Abstract

In the present contribution, we introduce a wireless optical communication-based system architecture that is shown to significantly improve the reliability and the spectral and power efficiency of the transcutaneous link in cochlear implants (CIs). We refer to the proposed system as an optical wireless cochlear implant (OWCI). In order to provide a quantified understanding of its design parameters, we establish a theoretical framework that takes into account the channel particularities, the integration area of the internal unit, the transceivers’ misalignment, and the characteristics of the optical units. To this end, we derive explicit expressions for the corresponding average signal-to-noise-ratio, outage probability, ergodic spectral efficiency and capacity of the transcutaneous optical link (TOL). These expressions are subsequently used to assess the dependence of the TOL’s communication quality on the transceivers’ design parameters and the corresponding channel’s characteristics. The offered analytic results are corroborated with respective results from Monte Carlo simulations. Our findings reveal that the OWCI is a particularly promising architecture that drastically increases the reliability and effectiveness of the CI TOL, whilst it requires considerably lower transmittal power when compared to the corresponding widely-used radio frequency (RF) solution.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

M. Z. Chowdhury, M. T. Hossan, A. Islam, and Y. M. Jang, “A comparative survey of optical wireless technologies: Architectures and applications,” IEEE Access 6, 9819–9840 (2018).
[Crossref]

2017 (4)

B. S. Wilson, “The modern cochlear implant: A triumph of biomedical engineering and the first substantial restoration of human sense using a medical intervention,” IEEE Pulse 8, 29–32 (2017).
[Crossref] [PubMed]

K. Agarwal, R. Jegadeesan, Y. X. Guo, and N. V. Thakor, “Wireless power transfer strategies for implantable bioelectronics,” IEEE Rev. Biomed. Eng. 10, 136–161 (2017).
[Crossref] [PubMed]

H. J. Kim, H. Hirayama, S. Kim, K. J. Han, R. Zhang, and J. W. Choi, “Review of near-field wireless power and communication for biomedical applications,” IEEE Access 5, 21264–21285 (2017).
[Crossref]

M. A. Esmail, H. Fathallah, and M. S. Alouini, “Outage probability analysis of FSO links over foggy channel,” IEEE Photonics J. 9, 1–12 (2017).

2016 (4)

J. Gao, Y. Zhang, M. Cheng, Y. Zhu, and Z. Hu, “Average capacity of ground-to-train wireless optical communication links in the non-Kolmogorov and gamma–gamma distribution turbulence with pointing errors,” Opt. Commun. 358, 147–153 (2016).
[Crossref]

I. S. Ansari, F. Yilmaz, and M. S. Alouini, “Performance analysis of free-space optical links over Málaga (ℳ) turbulence channels with pointing errors,” IEEE Trans. Wirel. Commun. 15, 91–102 (2016).
[Crossref]

M. N. Islam and M. R. Yuce, “Review of medical implant communication system (MICS) band and network,” ICT Express 2, 188–194 (2016).
[Crossref]

N. Kallweit, P. Baumhoff, A. Krueger, N. Tinne, A. Kral, T. Ripken, and H. Maier, “Optoacoustic effect is responsible for laser-induced cochlear responses,” Sci. Rep. 6, 1–10 (2016).
[Crossref]

2015 (5)

E. Zedini and M.-S. Alouini, “Multihop relaying over im/dd fso systems with pointing errors,” J. Light. Technol. 33, 5007–5015 (2015).
[Crossref]

Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. Cheng, “Emerging optical wireless communications-advances and challenges,” IEEE J. Sel. Areas Commun. 33, 1738–1749 (2015).
[Crossref]

T. Liu, J. Anders, and M. Ortmanns, “Bidirectional optical transcutaneous telemetric link for brain machine interface,” Electron. Lett. 51, 1969–1971 (2015).
[Crossref]

I. S. Ansari, M. S. Alouini, and J. Cheng, “Ergodic capacity analysis of free-space optical links with nonzero boresight pointing errors,” IEEE Trans. Wirel. Commun. 14, 4248–4264 (2015).
[Crossref]

E. Zedini, I. S. Ansari, and M. S. Alouini, “Performance analysis of mixed nakagami-m and gamma-gamma dual-hop FSO transmission systems,” IEEE Photonics J. 7, 1–20 (2015).

