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Journals >Chinese Optics Letters >Volume 16 >Issue 12 >Page 120101 > Article

Chinese Optics Letters
Vol. 16, Issue 12, 120101 (2018)

Average BER of coherent optical QPSK systems with phase errors over M turbulence channels

Jiashun Hu¹, Zaichen Zhang^1,*, Liang Wu¹, Jian Dang¹, and Guanghao Zhu²

Author Affiliations

¹National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China

²School of Electronic Science and Engineering, Nanjing University, Nanjing 210023, China

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DOI: 10.3788/COL201816.120101 Cite this Article Set citation alerts

Jiashun Hu, Zaichen Zhang, Liang Wu, Jian Dang, Guanghao Zhu, "Average BER of coherent optical QPSK systems with phase errors over M turbulence channels," Chin. Opt. Lett. 16, 120101 (2018) Copy Citation Text

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Abstract

The average bit-error-rate (BER) performance is studied for a coherent free-space optical communication system employing differentially encoded quadrature phase-shift keying (QPSK) with the Mth-power phase estimation method. A closed-form expression, considering the combined effects of the Málaga (M) turbulence fading, pointing errors, and phase estimation errors, is derived in terms of Meijer’s G function. Numerical and Monte Carlo simulation results are presented to verify the derived expression.

i (t) = i_{I} (t) + j i_{Q} (t) = R \sqrt{P_{s} P_{LO}} \exp {j [θ_{s} (t) + θ_{n} (t)]} + n (t),

(1)

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{\bar{i}}_{shot}^{2} = 2 e R \frac{P_{LO}}{2} \frac{B}{2},

(2)

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γ_{s} = \frac{{| i (t) |}^{2}}{2 σ_{n}^{2}} = \frac{R P_{s}}{e B} = \frac{η P_{s}}{ℏ f B},

(3)

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γ_{b} = \frac{γ_{s}}{\log_{2} M} = \frac{η A_{r}}{2 ℏ f B} I .

(4)

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{\bar{γ}}_{b} = E [γ_{b}] = \frac{η A_{r}}{2 ℏ f B} E [I],

(5)

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f_{I_{a}} (I_{a}) = A \sum_{k = 1}^{β} a_{k} I_{a}^{\frac{α + k}{2} - 1} K_{α - k} (2 \sqrt{\frac{α β I_{a}}{γ β + Ω^{'}}}),

(6)

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{\begin{cases} A = \frac{2 α^{\frac{α}{2}}}{γ^{1 + \frac{α}{2}} Γ (α)} {(\frac{γ β}{γ β + Ω^{'}})}^{\frac{α}{2} + β}; \\ a_{k} = (\begin{array}{l} β - 1 \\ k - 1 \end{array}) \frac{1}{(k - 1)!} {(\frac{Ω^{'}}{γ})}^{k - 1} {(\frac{α}{β})}^{\frac{k}{2}} {(γ β + Ω^{'})}^{1 - \frac{k}{2}} . \end{cases}

(7)

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f_{I_{p}} (I_{p}) = \frac{g^{2}}{A_{0}^{g^{2}}} I_{p}^{g^{2} - 1}, 0 \leq I_{p} \leq A_{0} .

(8)

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f_{I} (I) = \frac{g^{2} A}{2 I} \sum_{k = 1}^{β} b_{k} G_{1, 3}^{3, 0} (\frac{α β}{γ β + Ω^{'}} \frac{I}{A_{0} I_{l}} | \begin{array}{l} g^{2} + 1 \\ g^{2}, α, k \end{array}),

(9)

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f_{γ_{b}} (γ_{b}) = \frac{g^{2} A}{2 γ_{b}} \sum_{k = 1}^{β} b_{k} G_{1, 3}^{3, 0} (c \frac{γ_{b}}{{\bar{γ}}_{b}} | \begin{array}{l} g^{2} + 1 \\ g^{2}, α, k \end{array}),

(10)

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σ^{2} = \frac{N_{b}^{2} - 1}{6 N_{b}} σ_{δ}^{2} + \frac{σ_{n}^{2} (1 + 4.5 σ_{n}^{2})}{2 N_{b}},

(11)

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P_{b} (e | θ) = \frac{1}{2} erfc [\sqrt{γ_{b}} (\cos θ - \sin θ)] \times {1 - \frac{1}{2} erfc [\sqrt{γ_{b}} (\cos θ - \sin θ)]} + \frac{1}{2} erfc [\sqrt{γ_{b}} (\cos θ + \sin θ)] \times {1 - \frac{1}{2} erfc [\sqrt{γ_{b}} (\cos θ + \sin θ)]},

(12)

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P_{b} (e) = \int_{0}^{\infty} \int_{- \infty}^{\infty} P_{b} (e | θ) p (θ) f_{γ_{b}} (γ_{b}) d θ d γ_{b},

(13)

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P_{b} (e | θ) \approx erfc [\sqrt{γ_{b}} (\cos θ - \sin θ)] .

(14)

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P_{b} (e) = \frac{g^{2} A}{2 π} \sum_{i = 1}^{n} \sum_{k = 1}^{β} w_{i} b_{k} G_{3, 4}^{3, 2} (\frac{c}{d {\bar{γ}}_{b}} | \begin{array}{l} 1, \frac{1}{2}, g^{2} + 1 \\ g^{2}, α, k, 0 \end{array}),

(15)

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P_{b} (e) = \frac{g^{2}}{π Γ (α) Γ (β)} \sum_{i = 1}^{n} w_{i} G_{3, 4}^{3, 2} (\frac{α β g^{2}}{d {\bar{γ}}_{b} (1 + g^{2})} | \begin{array}{l} 1, \frac{1}{2}, g^{2} + 1 \\ g^{2}, α, β, 0 \end{array}) .

(16)

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Jiashun Hu, Zaichen Zhang, Liang Wu, Jian Dang, Guanghao Zhu, "Average BER of coherent optical QPSK systems with phase errors over M turbulence channels," Chin. Opt. Lett. 16, 120101 (2018)

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