higher modulation level M shows a better BER performance. However, at large zenith angles or under the condition of strong terrestrial turbulence, the improvement of the BER performance is not notable simply by increasing the modulation level M of PPM schemes. It indicates that the terrestrial turbulent strength is the key factor that affects the BER performance of the laser satellite communication system. Acknowledgments This work was supported by the National Nature Science Foundation of China grant No. 60902038, the 111 project under Grant No. B08038, the State Scholarship Fund under Grant 201406965012, and the Fundamental Research Funds for the Central Universities No. K50511010019. Moreover, Dr. Yunfei Hou who is with the Department of Computer Science and Engineering, State University of New York at Buffalo also contributes to this article. References [1] L.C. Andrews, R.L. Phillips, Laser Beam Propagation Through Random Media, SPIE Press, Bellingham, WA, 2005. [2] G.K. Varotsos, H.E. Nistazakis, C.K. Volos, G.S. Tombras, FSO links with diversity pointing errors and temporal broadening of the pulses over weak-to-strong atmospheric turbulence channels, Optik (2015). [3] A.N. Stassinakis, H.E. Nistazakis, K.P. Peppas, G.S. Tombras, Improving the availability of terrestrial FSO links over log normal atmospheric turbulence channels using dispersive chirped Gaussian pulses, Opt. Laser Technol. 54 (2013) 329–334. [4] G.K. Varotsos, A.N. Stassinakis, H.E. Nistazakis, A.D. Tsigopoulos, K.P. Peppas, C.J. Aidinis, G.S. Tombras, Probability of fade estimation for FSO links with time dispersion and turbulence modeled with the gamma-gamma or the IK distribution, Optik 125 (2014) 7191–7197. [5] A. Vats, H. Kaushal, Analysis of free space optical link in turbulent atmosphere, Optik 125 (2014) 2776–2779.
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[6] H. Guo, B. Luo, Y. Ren, S. Zhao, A. Dhang, Influence of beam wander on uplink of ground-to-satellite laser communication and optimization for transmitter beam radius, Opt. Lett. 35 (2010) 1977–1979. [7] H.G. Sandalidis, Performance of a laser earth-to-satellite link over turbulence and beam wander using the modulated gamma-gamma irradiance distribution, Appl. Opt. 50 (2011) 952–961. [8] A. Zilberman, E. Golbraikh, N.S. Kopeika, Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence, Opt. Commun. 283 (2010) 1229–1235. [9] A.S. Gurvich, M.S. Belen’kii, Influence of stratospheric turbulence on infrared imaging, JOSA A 12 (1995) 2517–2522. [10] B.E. Stribling, B.M. Welsh, M.C. Roggemann, Optical propagation in non-Kolmogorov atmospheric turbulence, in: SPIE’s Symposium on OE/Aerospace Sensing and Dual Use Photonics, Int. Soc. Opt. Photonics (1995) 181–196. [11] A. Zilberman, E. Golbraikh, N.S. Kopeika, Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model, Appl. Opt. 47 (2008) 6385–6391. [12] W.H. Du, Z.M. Yao, D.S. Liu, C.J. Cai, X.F. Du, R. Ai, Influence of non-Kolmogorov turbulence on intensity fluctuations in laser satellite communication, J. Russ. Laser Res. 33 (2012) 90–97. [13] W. Liu, W. Shi, J. Cao, et al., Bit error rate analysis with real-time pointing errors correction in free space optical communication systems, Optik 125 (2014) 324–328. [14] L.C. Andrews, R.L. Phillips, C.Y. Hopen, Scintillation model for a satellite communication link at large zenith angles, Opt. Eng. 39 (2000) 3272–3280. [15] L.C. Andrews, R.L. Phillips, P.T. Yu, Optical scintillation and fade statistics for a satellite-communication system, Appl. Opt. 34 (1995) 7742–7751. [16] X. Yi, Z. Liu, P. Yue, Uplink laser satellite-communication system performance for a Gaussian beam propagating through three-layer altitude spectrum of weak-turbulence, Optik 124 (2013) 2916–2919. [17] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, Courier Corporation, 1964. [18] K.S. Gochelashvily, V.I. Shishov, Saturated fluctuations in the laser radiation intensity in a turbulent medium, Zh. Eksp. Theor. Fiz. 66 (1974) 1237. [19] P. Deng, X.H. Yuan, D. Huang, Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence, Opt. Commun. 285 (2012) 880–887. [20] A. Viswanath, H. Kaushal, V.K. Jain, S. Kar, Evaluation of performance of ground to satellite free space optical link under turbulence conditions for different intensity modulation schemes, SPIE LASE Int. Soc. Opt. Photonics (2014), 897106-897106. [21] M. Kashani, M. Uysal, M. Kavehrad, A novel statistical model for turbulence-induced fading in free-space optical system, IEEE 15th International Conference on Transparent Optical Networks (2013) 1–5. [22] S. Hranilovic, Wireless optical communication systems, Springer Science & Business Media, 2006. [23] E. Jakeman, P.N. Pusey, Significance of K distributions in scattering experiments, Phys. Rev. Lett. 40 (1978) 546. [24] M.A. Al-Habash, L.C. Andrews, R.L. Phillips, Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media, Opt. Eng. 40 (2001) 1554–1562.