2014 (5)

M. Usman, H. C. Yang, and M. S. Alouini, “Practical switching-based hybrid FSO/RF transmission and its performance analysis,” IEEE Photonics J. 6, 1–13 (2014).
[Crossref]

J.-Y. Wang, J.-B. Wang, M. Chen, N. Huang, L.-Q. Jia, and R. Guan, “Ergodic capacity and outage capacity analysis for multiple-input single-output free-space optical communications over composite channels,” Opt. Eng. 53, 1–8 (2014).

M. Cheng, Y. Zhang, J. Gao, F. Wang, and F. Zhao, “Average capacity for optical wireless communication systems over exponentiated Weibull distribution non-Kolmogorov turbulent channels,” Appl. Opt. 53, 4011–4017 (2014).
[Crossref] [PubMed]

C. Goßler, C. Bierbrauer, R. Moser, M. Kunzer, K. Holc, W. Pletschen, K. Köhler, J. Wagner, M. Schwaerzle, P. Ruther, O. Paul, J. Neef, D. Keppeler, G. Hoch, T. Moser, and U. T. Schwarz, “GaN-based micro-LED arrays on flexible substrates for optical cochlear implants,” J. Phys. D: Appl. Phys. 47, 205401 (2014).
[Crossref]

C.-P. Richter and X. Tan, “Photons and neurons,” Hear. Res. 311, 72–88 (2014). Annual Reviews.
[Crossref] [PubMed]

2013 (4)

A. C. Thompson, S. A. Wade, N. C. Pawsey, and P. R. Stoddart, “Infrared neural stimulation: Influence of stimulation site spacing and repetition rates on heating,” IEEE Trans. Biomed. Eng. 60, 3534–3541 (2013).
[Crossref] [PubMed]

I. S. Ansari, F. Yilmaz, and M. S. Alouini, “Impact of pointing errors on the performance of mixed RF/FSO dual-hop transmission systems,” IEEE Wirel. Commun. Lett. 2, 351–354 (2013).
[Crossref]

M. Z. Hassan, M. J. Hossain, and J. Cheng, “Ergodic capacity comparison of optical wireless communications using adaptive transmissions,” Opt. Express 21, 20346–20362 (2013).
[Crossref] [PubMed]

F. Benkhelifa, Z. Rezki, and M. S. Alouini, “Low snr capacity of fso links over gamma-gamma atmospheric turbulence channels,” IEEE Commun. Lett. 17, 1264–1267 (2013).
[Crossref]

2012 (2)

W. H. Ko, “Early history and challenges of implantable electronics,” J. Emerg. Technol. Comput. Syst. 8, 1–9 (2012).
[Crossref]

Y. Gil, N. Rotter, and S. Arnon, “Feasibility of retroreflective transdermal optical wireless communication,” Appl. Opt. 51, 4232–4239 (2012).
[Crossref] [PubMed]

2011 (1)

A. N. Bashkatov, E. A. Genina, and V. V. Tuchin, “Optical properties of skin, subcutaneous, and muscle tissues: a review,” J. Innov. Opt. Heal. Sci. 4, 9–38 (2011).
[Crossref]

2010 (1)

2009 (3)

A. Lapidoth, S. M. Moser, and M. A. Wigger, “On the capacity of free-space optical intensity channels,” IEEE Trans. Inf. Theory 55, 4449–4461 (2009).
[Crossref]

W. O. Popoola and Z. Ghassemlooy, “Bpsk subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Light. Technol. 27, 967–973 (2009).
[Crossref]

A. R. Duke, J. M. Cayce, J. D. Malphrus, P. Konrad, A. Mahadevan-Jansen, and E. D. Jansen, “Combined optical and electrical stimulation of neural tissue in vivo,” J. Biomed. Opt. 14, 060501 (2009).
[Crossref]

2008 (5)

B. S. Wilson and M. F. Dorman, “Cochlear implants: A remarkable past and a brilliant future,” Hear. Res. 242, 3–21 (2008).
[Crossref] [PubMed]

F. G. Zeng, S. Rebscher, W. Harrison, X. Sun, and H. Feng, “Cochlear implants: System design, integration, and evaluation,” IEEE Rev. Biomed. Eng. 1, 115–142 (2008).
[Crossref] [PubMed]

D. M. Ackermann, B. Smith, X.-F. Wang, K. L. Kilgore, and P. H. Peckham, “Designing the optical interface of a transcutaneous optical telemetry link,” IEEE Trans. Biomed. Eng. 55, 1365–1373 (2008).
[Crossref] [PubMed]

