Diversity Coherent and Incoherent Receivers for Free-Space Optical Communication in the Presence and Absence of Interference

Etty J. Lee; Vincent W.S. Chan

doi:10.1364/JOCN.1.000463

Journal of Optical Communications and Networking
Vol. 1,
Issue 5,
pp. 463-483
(2009)
•https://doi.org/10.1364/JOCN.1.000463

Diversity Coherent and Incoherent Receivers for Free-Space Optical Communication in the Presence and Absence of Interference

Etty J. Lee and Vincent W.S. Chan

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Etty J. Lee and Vincent W.S. Chan, "Diversity Coherent and Incoherent Receivers for Free-Space Optical Communication in the Presence and Absence of Interference," J. Opt. Commun. Netw. 1, 463-483 (2009)

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Abstract

When communicating optically through the clear atmospheric channel, not only is there atmospheric turbulence and background noise, but there may also be interference from other sources such as other users in a multiple access system. Even if the interference is off axis, its signal can couple into the receiver through scattering. We would like our communication system to perform well in the presence of interference, background noise, and fading. This paper investigates the performance of diversity coherent and incoherent receivers in the presence of fading, background noise, and various interference types. We find that diversity coherent detection provides significant power gain over diversity direct detection and that most of the benefit of diversity coherent detection can be achieved with a small amount of diversity. Moreover, we find that diversity always improves the performance of coherent detection, whereas in the presence of worst-case interference, diversity degrades the performance of direct detection. This paper also describes a sensible way to select the amount of diversity and power margin to deal with atmospheric turbulence and interference and quantifies the amount of interference that the system can handle while still achieving a given outage probability.

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Interference Type	Worst-Case Duty Cycle in Absence of Turbulence [Corresponding to $P (e)$ ]	Worst-Case Duty Cycle in Presence of Turbulence (Corresponding to $P_{outage}$ )
Direct Detection:
Constant on for half symbol	$\frac{N N_{I}}{m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}} {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$


Gaussian on for entire symbol	$\frac{N N_{I}}{m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}} \sqrt{2 (- \ln 2 + θ_{thresh})} {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$


Gaussian on for half symbol	$\frac{2 N N_{I}}{m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}} (2 \sqrt{- \ln 2 + θ_{thresh}} + 1) {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$


Canceling on for entire symbol	$\frac{2 N N_{I}}{m N_{S}^{*}}$	$\frac{2 N N_{I}}{m N_{S}^{*}}$

Canceling on for half symbol	$\frac{N N_{I}}{m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}}$
Homodyne Detection:
Gaussian	$\frac{N_{I}}{2 m N_{S}^{*}}$	$\frac{N_{I}}{2 m N_{S}^{*}} (- \ln 2 + θ_{thresh}) {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$


Canceling	$\frac{N_{I}}{m N_{S}^{*}}$	$\frac{N_{I}}{m N_{S}^{*}} {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$

Interference Type	Error Probability (in Absence of Turbulence)	Outage Probability (in Presence of Turbulence)
Direct Detection:
Constant on for half symbol	$\frac{N N_{I}}{4 m N_{S}^{*}}$	$\frac{N N_{I}}{2 m N_{S}^{*}} (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$


Gaussian on for entire symbol	$\frac{N N_{I}}{2 m N_{S}^{*}} e^{- 1 ∕ 2}$	$\frac{N N_{I}}{2 m N_{S}^{*}} \sqrt{2 (- \ln 2 + θ_{thresh})} (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$


Gaussian on for half symbol	$\frac{N N_{I}}{m N_{S}^{*}} e^{- 1 ∕ 4}$	$\frac{N N_{I}}{2 m N_{S}^{*}} (2 \sqrt{- \ln 2 + θ_{thresh}} + 1) (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$


Canceling on for entire symbol	$\frac{N N_{I}}{m N_{S}^{*}}$	$\frac{2 N N_{I}}{m N_{S}^{*}}$

Canceling on for half symbol	$\frac{N N_{I}}{2 m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}}$
Homodyne Detection:
Gaussian	$\frac{N_{I}}{4 m N_{S}^{*}} e^{- 1}$	$\frac{N_{I}}{4 m N_{S}^{*}} (- \ln 2 + θ_{thresh}) (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$


Canceling	$\frac{N_{I}}{4 m N_{S}^{*}}$	$\frac{N_{I}}{2 m N_{S}^{*}} (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$

Diversity in DD and Homo	Power Gain
Presence of interference
DD and Homo use diversity N	$(2 \sqrt{- \ln 2 + θ_{thresh}} + 1) N$
DD uses diversity N, Homo uses diversity $N_{opt}$	$\frac{N_{opt} (2 \sqrt{- \ln 2 + θ_{thresh}} + 1) (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N_{opt}})}{(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}$



