Abstract

This paper analytically investigates a bit error rate (BER) performance of radio over free space optical (FSO) systems considering laser phase noise under Gamma-Gamma turbulence channels. An external modulation using a dual drive Mach-Zehnder modulator (DD-MZM) and a phase shifter is employed because a DD-MZM is robust against a laser chirp and provides high spectral efficiency. We derive a closed form average BER as a function of different turbulence strengths and laser diode (LD) linewidth, and investigate its analytical behavior under practical scenario. As a result, for a given average SNR with normalized perturbation, it is shown that the difference of average BER corresponding to two LDs (with linewidth of 624MHz and 10MHz) under weak turbulence is almost 3 times larger than that under strong turbulence.

©2009 Optical Society of America

1. Introduction

The volume of data traffic continues to increase due to the demand of subscribers for multimedia services that require the access network to support high data rates at any time, in any place inexpensively. Such demands require broadband communication systems. Free space optical (FSO) systems have been good candidates for next generation broadband services since FSO systems support large bandwidth, unlicensed spectrum, excellent security, and quick and inexpensive setup [1]. In spite of these advantages, the performance of FSO systems can be unreliable due to atmospheric turbulence. The Gamma-Gamma distribution well represents atmospheric turbulence channels with a multiplication of two parameters of small-scale and large-scale irradi-ance fluctuations-of which pdfs are independent Gamma distributions-and provides excellent agreement between theoretical and simulation results [2].

 

Fig. 1. Overall architecture of the FSO system considering of optical transmitter, turbulence channels, and optical receiver.

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One of the well-known modulators of FSO systems is the external modulator, with a dual drive Mach-Zehnder modulator (DD-MZM), since it is robust against a laser chirp and provides high spectral efficiency [3]. Figure 1 represents the FSO system employing a DD-MZM. In this system, the laser phase noise from a laser diode (LD) is one of the decisive factors limiting the performance of FSO systems because the optical system is sensitive to laser phase noise. However, to the best of our knowledge, an analysis of the BER of FSO systems, with a DD-MZM impaired by the laser phase noise under turbulence channels, has not been carried out in the research due to the complexity of the analysis.

Therefore, in this paper, we first represent the optical signal model from a DD-MZM with laser phase noise using the Bessel expansion. We then analyze an average BER according to an average SNR, with normalized perturbation under atmospheric channels, where Gamma-Gamma distribution describes the turbulence-induced fading. Also, numerical results are provided to illustrate the degradation of performance according to the depth of scintillation and LD linewidth.

2. FSO System Architecture and Signal Model

Figure 1 shows the overall architecture of the FSO system. Data is modulated to a binary phase shift keying (BPSK) signal by the RF modulator. A BPSK signal from the RF modulator is split by a π/2 phase shifter. This BPSK signal is optically modulated by a LD with a DD-MZM. The output signal of the DD-MZM is transmitted via atmospheric turbulence channels between telescopes. The received signals are detected by the photodetector (PD), and the photocurrent corresponding to the transmitted BPSK signal is extracted by the bandpass filter (BPF). Finally, data is extracted by the RF demodulator module. The optical signal, xLD(t) [4] from the laser and the BPSK signal, xRF(t) from the RF modulator are modeled respectively, as follows:

xLD(t)=VLDexp[j(ωct+ΦLD(t))],
xRF(t)=VRFcos(ωrft+θ(t)).

In (1), VLD and VRF are the optical carrier amplitude and the BPSK signal amplitude, respectively, and ωc and ωrf are angular frequencies of the signals from the LD and the RF modulator. The laser phase noise process Φ LD(t) is commonly characterized as a Wiener process [6]. θ(i) = ∑ m=-∞ dmP(t - mT) where P(t) is a unit amplitude pulse of the bit duration T and dm is the information of the mth bit duration, which takes on {0, π}.

After optically modulating XRF(t) by xLD(t) with a DD-MZM, the output signal of a DD-MZM is represented as [3], [5]

ET(t)=VLD210L20{exp[j(ωct+π2+ΦLD(t)+επcos[ωrft+θ(t)])]
+exp[j(ωct+ΦLD(t)+επcos[ωrft+θ(t)+π2])]}

where ε(= VRF/Vπ) defines a normalized ac value, Vπ is the switching voltage of a DD-MZM, and L is the modulator insertion loss in decibels. Using the Bessel function, (2) can be expanded to

ET(t)=VLDexp[j(ωct+ΦLD(t))]210L20{jn=[jnexp[jn(ωct+ΦLD(t))]Jn(επ)]
+n=[jnexp[jn(π2+ωRFt+θ(t))]Jn(επ)]}
VLD10L20{J0(επ)exp[j(ωct+π4+ΦLD(t))]
2J1(επ)exp[j(ωct+ωrft+ΦLD(t)+θ(t))]}.