J. J. Sit and R. Sarpeshkar, “A cochlear-implant processor for encoding music and lowering stimulation power,” IEEE Pervasive Comput. 7, 40–48 (2008).
[Crossref]

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “Ber performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12, 44–46 (2008).
[Crossref]

2007 (1)

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Light. Technol. 25, 1702–1710 (2007).
[Crossref]

2006 (2)

I. Hochmair, P. Nopp, C. Jolly, M. Schmidt, H. Schößer, C. Garnham, and I. Anderson, “MED-EL cochlear implants: State of the art and a glimpse into the future,” Trends Amplif. 10, 201–219 (2006).
[Crossref] [PubMed]

J. F. Patrick, P. A. Busby, and P. J. Gibson, “The development of the nucleus® freedom cochlear implant system,” Trends Amplif. 10, 175–200 (2006).
[Crossref] [PubMed]

2005 (2)

E. Okamoto, Y. Yamamoto, Y. Inoue, T. Makino, and Y. Mitamura, “Development of a bidirectional transcutaneous optical data transmission system for artificial hearts allowing long-distance data communication with low electric power consumption,” J. Artif. Organs 8, 149–153 (2005).
[Crossref] [PubMed]

A. Bashkatov, E. Genina, V. Kochubey, and V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm,” J. Phys. D: Appl. Phys. 38, 2543 (2005).
[Crossref]

2004 (2)

M. F. Dorman and B. S. Wilson, “The design and function of cochlear implants,” Am. Sci. 92, 436–445 (2004).
[Crossref]

J. L. Abita and W. Schneider, “Transdermal optical communications,” Johns Hopkins APL Tech. Dig. 25, 261 (2004).

2003 (1)

2002 (1)

S. L. Pinski and R. G. Trohman, “Interference in implanted cardiac devices, part i,” Pacing Clin. Electrophysiol 25, 1367–1381 (2002).
[Crossref] [PubMed]

2001 (2)

Y. Du, X. Hu, M. Cariveau, X. Ma, G. Kalmus, and J. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167 (2001).
[Crossref] [PubMed]

T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6, 167–176 (2001).
[Crossref] [PubMed]

1998 (1)

C. R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex-vivo human skin and subcutaneous tissues measured using the monte carlo inversion technique,” Phys. Med. Biol. 43, 2465 (1998).
[Crossref] [PubMed]

1996 (1)

E. K. Chan, B. Sorg, D. Protsenko, M. O’Neil, M. Motamedi, and A. J. Welch, “Effects of compression on soft tissue optical properties,” IEEE J. Sel. Top. Quantum Electron. 2, 943–950 (1996).
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C. R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex-vivo human skin and subcutaneous tissues measured using the monte carlo inversion technique,” Phys. Med. Biol. 43, 2465 (1998).
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A. Bashkatov, E. Genina, V. Kochubey, and V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm,” J. Phys. D: Appl. Phys. 38, 2543 (2005).
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A. N. Bashkatov, E. A. Genina, and V. V. Tuchin, “Optical properties of skin, subcutaneous, and muscle tissues: a review,” J. Innov. Opt. Heal. Sci. 4, 9–38 (2011).
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Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. Cheng, “Emerging optical wireless communications-advances and challenges,” IEEE J. Sel. Areas Commun. 33, 1738–1749 (2015).
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J. F. Patrick, P. A. Busby, and P. J. Gibson, “The development of the nucleus® freedom cochlear implant system,” Trends Amplif. 10, 175–200 (2006).
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Guo, Y. X.

K. Agarwal, R. Jegadeesan, Y. X. Guo, and N. V. Thakor, “Wireless power transfer strategies for implantable bioelectronics,” IEEE Rev. Biomed. Eng. 10, 136–161 (2017).
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F. G. Zeng, S. Rebscher, W. Harrison, X. Sun, and H. Feng, “Cochlear implants: System design, integration, and evaluation,” IEEE Rev. Biomed. Eng. 1, 115–142 (2008).
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Hirayama, H.