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Optik journal homepage: www.elsevier.de/ijleo
Original research article
Performance analysis of a laser satellite-communication system with a three-layer altitude spectrum over weak-to-strong turbulence Peng Yue, Le Wu ∗ , Xiang Yi, Zhangjie Fu State Key Laboratory on Integrated Services Networks, Xidian University, Xi’an, China
a r t i c l e
i n f o
Article history: Received 29 January 2016 Received in revised form 9 August 2017 Accepted 11 August 2017 Keywords: Atmospheric turbulence Laser-satellite communication Scintillation index Three-layer altitude spectrum Gamma–Gamma distribution
a b s t r a c t In this article, the downlink bit-error-rate (BER) performance of scintillation models for a laser satellite communication system influenced by weak-to-strong turbulence on a slant path is investigated, which take optical wavelength, zenith angle, terrestrial turbulence strength and modulation schemes into account. To describe the properties of atmospheric turbulence which vary with the altitude between a satellite station and a ground station, a three-layer altitude spectrum is used. By adopting the extended Rytov theory valid in weak-to-strong turbulence, scintillation models of an unbounded plane wave for downlink path and spherical wave for uplink path are derived, respectively. The optical fading channel influenced by weak-to-strong turbulence is modeled by Gamma–Gamma distribution. The numeric results show that when the system is under strong terrestrial turbulence conditions or at large zenith angles, disparities in BER performance among different kinds of level of pulse position modulation (PPM) schemes are quite small. Under real circumstances, a longer optical wavelength and a suitable zenith angle range can be selected to reduce turbulence effects for the satellite communication system through weak-to-strong turbulence. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction A great many studies of impacts of atmospheric turbulence on horizontal or slant propagation paths for Free Space Optical (FSO) communication system have been conducted [1–5]. Statistics of atmospheric turbulence is often described by Kolmogorov spectrum [6–8]. However, recent studies and experiments show that Kolmogorov spectrum cannot properly describe the turbulence statistical characteristics in a vertical or slant path in the laser satellite communication system, and turbulence of stratosphere and upper troposphere along the propagation path between a satellite station and a ground station exhibits non-Kolmogorov statistical properties which change with the altitude [9–12]. Therefore, authors in Ref. [12] developed a two-layer turbulence spectrum model, in which numerical values 11/3 and 5, respectively, correspond to the spectral exponents of the troposphere and stratosphere in two-layer spectrum. However, Arkadi Zilberman proposed a three-layer altitude spectrum which has been proved matching the experiment data better [8,11]. In Ref. [8], with a three-layer altitude spectrum, the bit-error rate (BER) performance analysis of the optical communication links under weak turbulence conditions is developed. Meanwhile, a log-normal (LN) model is extensively used to
∗ Corresponding author. Tel.: +86 15291915621. E-mail addresses: [email protected] (P. Yue), [email protected] (L. Wu). http://dx.doi.org/10.1016/j.ijleo.2017.08.074 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
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Fig. 1. The power law exponent ˛(h) as a function of the altitude h.