We assume that high-order components of the Bessel function, can be negligible since the value of επ in the Bessel function is very small due to the fact that VπVRF in general. The output signal at a DD-MZM is transmitted via turbulence channels experiencing different group delays due to the chromatic dispersion. After the transmission of turbulence channels, the received optical signal is expressed as

ER(t)VLD·δ(t)10L20{J0(επ)exp[j(ωct+π4+ΦLD(tτ0)ψ0)]
2J1(επ)exp[j(ωct+ωrft+ΦLD(tτ1)+θ(t)ψ1)]}

where δ(t) = ∑ m=-∞ δmP(t - mT), and δm is the turbulence channel coefficient for the mth data duration, τ 0 and τ 1 are group delays, and ψ 0 and ψ 1 are phase-shift parameters.

Through direct detection, the optical signal can be detected at the PD, and the BPSK signal is extracted by the BPF. After the BPF, the photocurrent i(t) can be obtained as follows:

i(t)=RER(t)2+nth(t)
22R(VLD·δ(t)10L20)2J0(επ)J1(επ)cos(ωrft+ΦLD(tτ1)
ΦLD(tτ0)+θ(t)ψ1+ψ0)+nth(t)

where R is the responsivity of the PD and nth,(t) is a random fluctuation which has an unit of current due to thermal noise in the load resistance. In general FSO systems, as thermal noise is dominant due to the high operating temperature [7], nth(t) is modeled as a Gaussian process [8]. Also, Φ LD(t - τ 0) and Φ LD(t - τ 1) are independent–not of t–but of the differential delay (τ 1 - τ 0). Thus, using a Wienerprocess property, the difference of two random processes can be simplified as a zero mean Gaussian random variable Ψ with the variance σ 2 of 2πΔνΔτ where Δν is LD linewidth and Δτ is the differential delay [6]. Accordingly, (5) can be represented as

i(t)=δ·ϒ·cos(ωrft+Ψ+θ(t)+ψ2)+nth(t),mTt(m+1)T

where ϒ is 2√2R (VLD /10L/20)2 J 0(επ) J 1 (επ), ψ 2 = ψ 0 - ψ 1. When we assume that the coherence time is larger than a bit time duration, {δm} can be modeled as independent and identically distributed (i.i.d.) random variables, i.e., δm = δ with Gamma-Gamma distribution. After the matched filter in the RF demodulator, ∫T 0 i(t)dt, thermal noise becomes a random variable, nth with normal distribution, N[0, σ 2 th = (4kT/RLf] where k is the Boltzmann constant, RL is the load resistance, and Δf is the effective noise bandwidth [8].

Gamma-Gamma distribution [2] is

fδ(δ)=2(αβ)α+β2Γ(α)Γ(β)(δ)α+β21Kαβ(2αβδ)

where δ > 0, α and β are the scintillation parameters, Kε(·) is the modified Bessel function of the second kind of order ε, and Γ(·) is the Gamma function. According to the atmospheric conditions, α and β are defined as in [2]

α=(exp[0.49σR2(1+0.18d2+0.56σR12/5)7/6]1)1
β=(exp[0.51σR2(1+0.69σ12/5)5/6(1+0.18d2+0.56σR12/5)7/6]1)1

where, σ 2 R = 0.5C 2 n κ 7/6 L 11/6 is Rytov variance which has been used as an estimate of the intensity variance, σR is the turbulence strength d = (κD 2/4L)1/2, D is the diameter of the receiver collecting lens aperture, κ is the optical wave number, L is the propagation distance, and C 2 stands for the altitude-dependent index of the refractive structure parameter and varies from 10-13/m -2/3 for strong turbulence to 10-17/m -2/3 for weak turbulence. Moreover, according to α and β the scintillation index (SI) is defined as [2]

SI=1α+1β+1αβ.