H. J. Kim, H. Hirayama, S. Kim, K. J. Han, R. Zhang, and J. W. Choi, “Review of near-field wireless power and communication for biomedical applications,” IEEE Access 5, 21264–21285 (2017).
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C. Goßler, C. Bierbrauer, R. Moser, M. Kunzer, K. Holc, W. Pletschen, K. Köhler, J. Wagner, M. Schwaerzle, P. Ruther, O. Paul, J. Neef, D. Keppeler, G. Hoch, T. Moser, and U. T. Schwarz, “GaN-based micro-LED arrays on flexible substrates for optical cochlear implants,” J. Phys. D: Appl. Phys. 47, 205401 (2014).
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I. Hochmair, P. Nopp, C. Jolly, M. Schmidt, H. Schößer, C. Garnham, and I. Anderson, “MED-EL cochlear implants: State of the art and a glimpse into the future,” Trends Amplif. 10, 201–219 (2006).
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C. Goßler, C. Bierbrauer, R. Moser, M. Kunzer, K. Holc, W. Pletschen, K. Köhler, J. Wagner, M. Schwaerzle, P. Ruther, O. Paul, J. Neef, D. Keppeler, G. Hoch, T. Moser, and U. T. Schwarz, “GaN-based micro-LED arrays on flexible substrates for optical cochlear implants,” J. Phys. D: Appl. Phys. 47, 205401 (2014).
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A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Light. Technol. 25, 1702–1710 (2007).
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Y. Du, X. Hu, M. Cariveau, X. Ma, G. Kalmus, and J. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167 (2001).
[Crossref] [PubMed]

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J. Gao, Y. Zhang, M. Cheng, Y. Zhu, and Z. Hu, “Average capacity of ground-to-train wireless optical communication links in the non-Kolmogorov and gamma–gamma distribution turbulence with pointing errors,” Opt. Commun. 358, 147–153 (2016).
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J.-Y. Wang, J.-B. Wang, M. Chen, N. Huang, L.-Q. Jia, and R. Guan, “Ergodic capacity and outage capacity analysis for multiple-input single-output free-space optical communications over composite channels,” Opt. Eng. 53, 1–8 (2014).

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I. Oppermann, M. Hämäläinen, and J. Iinatti, UWB: Theory and applications (John Wiley & Sons, 2005).

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Jegadeesan, R.

K. Agarwal, R. Jegadeesan, Y. X. Guo, and N. V. Thakor, “Wireless power transfer strategies for implantable bioelectronics,” IEEE Rev. Biomed. Eng. 10, 136–161 (2017).
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Jolly, C.

I. Hochmair, P. Nopp, C. Jolly, M. Schmidt, H. Schößer, C. Garnham, and I. Anderson, “MED-EL cochlear implants: State of the art and a glimpse into the future,” Trends Amplif. 10, 201–219 (2006).
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N. Kallweit, P. Baumhoff, A. Krueger, N. Tinne, A. Kral, T. Ripken, and H. Maier, “Optoacoustic effect is responsible for laser-induced cochlear responses,” Sci. Rep. 6, 1–10 (2016).
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M. Schultz, P. Baumhoff, N. Kallweit, M. Sato, A. Krüger, T. Ripken, T. Lenarz, and A. Kral, “Optical stimulation of the hearing and deaf cochlea under thermal and stress confinement condition,” in Optical Techniques in Neurosurgery, Neurophotonics, and Optogenetics, (International Society for Optics and Photonics, 2014), p. 892816.

Kalmus, G.

Y. Du, X. Hu, M. Cariveau, X. Ma, G. Kalmus, and J. Lu, “Optical properties of porcine skin dermis between 900 nm and 1500 nm,” Phys. Med. Biol. 46, 167 (2001).
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D. M. Ackermann, B. Smith, X.-F. Wang, K. L. Kilgore, and P. H. Peckham, “Designing the optical interface of a transcutaneous optical telemetry link,” IEEE Trans. Biomed. Eng. 55, 1365–1373 (2008).
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H. J. Kim, H. Hirayama, S. Kim, K. J. Han, R. Zhang, and J. W. Choi, “Review of near-field wireless power and communication for biomedical applications,” IEEE Access 5, 21264–21285 (2017).
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H. J. Kim, H. Hirayama, S. Kim, K. J. Han, R. Zhang, and J. W. Choi, “Review of near-field wireless power and communication for biomedical applications,” IEEE Access 5, 21264–21285 (2017).
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J. Gao, Y. Zhang, M. Cheng, Y. Zhu, and Z. Hu, “Average capacity of ground-to-train wireless optical communication links in the non-Kolmogorov and gamma–gamma distribution turbulence with pointing errors,” Opt. Commun. 358, 147–153 (2016).
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Figures (7)

