Researching | Average bit error rate of multi-hop parallel decode-and-forward-based FSO cooperative system with the max–min criterion under the gamma–gamma distribution

Journals >Chinese Optics Letters >Volume 13 >Issue 8 >Page 080101 > Article

Chinese Optics Letters
Vol. 13, Issue 8, 080101 (2015)

Average bit error rate of multi-hop parallel decode-and-forward-based FSO cooperative system with the max–min criterion under the gamma–gamma distribution

Tian Cao¹, Ping Wang^1、2、*, Lixin Guo², Bensheng Yang¹, Jing Li¹, and Yintang Yang³

Author Affiliations

¹State Key Laboratory of Integrated Service Networks, School of Telecommunications Engineering, Xidian University, Xi’an 710071, China

²School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China

³Key Laboratory of the Ministry of Education for Wide Band-Gap Semiconductor Materials and Devices, School of Microelectronics, Xidian University, Xi’an 710071, China

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DOI: 10.3788/COL201513.080101 Cite this Article Set citation alerts

Tian Cao, Ping Wang, Lixin Guo, Bensheng Yang, Jing Li, Yintang Yang. Average bit error rate of multi-hop parallel decode-and-forward-based FSO cooperative system with the max–min criterion under the gamma–gamma distribution[J]. Chinese Optics Letters, 2015, 13(8): 080101 Copy Citation Text

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Abstract

The average bit error rate (BER) performance of a free-space optical (FOS) system based on the multi-hop parallel decode-and-forward cooperative communication method with an M-ary phase shift keying subcarrier intensity modulation is studied systematically. With the max–min criterion as the best path selection scheme, the probability density function and the cumulative distribution function of the gamma–gamma distribution random variable signal-to-noise ratio are derived. The analytical BER expression is then obtained in terms of the Gauss–Laguerre quadrature rule. Monte Carlo simulation is also provided to confirm the validity of the presented average BER model.

y_{m, n} = R_{p} h_{m, n} x_{m, n} + n_{m, n},

(1)

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f_{h_{m, n}} (h) = \frac{2 (α β)^{(α + β) / 2}}{Γ (α) Γ (β)} h^{(α + β) / 2 - 1} K_{α - β} [2 {(α β h)}^{1 / 2}],

(2)

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α = {\exp [0.49 σ_{R}^{2} / {(1 + 1.11 σ_{R}^{12 / 5})}^{7 / 6}] - 1}^{- 1},

(3)

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β = {\exp [0.51 σ_{R}^{2} / {(1 + 0.69 σ_{R}^{12 / 5})}^{5 / 6}] - 1}^{- 1},

(4)

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f_{γ_{m, n}} (γ) = \frac{B}{4} A_{m, n}^{α + β} γ^{\frac{α + β - 4}{4}} K_{α - β} (A_{m, n} γ^{1 / 4}),

(5)

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F_{γ_{m, n}} (γ) = B \int_{0}^{A_{m, n} γ^{1 / 4}} x^{(α + β - 1)} K_{α - β} (x) d x,

(6)

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F_{γ_{m, n}} (γ) = B \cdot 2^{α + β - 2} G_{1, 3}^{2, 1} (\frac{A_{m, n}^{2}}{4} γ^{1 / 2} | \begin{matrix} 1 \\ α, β, 0 \end{matrix}),

(7)

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γ_{\max, \min} = {{\max (\min (γ_{m, n}) |}_{n = 1}^{C}) |}_{m = 1}^{R} .

(8)

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{\hat{f}}_{γ_{m, n}} (γ) = \frac{B}{4} A^{α + β} γ^{\frac{α + β - 4}{4}} K_{α - β} (A γ^{1 / 4}),

(9)

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{\hat{F}}_{γ_{m, n}} (γ) = B \cdot 2^{α + β - 2} G_{1, 3}^{2, 1} (\frac{A^{2}}{4} γ^{1 / 2} | \begin{matrix} 1 \\ α, β, 0 \end{matrix}) .

(10)

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F_{γ_{\max, \min}} (γ) = {1 - {[1 - {\hat{F}}_{γ_{m, n}} (γ)]}^{C}}^{R} .

(11)

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f_{γ_{\max, \min}} (γ) = R C \cdot {1 - {[1 - {\hat{F}}_{γ_{m, n}} (γ)]}^{C}}^{R - 1} \times {[1 - {\hat{F}}_{γ_{m, n}} (γ)]}^{C - 1} {\hat{f}}_{γ_{m, n}} (γ) .

(12)

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F_{γ_{s, d}} (γ) = B \cdot 2^{α + β - 2} G_{1, 3}^{2, 1} (\frac{A^{2}}{4} γ^{1 / 2} | \begin{matrix} 1 \\ α, β, 0 \end{matrix}) .

(13)

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\begin{array}{l} if γ_{\max, \min} > γ_{s, d}, & using cooperative path \\ if γ_{s, d} > γ_{\max, \min}, & using SD path . \end{array}

(14)

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γ_{\max} = \max (γ_{\max, \min}, γ_{s, d}) .

(15)

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F_{γ_{\max}} (γ) = F_{γ_{\max, \min}} (γ) \cdot F_{γ_{s, d}} (γ) = {1 - {[1 - B \cdot 2^{α + β - 2} G_{1, 3}^{2, 1} (\frac{A^{2}}{4} γ^{1 / 2} | \begin{matrix} 1 \\ α, β, 0 \end{matrix})]}^{C}}^{R} \times B \cdot 2^{α + β - 2} G_{1, 3}^{2, 1} (\frac{A^{2}}{4} γ^{1 / 2} | \begin{matrix} 1 \\ α, β, 0 \end{matrix}),

(16)

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f_{γ_{\max}} (γ) = [\frac{B}{4} A^{α + β} γ^{\frac{α + β - 4}{4}} K_{α - β} (A γ^{1 / 4})] \times {{[1 - {(1 - {\hat{F}}_{γ_{m, n}} (γ))}^{C}]}^{R} + R C \cdot {\hat{F}}_{γ_{m, n}} (γ) \cdot {[1 - {(1 - {\hat{F}}_{γ_{m, n}} (γ))}^{C}]}^{R - 1} \cdot {(1 - {\hat{F}}_{γ_{m, n}} (γ))}^{C - 1}} .

(17)

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p_{e} (γ) = D (M) \cdot erfc (\sqrt{γ} \sin \frac{π}{M}),

(18)

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P_{e} = \int_{0}^{\infty} p_{e} (γ) \times f_{γ_{\max}} (γ) d γ = \frac{D (M)}{\sqrt{π}} \int_{0}^{\infty} γ^{- 1 / 2} e^{- γ} F_{γ_{\max}} (\frac{γ}{\sin^{2} (π / M)}) d γ .

(19)

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P_{e} \approx \frac{D (M)}{\sqrt{π}} \sum_{k = 1}^{n} Y_{k} F_{γ_{\max}} (\frac{x_{k}}{\sin^{2} (π / M)}),

(20)

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

(21)

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References (28)

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Tian Cao, Ping Wang, Lixin Guo, Bensheng Yang, Jing Li, Yintang Yang. Average bit error rate of multi-hop parallel decode-and-forward-based FSO cooperative system with the max–min criterion under the gamma–gamma distribution[J]. Chinese Optics Letters, 2015, 13(8): 080101

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