describe the weak turbulent fluctuation channel [1,13]. In effect, when zenith angles exceed 60◦ , optical links will experience moderate-to-strong refractive fluctuations [14,15]. In this situation, Gamma–Gamma distribution is being adopted as it is applicable to weak-to-strong turbulence. In this article, based on a three-layer altitude spectrum and an extended Rytov theory, theoretical expressions governing the scintillation index are derived, respectively, when the unbounded plane wave and spherical wave propagates through weak-to-strong turbulence along a slant path. Ignoring inner-scale and outer-scale effects, we investigate the BER performance of a laser satellite communication system with some restrictions of parameters such as the turbulent strength, zenith angles and wavelengths. 2. The three-layer altitude spectrum The three-layer altitude spectrum model ˚n (, ˛, h) [11,16] is defined by
k ˛/2−11/6
˚n (, ˛, h) = 0.033
L
Cn2 (h) −˛ ,
(1)
where is the spatial frequency, ˛ is the spectral exponent varying with the altitude h, k is the electromagnetic wave number which can be written as k = 2/ at the wavelength of , Cn2 (h) is the refractive index structure parameter, and L is the propagation distance expressed as L = (H3 − H0 ) sec () in meters(m). In this formula, H3 is the altitude of the satellite or the space station, H0 is the height of a ground station. is the zenith angle. Cn2 (h) varies with h and gets the unit of m−2/3 , and the H − V5/7 model [1,11] is used to describe Cn2 (h)
w 2
Cn2 (h) = 0.00594 −16
+2.7 × 10
27
10
10−5 h
h exp − 1500
exp −
h 1000
h + A0 exp − 100
,
(2)
where w denotes the wind speed and has the units of m/s (generally w = 21m/s). A0 is the terrestrial refractive index structure parameter in m−2/3 . ˛ (h) takes the form [11]
˛ (h) =
˛1
1 + h/H1
b1 +
b1
b1 ·
˛2 · h/H1 1 + h/H1
1
1 + h/H2
b2 +
b2
b2 ,
˛3 · h/H2 1 + h/H2
(3)
where ˛1 = 11/3, ˛2 = 10/3, ˛3 = 5 represent the numerical values of the spectral exponents of the surface layer, troposphere and upper stratosphere above them, respectively. H1 and H2 are boundary layer heights. b1 and b2 are parameters of flatness between layers. There are no exact values for H1 , H2 , b1 and b2 . In the following parts, we set those values as [11], H1 = 2km, H2 = 8km, b1 = 8 and b2 = 10. The curve of ˛ (h) is showed in Fig. 1. It is observed in Fig. 1 that for the atmosphere under the altitude of 2 km, the spectral exponent of the three-layer altitude spectrum is 11/3, which is equal to the Kolmogorov spectrum. This means that the three-layer altitude spectrum presents the Kolmogorov properties for near-ground troposphere. For the upper troposphere with the altitude between 2 and 8 km, the spectral exponent of the three-layer altitude spectrum falls to 10/3, representing the non-Kolmogorov characteristics. Then the spectral exponent goes to 5 for the stratosphere with the altitude higher than 8 km, also showing the non-Kolmogorov properties in range. In Fig. 2 we also see that the flatness factors b1 and b2 make the spectral exponent curve transit smoothly between layers.
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285
Fig. 2. BER versus the zenith angle with different terrestrial turbulent strengths and modulation schemes for the wavelength of 1550 nm.