Fig. 1
Fig. 1 System architecture of OWCI. In this figure ‘DSP unit’ and ‘STM unit’ denote the digital signal processing and stimulation units, respectively.
Fig. 2
Fig. 2 Impact of skin thickness on the OWCI’s effectiveness.
Fig. 3
Fig. 3 Impact of misalignment on OWCI’s performance.
Fig. 4
Fig. 4 The joint effect of skin thickness and misalignment.
Fig. 5
Fig. 5 The impact of channel wavelength selectivity on the performance of OWCIs.
Fig. 6
Fig. 6 The impact of TX design parameters on the performance of OWCI.
Fig. 7
Fig. 7 The impact of RX design parameters on the performance of OWCI.

Tables (1)

Tables Icon

Table 1 Comparison between OWCIs and RFCIs.

Equations (57)

Equations on this page are rendered with MathJax. Learn more.

y = R h x + n
R = η q p v .
h = h l h p
h l = exp ( 1 2 α ( λ ) δ )
n = n s + n th
n s = n b + n DC
σ b 2 = 2 q RBP b
σ DC 2 = 2 q BI DC
σ 2 = σ b 2 + σ DC 2 + σ th 2 .
γ = R 2 exp ( α ( λ ) δ ) h p 2 P s 2 q R B P b + 2 q B I DC + σ th 2
γ = R 2 exp ( α ( λ ) δ ) h p 2 P ˜ s 2 q R P b + 2 q I DC + N 0
γ ˜ = R 2 exp ( α ( λ ) δ ) P ˜ s 2 q R P b + 2 q I D C + N 0 ξ A 0 2 ξ + 2
A 0 = [ erf ( υ ) ] 2
υ = π β 2 w δ .
ξ = w eq 2 4 σ s 2
w eq 2 = w δ 2 π erf ( υ ) 2 υ exp ( υ 2 ) .
C = 1 2 log 2 ( 1 + ( λ ) A 0 2 ) 1 2 A 0 2 ( λ ) ln ( 2 ) Φ ( A 0 2 ( λ ) , 1 , 1 + ξ 2 )
( λ ) = ψ R 2 exp ( α ( λ ) δ ) P ˜ s 2 q R P b + 2 q I DC + N 0 .
C > 1 2 log 2 ( 1 + ( λ ) A 0 2 ) 1 ξ ln ( 2 )
C = 1 2 log 2 ( 1 + ψ ξ + 2 ξ γ ˜ ) ψ 2 ln ( 2 ) ξ + 2 ξ γ ˜ Φ ( ψ ξ + 2 ξ γ ˜ , 1 , 1 + ξ 2 )
C = 1 2 log 2 ( 1 + ψ ξ + 2 ξ γ ˜ ) 1 ξ ln ( 2 ) .
C B = 1 2 i = 1 K Δ f ( log 2 ( 1 + ( λ i ) A 0 2 ) A 0 2 ( λ i ) ln ( 2 ) Φ ( A 0 2 ( λ i ) , 1 , 1 + ξ 2 ) ) .
C B = 1 2 B log 2 ( 1 + ( λ ) A 0 2 ) 1 2 B A 0 2 ( λ ) ln ( 2 ) Φ ( A 0 2 ( λ ) , 1 , 1 + ξ 2 )
C B > 1 2 i = 1 K Δ f log 2 ( 1 + ( λ i ) A 0 2 ) 1 ξ ln ( 2 )
C B > 1 2 log 2 ( 1 + ( λ ) A 0 2 ) B ξ ln ( 2 )
P o ( γ th ) = { 1 A 0 ξ ( 2 q R P b + 2 q I DC + N 0 R 2 exp ( α ( λ ) δ ) P ˜ s γ th ) ξ 2 , γ th R 2 exp ( α ( λ ) δ ) A 0 2 P ˜ s 2 q R P b + 2 q I DC + N 0 , 1 , o t h e r w i s e .