3. BER Analysis of FSO systems

In this section, we derive a closed form of the average BER considering turbulence channels and the laser phase noise. We can define the instantaneous electrical SNR as Λ = (ϒδ)2 / σ 2 th. For the convenience of the analysis, we introduce a new measure, μ(= ϒ2/σ 2 th) which corresponds to the average SNR with E[δ 2] = 1, i.e., the average SNR E[Λ] = μ · E[δ 2] = μ. When we assume that the random phase noise Ψ is fixed over the bit duration, the conditional BER is represented as [9]

Pb(EΨ,Λ)=12erfc(Λcos2[Ψ])

where er fc is the complementary error function. Consequently, the average BER (E[Pb]) can be obtained by using the following integral calculus as

E[Pb]=0Pb(EΨ,Λ)fΨ(Ψ)fδ(δ)dΨdδ

Using the change of variable x = Ψ/ √2σ and Gauss-Hermite quadrature formula, (12) can be simplified to (13)

E[Pb]=12π0erfc(μδ2cos2[2σx])exp[x2]fδ(δ)dxdδ
=12π0i=1Nwierfc(μδ2cos2[2σxi]fδ(δ)dδ

where N is the order of approximation, xi, i = 1,….,N are the zeros of the Nth-order Hermite polynormial, and wi, i = 1,….,N are weight factors for the Nth-order approximation. Using the following Meijer G functions [10]:

erfc(x)=1πG1,22,0[x0,121]
Kν(x)=12G0,22,0[x24ν2,ν2]

the average BER is represented as

E[Pb]=12πi=1Nwi0erfc(μδ2cos2[2σxi]2(αβ)α+β2Γ(α)Γ(β)(δ)α+β21Kαβ(2αβδ))dδ
=(αβ)α+β22πΓ(α)Γ(β)i=1Nwi0(δ)α+β21G1,22,0[μδ2cos2[2σxi]0,121]
×G0,22,0[αβ(δ)αβ2,βα2]dδ.

Finally, by using the classic Meijer integral of the two G functions [10, Eq. 07.34.21.0011.01], (16) is simplified as

E[Pb]=2α+β3π2Γ(α)Γ(β)i=1NwiG5,22,4[24μcos2[2σxi](αβ)20,121α2,2α2,1β2,2β2,1].

whichis as a function of LD linewidth (Δν) and scintillation parameters (α,β).

4. Numerical results

Figure 2 illustrates the results of the average BER according to the average SNR with E [δ 2] = 1. We show two examples of practical LDs with Δν=10MHz and Δν=624MHz, which correspond to typical values of a DFB LD and a FP LD, respectively. We consider that the optical wavelength is 1.55μm, the responsivity is 0.8 A/W, and the differential delay Δτ is 10-10 sec.

We numerically evaluate performance under three different turbulence channel conditions:(α,β,SI) ∈{(4,1,1.5),(4,2,0.875),(4,4,0.5625)}. As β decreases and SI increases, the turbulence effects become stronger. The numerical results show that the difference of average BER corresponding to two LDs (with linewidth of 624MHz and 10MHz) under weak turbulence is almost 3 times larger than that under strong turbulence and the effect of the laser phase noise of LD linewidth 624MHz degrades almost 5dB to the SNR than that of LD linewidth 10MHz at all turbulence conditions. It is noteworthy that in (17), we use N = 10 for the Gauss-Hermite formula.

 

Fig. 2. Average BER according to average SNR when E[δ 2] = 1 according to (α, β,SI) ∈ {(4,1,1.5),(4,2,0.875),(4,4,0.5625)} and 624MHz and 10MHz of LD linewidth.

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5. Conclusions

In this paper, we present an optical signal model from the transmitter output and the receiver input using the Bessel expansion. Also, we derive a close form average BER performance by considering the laser phase noise from LD under atmospheric turbulence channels with Gamma-Gamma distribution using the Gauss-Hermite quadrature formula and Meijer G function. As a result, we can more easily predict BER performance without complicated calculations. In practical terms, when we establish FSO systems, we can make an engineering table by using this derived BER formula according to each LD, which enables us to determine efficient LD under turbulence channel conditions.

6. Acknowledgment

This work was partially supported by the Center for Distributed Sensor Network at GIST, and by a grant (B01-03) from the Plant Technology Advancement Program funded by the Ministry of Construction and Transportation of the Korean government.

References and links

1. V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol. 24, 4750–4762 (2006). [CrossRef]  

2. 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, 1554–1562 (2001). [CrossRef]  

3. G. H. Smith and D. Novak, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech. 45, 1410–1415 (1997). [CrossRef]  

4. R. W. Tkach and A. R. Chraplyvy, “Phase noise and linewidth in an InGaAsP DFB laser,” J. Lightwave Technol. LT-4, 1711–1716 (1986). [CrossRef]  

5. T. Cho, C. Yun, J. Song, and K. Kim, “Analysis of CNR penalty of radio-over-fiber systems including the effects of phase noise from laser and RF oscillator,” J. Lightwave Technol. 23, 4093–4100 (2005). [CrossRef]  

6. J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE 78, 1369–1394 (1990). [CrossRef]  

7. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54, 604–607 (2006). [CrossRef]  