3. Scintillation index models in a weak-to-strong turbulence region Under weak turbulence circumstances, an unbounded plane wave is applied to approximating a downlink propagation beam, and for an uplink path, we consider it a spherical beam model. In this situation, expressions of the scintillation index for both downlink and uplink path are given by Ref. [16]:
⎧ 7 5 11 ⎪ ⎪ 1.303 ⎪ ⎪ d2 = − d k 6 (H3 − H0 ) 6 sec 6 ⎨ 2 ˛/2 cos ˛/4 ,
H3 ⎪ ⎪ ⎪ 2 ˛/2−1 ⎪ Cn (h) dh ⎩ d = Re
(4)
H0
⎧ 7 5 11 ⎪ ⎪ 1.303 ⎪ 2 =− 6 6 6 ⎪ k − H ) (H sec u 3 0 u ⎨ 2 ˛/2 cos ˛/4 ,
H3 ⎪ ⎪ ˛/2−1 ⎪ ⎪ Cn2 (h) ˛/2−1 (1 − ) dh ⎩ u = Re
(5)
H0
where d2 and u2 denote the downlink and uplink scintillation index, respectively. is the normalized variable [1] written as = (h − H0 ) / (H3 − H0 ). is the zenith angle. Re ( ) takes the real part of the expressions. ( ) is the Gamma function [17]. Cn2 (h) is the refractive index structure parameter provided by Eq. (2). ˛ (h) is the spectral exponent described by Eq. (3). In a strong turbulence regime, the asymptotic theory predicts that scintillation I2 (L) for an unbounded plane wave or a spherical wave can be expressed in the form [18,19]
H3
∞
I2 (L) = 1 + 322 k2 sec
× exp
−
1
DS 0
H0
L k
w ,
0
˚n (, ˛, h) sin2
d
L2 2k
(6)
ddh,
where is the normalized distance variable, and w , is defined by
1 − , < , w , =
(7)
1 − , >
¯ = 1 − . In Eq. (6), the exponential function is a low pass space filter defined where is the optical beam parameter, and by the phase structure function DS ( , ˛), and is the distance of two observation sites. Owing to L2 /k «1, a geometrical optics approximation can be used [1]. At the same time, the phase structure function can be reduced to the wave structure function Dpl ( , ˛) Ds ( , ˛)
∼ = Dpl ( , ˛)
H3
= 82 k2 sec
∞
˚n (, ˛, h) [1 − J0 ( )] ddh H0
0
.
(8)
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Replacing ˚n (, ˛, h) with a three-layer altitude spectrum (Eq. (1)), and using Lw , /k instead of , we get
DS
L
L w , , ˛ ∼ w , , ˛ = Dpl k k
H3
= 2.606k1/6 L11/6 sec H0
k (2−˛)/2 −˛ ˛−2 Cn2 (h) ( ) 2 ˛ . L
(9)
˛−2 −˛/2 dh × w , ˛/2
By substituting Eq. (8) into Eq. (6), the scintillation index for an unbounded plane wave can be written as
⎧ H3 ∞ k ˛/2−11/6 ⎪ 2 ⎪ = 1 + 2.606 sec Cn2 (h) L2 2 5−˛ ⎪ (L) I ⎪ L ⎪ ⎪ H0 0 ⎪ ⎪
1−˛/2 ⎪ ⎪ k ⎪ 2 ˛−2 ˛−2 ⎪ × exp −E (˛) d ddh − +1 ⎪ ⎪ ˛−1 L ⎪ ⎪ ⎪ ⎪ 2(˛−4)/(˛−2) ⎪ ⎪ ⎪ = 1 + E (˛) × d2 ⎪ ⎪ ⎨ , 4 (6 − ˛) / (˛ − 2) ˛/2 cos ˛/4 [E (˛)](˛−6)/(˛−2) ⎪ ⎪ E − = (˛) ⎪ ⎪ (˛ − 2) d ⎪ ⎪ ⎪ ⎪ (˛−6)/(˛−2) H3 ⎪ ⎪ ⎪ ⎪ 2 ˛−4 ⎪ × Cn (h) dh − +1 ⎪ ⎪ ˛−1 ⎪ H0 ⎪ ⎪ H ⎪ ⎪ 3 ⎪ 22−˛ ˛ −˛/2 cos ˛/4 ⎪ ⎪ − Cn2 (h) dh E = ⎩ (˛) d
(10)
H0
∼ L2 4 2 /4k2 and exp − L2 2 /k = ∼ 1 are used. where the geometrical optics approximation sin2 L2 /2k = A conventional Rytov approximation is limited to weak turbulent fluctuations, which cannot properly characterize the coherence loss caused by the increase of propagation distances and the turbulence strength. Adopting an extended Rytov theory, we can deduce the scintillation index models with large-scale and small-scale turbulence effects considered through weak-to-strong turbulence. With inner-scale and outer-scale effects neglected, the space filter functions of large-scale GX () and small-scale GY () are expressed as [1]
GX () = exp
GY () =
−
2
,
X2
(11)
11/3 2 + Y2
11/6 ,
(12)
where X and Y , respectively, represent the large-scale and small-scale spatial cut-off frequencies, both of which are used to cut off the effects caused by turbulence whose scales are between a spatial coherence radius 0 and a scatter disk L/ (k 0 ). 2 Along a downlink propagation path, scintillation index models of the log-irradiance for large-scale ln and small-scale X 2 ln Y take the form as follows [1]
2 ln X
H3
= 8 k sec
˚n (, ˛, h) GX () 1 − cos H0
2 ln Y
∞
2 2
H3
0
2 2
= 8 k sec
∞
˚n (, ˛, h) GY () 1 − cos
H0
0
2 h sec k 2 h sec k
ddh,
(13)
ddh.