DD uses diversity N, Homo uses diversity $N_{optGaussianHalfSymbolInterf.}$	$\frac{(2 \sqrt{- \ln 2 + θ_{thresh}} + 1) e^{4 σ_{χ}^{2}}}{(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}$


Absence of interference
DD and Homo use diversity N	$2 \frac{θ_{thresh} + \sqrt{2 θ_{thresh} N N_{n}}}{(1 + N_{n}) (- \ln 2 + θ_{thresh})}$

DD uses diversity N, Homo uses diversity $N_{opt}$	$k_{1} \exp {- \sqrt{- 2 \ln (2 P_{outage}) \ln (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})} - \frac{1}{2} \ln (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})},$
where

	$k_{1} = 2 (\frac{θ^{thresh} + \sqrt{2 θ^{thresh} N_{opt} N_{n}}}{(1 + N_{n}) (- \ln 2 + θ^{thresh})}) \sqrt{1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N_{opt}}} \exp {\sqrt{- 2 \ln (2 P_{outage}) \ln (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N_{opt}})}}$

DD uses diversity N, Homo uses diversity $N_{optGaussianHalfSymbolInterf.}$	$c_{1} \exp {\begin{cases} - \sqrt{- 2 \ln (2 P_{outage}) \ln (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})} \\ - \frac{1}{2} \ln (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N}) \end{cases}},$
where

	$c_{1} = 2 (\frac{θ^{thresh} + \sqrt{2 θ^{thresh} N_{n}}}{(1 + N_{n}) (- \ln 2 + θ^{thresh})}) e^{2 σ_{χ}^{2}} \exp {\sqrt{- 2 \ln (2 P_{outage}) 4 σ_{χ}^{2}}}$

Interference Type	Worst-Case Duty Cycle in Absence of Turbulence [Corresponding to $P (e)$ ]	Worst-Case Duty Cycle in Presence of Turbulence (Corresponding to $P_{outage}$ )
Direct Detection:
Constant on for half symbol	$\frac{N N_{I}}{m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}} {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$


Gaussian on for entire symbol	$\frac{N N_{I}}{m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}} \sqrt{2 (- \ln 2 + θ_{thresh})} {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$


Gaussian on for half symbol	$\frac{2 N N_{I}}{m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}} (2 \sqrt{- \ln 2 + θ_{thresh}} + 1) {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$


Canceling on for entire symbol	$\frac{2 N N_{I}}{m N_{S}^{*}}$	$\frac{2 N N_{I}}{m N_{S}^{*}}$

Canceling on for half symbol	$\frac{N N_{I}}{m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}}$
Homodyne Detection:
Gaussian	$\frac{N_{I}}{2 m N_{S}^{*}}$	$\frac{N_{I}}{2 m N_{S}^{*}} (- \ln 2 + θ_{thresh}) {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$


Canceling	$\frac{N_{I}}{m N_{S}^{*}}$	$\frac{N_{I}}{m N_{S}^{*}} {(1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})}^{3 ∕ 2}$

Interference Type	Error Probability (in Absence of Turbulence)	Outage Probability (in Presence of Turbulence)
Direct Detection:
Constant on for half symbol	$\frac{N N_{I}}{4 m N_{S}^{*}}$	$\frac{N N_{I}}{2 m N_{S}^{*}} (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$


Gaussian on for entire symbol	$\frac{N N_{I}}{2 m N_{S}^{*}} e^{- 1 ∕ 2}$	$\frac{N N_{I}}{2 m N_{S}^{*}} \sqrt{2 (- \ln 2 + θ_{thresh})} (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$


Gaussian on for half symbol	$\frac{N N_{I}}{m N_{S}^{*}} e^{- 1 ∕ 4}$	$\frac{N N_{I}}{2 m N_{S}^{*}} (2 \sqrt{- \ln 2 + θ_{thresh}} + 1) (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$


Canceling on for entire symbol	$\frac{N N_{I}}{m N_{S}^{*}}$	$\frac{2 N N_{I}}{m N_{S}^{*}}$

Canceling on for half symbol	$\frac{N N_{I}}{2 m N_{S}^{*}}$	$\frac{N N_{I}}{m N_{S}^{*}}$
Homodyne Detection:
Gaussian	$\frac{N_{I}}{4 m N_{S}^{*}} e^{- 1}$	$\frac{N_{I}}{4 m N_{S}^{*}} (- \ln 2 + θ_{thresh}) (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$


Canceling	$\frac{N_{I}}{4 m N_{S}^{*}}$	$\frac{N_{I}}{2 m N_{S}^{*}} (1 + \frac{e^{4 σ_{χ}^{2}} - 1}{N})$

Abstract

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Equations (62)

Journal of Optical Communications and Networking