8. G. P. Agrawal, Fiber-Optic Communication Systems. (John Wiley and Sons, New York, 2002). [CrossRef]  

9. A. J. Viterbi, Principles of Coherent Communication. (McGraw-Hill, New York, 1966).

10. The Wolfram function site (2004), http://functions.wolfram.com/.

References

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  1. V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol. 24, 4750–4762 (2006).
    [Crossref]
  2. 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, 1554–1562 (2001).
    [Crossref]
  3. G. H. Smith and D. Novak, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech. 45, 1410–1415 (1997).
    [Crossref]
  4. R. W. Tkach and A. R. Chraplyvy, “Phase noise and linewidth in an InGaAsP DFB laser,” J. Lightwave Technol. LT-4, 1711–1716 (1986).
    [Crossref]
  5. T. Cho, C. Yun, J. Song, and K. Kim, “Analysis of CNR penalty of radio-over-fiber systems including the effects of phase noise from laser and RF oscillator,” J. Lightwave Technol. 23, 4093–4100 (2005).
    [Crossref]
  6. J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE 78, 1369–1394 (1990).
    [Crossref]
  7. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54, 604–607 (2006).
    [Crossref]
  8. G. P. Agrawal, Fiber-Optic Communication Systems. (John Wiley and Sons, New York, 2002).
    [Crossref]
  9. A. J. Viterbi, Principles of Coherent Communication. (McGraw-Hill, New York, 1966).
  10. The Wolfram function site (2004), http://functions.wolfram.com/.

2006 (2)

V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol. 24, 4750–4762 (2006).
[Crossref]

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54, 604–607 (2006).
[Crossref]

2005 (1)

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, 1554–1562 (2001).
[Crossref]

1997 (1)

G. H. Smith and D. Novak, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech. 45, 1410–1415 (1997).
[Crossref]

1990 (1)

J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE 78, 1369–1394 (1990).
[Crossref]

1986 (1)

R. W. Tkach and A. R. Chraplyvy, “Phase noise and linewidth in an InGaAsP DFB laser,” J. Lightwave Technol. LT-4, 1711–1716 (1986).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems. (John Wiley and Sons, New York, 2002).
[Crossref]

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, 1554–1562 (2001).
[Crossref]

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, 1554–1562 (2001).
[Crossref]

Barry, J. R.

J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE 78, 1369–1394 (1990).
[Crossref]

Chan, V. W. S.

Cho, T.

Chraplyvy, A. R.

R. W. Tkach and A. R. Chraplyvy, “Phase noise and linewidth in an InGaAsP DFB laser,” J. Lightwave Technol. LT-4, 1711–1716 (1986).
[Crossref]

Kiasaleh, K.

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54, 604–607 (2006).
[Crossref]

Kim, K.

Lee, E. A.

J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE 78, 1369–1394 (1990).
[Crossref]

Novak, D.

G. H. Smith and D. Novak, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech. 45, 1410–1415 (1997).
[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, 1554–1562 (2001).
[Crossref]

Smith, G. H.

G. H. Smith and D. Novak, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech. 45, 1410–1415 (1997).
[Crossref]

Song, J.

Tkach, R. W.

R. W. Tkach and A. R. Chraplyvy, “Phase noise and linewidth in an InGaAsP DFB laser,” J. Lightwave Technol. LT-4, 1711–1716 (1986).
[Crossref]

Viterbi, A. J.

A. J. Viterbi, Principles of Coherent Communication. (McGraw-Hill, New York, 1966).

Yun, C.

IEEE Trans. Commun. (1)

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54, 604–607 (2006).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

G. H. Smith and D. Novak, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech. 45, 1410–1415 (1997).
[Crossref]

J. Lightwave Technol. (3)

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, 1554–1562 (2001).
[Crossref]

Proc. IEEE (1)

J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE 78, 1369–1394 (1990).
[Crossref]

Other (3)

G. P. Agrawal, Fiber-Optic Communication Systems. (John Wiley and Sons, New York, 2002).
[Crossref]

A. J. Viterbi, Principles of Coherent Communication. (McGraw-Hill, New York, 1966).

The Wolfram function site (2004), http://functions.wolfram.com/.

Cited By

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Figures (2)

Fig. 1.
Fig. 1. Overall architecture of the FSO system considering of optical transmitter, turbulence channels, and optical receiver.
Fig. 2.
Fig. 2. Average BER according to average SNR when E[δ 2] = 1 according to (α, β,SI) ∈ {(4,1,1.5),(4,2,0.875),(4,4,0.5625)} and 624MHz and 10MHz of LD linewidth.