(14)
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287
By substituting the space filter function Eqs. (11) and (12) into Eqs. (13) and (14), respectively, and replacing ˚n (, ˛, h) with a three-layer altitude spectrum, scintillation index models of the log-irradiance for large-scale and small-scale turbulence become
⎧ 2 L ∞ 2 2 z ⎪ ˛/2−11/6 1−˛ 2 2 2 ⎪ Cn (h)(k/L) exp − 2 ddz ⎪ ⎪ ln X = 1.303k k ⎪ X ⎪ 0 0 ⎪ ⎪ H ⎪ 3 ⎪ 3−˛/2 3 − ˛/2 ˛/2 cos ˛/4 ⎪ ⎪ ⎨ Cn2 (h) 2 dh d2 LX2 /k =− d
⎪ ⎪ = EX (˛) × d2 × X 3−˛/2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 − ˛/2 ˛/2 cos ˛/4 ⎪ ⎪ ⎩ EX (˛) = − d
(15)
,
H0
H3
Cn2 (h) 2 dh
H0
⎧ L ∞ 11/3 ⎪ ˛/2−11/6 1−˛ 2 2 ⎪ = 2.606k Cn2 (h)(k/L) ⎪ ln Y 11/6 ddz ⎪ 2 + 2 ⎪ 0 0 ⎪ Y ⎪ ⎪ ⎪ ⎪ 2 ˛/2 cos ˛/4 17/6 − ˛/2 ˛/2 − 1 ⎪ ⎪ =− ⎨ (11/6) d
⎪ 1−˛/2 ⎪ = EY (˛) × d2 × Y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ˛/2 17/6 − ˛/2 ˛/2 − 1 cos ˛/4 ⎪ ⎪ ⎩ EY (˛) = − d (11/6)
H3
Cn2 (h)dh
d2 (LY2 /k)
1−˛/2
H0
(16)
,
H3
Cn2 (h)dh
H0
where X = LX2 /k and Y = LY2 /k are the non-dimension spatial cut-off frequencies of large-scale turbulence and small-
scale turbulence, respectively. Geometrical optics approximations are used as 1 − cos 2 z/k ∼ = 0.5 2 z/k 2 ∼ 1, > Y >> k/L. 1 − cos z/k =
2
, « X , and
To determine X and Y , we should utilize the scintillation index in the weak turbulence and saturated region. The expression of the scintillation index in a weak and saturated strong turbulence region is deduced from Eq. (10)
I2
∼ =
2 2 , + ln ln X Y
2 2(˛−4)/(˛−2)
2 = 1 + E (˛) × d 1 + 2ln X
d2 << 1 ,
.
(17)
d2 >> 1
And the non-dimension spatial cut-off frequencies of large-scale turbulence and small-scale turbulence X and Y are, respectively, described as
X =
2 L kl
L 2 = k X k
X
L
=
1 c1 + c2 L/k 02
∼ =
⎧ 1 ⎪ ⎪ ⎨ c1 , k 2 ⎪ ⎪ ⎩ c L0 , 2
L 1 2
L Y = Y2 = k k
lY
⎧ ⎪ ⎨ c3 ,
c4 L ∼ = c3 + = c L k 02 ⎪ ⎩ 42, k 0
L k 02 L k 02
L k 02 L k 02
<< 1 ,
(18)
>> 1
<< 1 ,
(19)
>> 1
where lX denotes the lower bound of the largest scales, and lY represents the upperbound of the smallest scales. c1 , c2 , L/k, lY ∼ L/k. In a strong turbulence regime, c3 , c4 are constants to be determined. In a weak turbulence regime, lX ∼ = = lX ∼ = L/ (k 0 ), lY ∼ = 0 . According to Eq. (17), in a weak turbulence regime, scintillation index models of the log-irradiance for large-scale and small-scale turbulence is set as
2 = 0.49d2 ln X 2 ln = 0.51d2 Y
.