P o ( r th ) = { ( ξ ξ + 2 γ t h γ ˜ ) ξ 2 , γ th ξ + 2 ξ γ ˜ , 1 , otherwise
σ s 2 = w eq 2 8 log ( P o ( γ th ) ) = w eq 2 ln ( ) 8 ln ( P o ( γ th ) ) , γ th R 2 exp ( α ( λ ) δ ) A 0 2 P ˜ s 2 q R P b + 2 q I DC + N 0
σ s = w eq ln ( ) 2 2 ln ( P o ( γ th ) ) = w e q 2 2 α ( λ ) δ + ln ( γ th ) + ln ( 2 q R P b + 2 q I DC + N 0 ) 2 ln ( A 0 ) 2 ln ( R ) ln ( P ˜ s ) ln ( P o ( γ th ) )
= 2 q R P b + 2 q I DC + N 0 A 0 2 R 2 exp ( α ( λ ) δ ) P ˜ s γ th .
C B = B 2 log 2 ( 1 + ( λ ) A 0 2 ) B A 0 2 ( λ ) 2 ln ( 2 ) Φ ( A 0 2 ( λ ) , 1 , 1 + ln ( P o ( γ th ) ) ln ( ) )
C B > B 2 log 2 ( 1 + ( λ ) A 0 2 ) B ln ( ) 2 ln ( 2 ) ln ( P o ( γ th ) .
h p A 0 exp ( 2 r 2 w eq 2 ) .
f r ( r ) = r σ s 2 exp ( r 2 2 σ s 2 ) , r > 0
f h p ( x ) = ξ A 0 ξ x ξ 1 , 0 x A 0
F h p ( x ) 0 x f y ( x ) d y
F h p ( x ) = { 1 A 0 ξ x ξ 0 x A 0 1 , x A 0
F h p 2 ( x ) = P ( h p 2 x ) = P ( h p x ) = P ( h p x ) = F h p ( x )
F h p 2 ( x ) = { 1 A 0 ξ x ξ / 2 0 x A 0 2 1 , x A 0 2 .
f | p 2 ( x ) = d F h p 2 ( x ) d x
f h p 2 ( x ) = ξ 2 A 0 ξ x ξ / 2 1 .
γ ˜ = 𝔼 [ γ ] .
γ ˜ = R exp ( α ( λ ) δ ) P ˜ s 2 q R P b + 2 q I DC + N 0 ξ 2 A 0 ξ 0 A 0 2 x ξ / 2 d x
C = 1 2 𝔼 [ log 2 ( 1 + ψ γ ) ]
ψ = { 1 , for heterodyne RX e 2 π , for IM / DD RX
C = 1 2 0 A log 2 ( 1 + ψ R 2 exp ( α ( λ ) δ ) x 2 P ˜ s 2 q R P b + 2 q I DC + N 0 ) f h p ( x ) d x .
C = ξ 2 A 0 ξ ln ( 2 )
= 0 A 0 x ξ 1 ln ( 1 + ( λ ) x 2 ) d x
= 1 ξ 0 A 0 ln ( 1 + ( λ ) x 2 ) d x ξ d x d x .
= 1 ξ A 0 ξ ln ( 1 + ( λ ) A 0 2 ) 1 ξ 0 A 0 x ξ 1 + ( λ ) x 2 d x
= 1 ξ A 0 ξ ln ( 1 + ( λ ) A 0 2 ) 1 ξ A 0 ξ + 2 ( λ ) Φ ( A 0 2 ( λ ) , 1 , 1 + ξ 2 ) .
Φ ( a , b , c ) 1 Γ ( b ) 0 y b 1 e x y 1 a e y d y .
C = 1 2 log 2 ( 1 + ( λ ) A 0 2 ) 1 2 A 0 2 ( λ ) ln ( 2 ) 0 e ( 1 + ξ 2 ) y 1 + A 0 2 ( λ ) e y d y .
C > 1 2 log 2 ( 1 + ( λ ) A 0 2 1 2 A 0 2 ( λ ) ln ( 2 ) 0 e ( 1 + ξ 2 ) y + y A 0 2 ( λ ) d y .
P o ( r th ) = P r ( C r th ) = P r ( γ γ th )
γ th = 2 2 r th 1 ψ .
P o ( γ th ) = P r ( h p 2 2 q R P b + 2 q I DC + N 0 R 2 exp ( α ( λ ) δ ) P ˜ s γ th ) = F h p 2 ( 2 q R P b + 2 q I DC + N 0 R 2 exp ( α ( λ ) δ ) P ˜ s γ th )

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