Equations (28)

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x LD ( t ) = V LD exp [ j ( ω c t + Φ LD ( t ) ) ] ,
x RF ( t ) = V RF cos ( ω rf t + θ ( t ) ) .
E T ( t ) = V LD 2 10 L 20 { exp [ j ( ω c t + π 2 + Φ LD ( t ) + επ cos [ ω rf t + θ ( t ) ] ) ]
+ exp [ j ( ω c t + Φ LD ( t ) + επ cos [ ω rf t + θ ( t ) + π 2 ] ) ] }
E T ( t ) = V LD exp [ j ( ω c t + Φ LD ( t ) ) ] 2 10 L 20 { j n = [ j n exp [ j n ( ω c t + Φ LD ( t ) ) ] J n ( επ ) ]
+ n = [ j n exp [ jn ( π 2 + ω RF t + θ ( t ) ) ] J n ( επ ) ] }
V LD 10 L 20 { J 0 ( επ ) exp [ j ( ω c t + π 4 + Φ LD ( t ) ) ]
2 J 1 ( επ ) exp [ j ( ω c t + ω rf t + Φ LD ( t ) + θ ( t ) ) ] } .
E R ( t ) V LD · δ ( t ) 10 L 20 { J 0 ( επ ) exp [ j ( ω c t + π 4 + Φ LD ( t τ 0 ) ψ 0 ) ]
2 J 1 ( επ ) exp [ j ( ω c t + ω rf t + Φ LD ( t τ 1 ) + θ ( t ) ψ 1 ) ] }
i ( t ) = R E R ( t ) 2 + n th ( t )
2 2 R ( V LD · δ ( t ) 10 L 20 ) 2 J 0 ( επ ) J 1 ( επ ) cos ( ω rf t + Φ LD ( t τ 1 )
Φ LD ( t τ 0 ) + θ ( t ) ψ 1 + ψ 0 ) + n th ( t )
i ( t ) = δ · ϒ · cos ( ω rf t + Ψ + θ ( t ) + ψ 2 ) + n th ( t ) , mT t ( m + 1 ) T
f δ ( δ ) = 2 ( αβ ) α + β 2 Γ ( α ) Γ ( β ) ( δ ) α + β 2 1 K α β ( 2 αβδ )
α = ( exp [ 0.49 σ R 2 ( 1 + 0.18 d 2 + 0.56 σ R 12 / 5 ) 7/6 ] 1 ) 1
β = ( exp [ 0.51 σ R 2 ( 1 + 0.69 σ 12 / 5 ) 5 / 6 ( 1 + 0.18 d 2 + 0.56 σ R 12 / 5 ) 7/6 ] 1 ) 1
SI = 1 α + 1 β + 1 αβ .
P b ( E Ψ , Λ ) = 1 2 erfc ( Λ cos 2 [ Ψ ] )
E [ P b ] = 0 P b ( E Ψ , Λ ) f Ψ ( Ψ ) f δ ( δ ) d Ψ d δ
E [ P b ] = 1 2 π 0 erfc ( μ δ 2 cos 2 [ 2 σx ] ) exp [ x 2 ] f δ ( δ ) d x d δ
= 1 2 π 0 i = 1 N w i erfc ( μ δ 2 cos 2 [ 2 σ x i ] f δ ( δ ) d δ
erfc ( x ) = 1 π G 1,2 2,0 [ x 0 , 1 2 1 ]
K ν ( x ) = 1 2 G 0,2 2,0 [ x 2 4 ν 2 , ν 2 ]
E [ P b ] = 1 2 π i = 1 N w i 0 erfc ( μ δ 2 cos 2 [ 2 σ x i ] 2 ( αβ ) α + β 2 Γ ( α ) Γ ( β ) ( δ ) α + β 2 1 K α β ( 2 αβδ ) ) d δ
= ( αβ ) α + β 2 2 π Γ ( α ) Γ ( β ) i = 1 N w i 0 ( δ ) α + β 2 1 G 1,2 2,0 [ μ δ 2 cos 2 [ 2 σ x i ] 0 , 1 2 1 ]
× G 0,2 2,0 [ αβ ( δ ) α β 2 , β α 2 ] d δ .
E [ P b ] = 2 α + β 3 π 2 Γ ( α ) Γ ( β ) i = 1 N w i G 5,2 2,4 [ 2 4 μ cos 2 [ 2 σ x i ] ( αβ ) 2 0 , 1 2 1 α 2 , 2 α 2 , 1 β 2 , 2 β 2 , 1 ] .

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