(20)
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Since the second of Eq. (17) shows the limit of the small-scale intensity scintillation in a saturated region, 2expression namely Y2 = exp ln − 1 → 1, for a moderate-to-strong turbulence regime, we have Y
2(˛−4)/(˛−2)
2 ln = 0.5E (˛) × d2 X
.
(21)
2 ln → ln 2 Y
Combining Eqs. (15) and (16) with Eqs. (20) and (21), constants c1 , c2 , c3 , c4 in Eqs. (18) and (19) can be determined. Hence, X and Y are derived X =
1 c1 + c2 L/k 02
Y = c3 +
c4 L k 02
=
=
0.49 EX (˛)
1/(˛/2−3)
0.51 1/(1−˛/2) EY (˛)
+
+ 0.5
2(˛−4)/(˛−2)−1 E (˛) × d2 EX (˛)
1/(˛/2−3) −1
,
(22)
1/(1−˛/2)
ln 2
(23)
.
EY (˛) × d2
If we insert Eqs. (22) and (23) into Eqs. (15) and (16), respectively, the large-scale and small-scale scintillation index models of the log-irradiance for a downlink propagation path become 2 ln = EX (˛) × d2 × X 3−˛/2 X
= EX (˛) × d2 ×
0.49 EX (˛)
1/(˛/2−3)
+
E (˛)
2(˛−4)/(˛−2)−1 1/(˛/2−3)
˛/2−3 ,
(24)
× d2
2EX (˛)
1−˛/2
2 ln = EY (˛) × d2 × Y Y
=
EY (˛) × d2
×
0.51 1/(1−˛/2) EY (˛)
+
ln 2
1/(1−˛/2) 1−˛/2 .
(25)
EY (˛) × d2
Merging Eqs. (24) and (25), the scintillation index model of a downlink path through weak-to-strong turbulence is represented as
2 2 2 I,pl + ln −1 (L) = exp ln X Y
1−˛/2
= exp EX (˛) × d2 × X 3−˛/2 + EY (˛) × d2 × Y
−1
(26)
,
where ˛ is the spectral exponent described by Eq. (3). d2 denotes the downlink scintillation index presented by Eq. (4). X is the non-dimension spatial cut-off frequencies of large-scale written as Eq. (22). In a similar way, the scintillation index model of an uplink path through weak-to-strong turbulence is given by
1−˛/2
2 I,sp (L) = exp EX (˛) × u2 × X 3−˛/2 + EY (˛) × u2 × Y
− 1,
(27)
where u2 denotes the uplink scintillation index presented by Eq. (5). Y is the non-dimension spatial cut-off frequencies of small-scale written as Eq. (23). 4. BER analysis In a wireless laser communication system with the Intensity Modulation/Direct Detection (IM/DD) technology, the received optical beam is converted to an electronic signal through a photo diode. Shot noise and thermal noise are introduced by the receiver, and the noise obeys a Gaussian distribution with the mean zero and the variance. A statistical channel model is defined as follows [20,21] y = RIx + n,
(28)
where x denotes the modulated signal. y denotes the received signal. R represents the photoelectric responsivity of the positive intrinsic-negative (PIN) photodetector. I is the channel fading. n denotes additive Gaussian white noise. The on-off keying (OOK) scheme has been extensively used to represent the weak turbulence channel. The conditional BER of OOK scheme can be written as BEROOK (I) =
1 erfc 2
SNR
√ I , 2 2
where SNR is the signal-to-noise ratio, and erfc ( ) is complementary error function [17].
(29)
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289
Table 1 Parameters for the numerical analysis. Symbol
Description
Value
H3 H2 H1 H0 A0
Height of the satellite station Boundary layer heights between the surface layer and the troposphere Boundary layer heights between the troposphere and the stratosphere Height of the ground station Wavelength Zenith angle Terrestrial refractive index structure parameter for weak turbulence Terrestrial refractive index structure parameter for moderate turbulence Terrestrial refractive index structure parameter for strong turbulence
35800 km 8 km 2 km 0 km 1550 nm, 1310 nm, 850 nm 0◦ –80◦ 1.7 × 10 − 14 m−2/3 3.0 × 10 − 13 m−2/3 1.7 × 10 − 12 m−2/3
Compared with OOK, M-level pulse position modulation (M-PPM) requires a lower average transmitted power. In a MPPM scheme, there is no thresholds at the receiver. Moreover, to a certain degree, M-PPM can resist background noise. In the absence of turbulence, the conditional BER of an M-PPM system can be written as [22]
SNR
M BERM−PPM (I) ≈ erfc 4
Mlog2 M 4
I
(30)
.
In the study of atmospheric turbulence, many density functions are adopted to describe the scintillation, such as lognormal (LN) distribution [1] which is used in a weak turbulence region, and K-distribution [23] is commonly used in a strong turbulence region. However, Gamma–Gamma distribution [2,24] is a model that can describe the probability density function (PDF) of the irradiance over weak-to-strong turbulence. The PDF of Gamma–Gamma distribution takes the form
⎧ (a+b)/2 2(ab) (a+b)/2−1 ⎪ ⎪ Ka−b 2 abI , I > 0 ⎨ fI (I, a, b) = (a) (b) I ⎪ 1 ⎪ ⎩a = 2 = X
1
2 exp ln −1 X
,
b=
1 Y2
=
1
,
(31)
2 exp ln −1 Y
where Kv ( ) is the second kind of modified Bessel function in v-order [17], and a and b denote the large-scale and small-scale turbulence cells. Hence, the average BER for both OOK and M-PPM schemes can be written as
⎧ ∞ SNR ⎪ 1 ⎪ ⎪ BER = erfc √ I × fI (I)dI ⎪ ⎪ OOK 2 2 2 ⎨ 0 , ∞ ⎪ ⎪ SNR Mlog2 M M ⎪ ⎪ BER I × fI (I)dI = erfc ⎪ M−PPM ⎩ 4 4
(32)
0
where fI (I) denotes the PDF of Gamma–Gamma distribution, and M is the level of a PPM modulation scheme. 5. Numeric results In this section, using the previously derived formula (Eq. (32)), we estimate the BER performance of a laser satellite communication system on a downlink propagation path employing the M-PPM schemes over a Gamma–Gamma distribution channel. In a subsequent section, we make a numerical analysis of downlink BER performance under various conditions. The parameters used in the numerical analyses are listed in Table 1. In Figs. 2–4, downlink BER performance for OOK, 2-PPM, 4-PPM, and 8-PPM schemes are presented as a function of the zenith angle. Here, we make a comparison among three kinds of terrestrial turbulent strength at different wavelengths ( = 850nm, = 1310nm, = 1550nm). SNR is set as 20 dB. Figs. 2–4 show that the BER performance for OOK scheme is equal to 2-PPM in value because they have the same BER formula. As it is shown in Figs. 2–4, BER rises with the increase of the zenith angle. We show that the BER increases slowly until the zenith angle approach 60◦ , and then increases rapidly as the zenith angle continue to increase. This appears for various wavelengths. For zenith angle larger than 60◦ , the effect of atmospheric turbulence reinforces, the BER performance severely degraded. In an actual system, we can select a suitable zenith angle range to help improve the system reliability. It is also observed from Figs. 2–4 that for weak, moderate and strong terrestrial turbulent strength, when zenith angle exceeds 60◦ , the disparities in these three kinds of level of PPM schemes decrease gradually with the increase of the zenith angle. But when the zenith angle is under 60◦ , a higher modulation level M of PPM scheme shows a better BER performance. However, for strong terrestrial turbulent strength, differences of BER performance among those PPM schemes are quite small. It indicates that at large zenith angles or under the condition of strong terrestrial turbulence, the improvement of the BER performance is not obvious simply by increasing the modulation
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Fig. 3. BER versus the zenith angle with different terrestrial turbulent strengths and modulation schemes for the wavelength of 1310 nm.
Fig. 4. BER versus the zenith angle with different terrestrial turbulent strengths and modulation schemes for the wavelength of 850 nm.
Fig. 5. BER versus SNR with different terrestrial turbulent strengths and modulation schemes for the wavelength of 1550 nm.
level M. Furthermore, numeric results obtained from Figs. 2–4 show that BER rises with the decrease of the wavelength for different terrestrial turbulence strengths. Although BER performance of a long optic wavelength is better than that of a short one, considering the limitation of the power of a laser device, an appropriate wavelength should be selected in the actual laser communication system. Downlink BER performance for OOK, 2-PPM, 4-PPM and 8-PPM schemes under different terrestrial turbulent strength conditions are given in Figs. 5–7, as a function of SNR. Here, we make a comparison among three kinds of wavelengths ( = 850nm, = 1310nm, = 1550nm). The zenith angle takes = 0◦ . As it is shown in Figs. 5–7, no matter which kind of wavelength we choose, there is a rapid decline in the differences in BER among these three kinds of PPM schemes at a certain SNR, when the optical signal go through from weak-to-strong terrestrial turbulence. Furthermore, it is found that for a certain modulation level of PPM schemes, a longer wavelength helps us get a lower BER. With respect to the 8-PPM scheme, for example, when we set SNR as 12 dB for weak terrestrial turbulence, Fig. 5 shows that the value of BER can reach nearly 10−12 at 1550 nm, but in Fig. 7 for the wavelength at 850 nm, the value of BER only gets 10−8 . Moreover, for both weak and moderate terrestrial turbulent strength, 8-PPM gets the minimum of BER among these three kinds of level of PPM schemes. In other words, under weak and moderate terrestrial turbulent strength conditions, a higher modulation level M of PPM scheme can help improve the BER performance. However, when the optical beam propagates through strong terrestrial turbulence on a slant path, the improvement of BER performance is not notable just by increasing the modulation level M.
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Fig. 6. BER versus SNR with different terrestrial turbulent strengths and modulation schemes for the wavelength of 1310 nm.
Fig. 7. BER versus SNR with different terrestrial turbulent strengths and modulation schemes for the wavelength of 850 nm.
It indicates that the terrestrial turbulent strength is the key factor that affects the BER performance of this communication system. 6. Conclusion In this article, based on a three-layer altitude spectrum and an extended Rytov theory, the uplink and downlink scintillation index models in weak-to-strong turbulence are derived. Then, using Gamma–Gamma distribution, we theoretically investigate the BER performance of the different (OOK and M-PPM) schemes for a satellite communication system through weak-to-strong turbulence. In conclusion, M-PPM shows a better BER performance than OOK in the same condition. Under real circumstances, a longer optical wavelength and a suitable zenith angle range can be selected to reduce turbulence effects. What is more, when the zenith angle falls below 60◦ and the system is under the condition of weak and moderate terrestrial turbulent strength, PPM scheme of a higher modulation level M shows a better BER performance. However, at large zenith angles or under the condition of strong terrestrial turbulence, the improvement of the BER performance is not notable simply by increasing the modulation level M of PPM schemes. It indicates that the terrestrial turbulent strength is the key factor that affects the BER performance of the laser satellite communication system. Acknowledgments This work was supported by the National Nature Science Foundation of China grant No. 60902038, the 111 project under Grant No. B08038, the State Scholarship Fund under Grant 201406965012, and the Fundamental Research Funds for the Central Universities No. K50511010019. Moreover, Dr. Yunfei Hou who is with the Department of Computer Science and Engineering, State University of New York at Buffalo also contributes to this article. References [1] L.C. Andrews, R.L. Phillips, Laser Beam Propagation Through Random Media, SPIE Press, Bellingham, WA, 2005. [2] G.K. Varotsos, H.E. Nistazakis, C.K. Volos, G.S. Tombras, FSO links with diversity pointing errors and temporal broadening of the pulses over weak-to-strong atmospheric turbulence channels, Optik (2015). [3] A.N. Stassinakis, H.E. Nistazakis, K.P. Peppas, G.S. Tombras, Improving the availability of terrestrial FSO links over log normal atmospheric turbulence channels using dispersive chirped Gaussian pulses, Opt. Laser Technol. 54 (2013) 329–334. [4] G.K. Varotsos, A.N. Stassinakis, H.E. Nistazakis, A.D. Tsigopoulos, K.P. Peppas, C.J. Aidinis, G.S. Tombras, Probability of fade estimation for FSO links with time dispersion and turbulence modeled with the gamma-gamma or the IK distribution, Optik 125 (2014) 7191–7197. [5] A. Vats, H. Kaushal, Analysis of free space optical link in turbulent atmosphere, Optik 125 (2014) 2776–2779.
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