Abstract

We analyze the information efficiency of a deep-space optical communication link with background noise, employing the pulse position modulation (PPM) format and a direct-detection receiver based on Geiger-mode photon counting. The efficiency, quantified using Shannon mutual information, is optimized with respect to the PPM order under the constraint of a given average signal power in simple and complete decoding scenarios. We show that the use of complete decoding, which retrieves information from all combinations of detector photocounts occurring within one PPM frame, allows one to achieve information efficiency scaling as the inverse of the square of the distance, i.e. proportional to the received signal power. This represents a qualitative enhancement compared to simple decoding, which treats multiple photocounts within a single PPM frame as erasures and leads to inverse-quartic scaling with the distance. We provide easily computable formulas for the link performance in the limit of diminishing signal power.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. W. D. Williams, M. Collins, D. M. Boronson, J. Lesch, and A. Biswas, “RF and optical communications: a comparison of high data rate returns from deep space in the 2020 timeframe,” NASA/TM-2007-214459, NASA Glenn Research Center, Cleveland, Ohio, 1–16 (2007)
  2. H. Hemmati, Deep-Space Optical Communication (Wiley, 2005) Chap. 4.
  3. A. Waseda, M. Sasaki, M. Takeoka, M. Fujiwara, M. Toyoshima, and A. Assalini, “Numerical evaluation of PPM for deep-space links,” J. Opt. Commun. Netw. 3(6), 514–521 (2011).
    [Crossref]
  4. S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3), 257–265 (2011)
    [Crossref]
  5. L. Rizzo, “Effective erasure codes for reliable computer communication protocols,” ACM SIGCOMM Computer Communication Review 27(2), 24–36 (1997).
    [Crossref]
  6. V. Giovannetti, R. Garcia-Patron, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Phot. 8, 796–800 (2014).
    [Crossref]
  7. S. Dolinar, K. M. Birnbaum, B. I. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2011), pp. 269–278.
  8. Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
    [Crossref]
  9. M. Jarzyna, P. Kuszaj, and K. Banaszek, “Incoherent on-off keying with classical and non-classical light,” Opt. Express 23(3), 3170–3175 (2015).
    [Crossref] [PubMed]
  10. H. W. Chung, S. Guha, and L. Zheng, “On capacity of optical communications over a lossy bosonic channel with a receiver employing the most general coherent electro-optic feedback control,” Phys. Rev. A 96, 012320 (2017).
    [Crossref]
  11. M. Toyoshima, W. R. Leeb, H. Kunimori, and T. Takano, “Comparison of microwave and light wave communication systems in space applications,” Optical Engineering 46(1), 015003 (2007).
    [Crossref]
  12. B. Moisson and W. Farr, “Range dependence of the optical communications channel,” IPN Progerss Report 42–199 (2014).
  13. J. G. Proakis and M. Salehi, Communication systems engineering (Prentice-Hall, Inc., 1994) pp. 11.
  14. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 2006), Chap. 8.
  15. S. Guha, “Structured optical receivers to attain superadditive capacity and the Holevo limit,” Phys. Rev. Lett. 106(24), 240502 (2011).
    [Crossref] [PubMed]
  16. M. Rosati, A. Mari, and V. Giovannetti, “Multiphase Hadamard receivers for classical communication on lossy bosonic channels,” Phys. Rev. A 94(6), 062325 (2016).
    [Crossref]
  17. K. Banaszek and M. Jachura, “Structured optical receivers for efficient deep-space communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2017), pp. 176–181.
  18. P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316–A334 (1964).
    [Crossref]
  19. M. Jarzyna, W. Zwoliński, M. Jachura, and K. Banaszek, “Optimizing deep-space optical communication under power constraints,” Proc. SPIE 10524 Free-Space Laser Communication and Atmospheric Propagation XXX, 105240A (15 February 2018).
  20. S. Verdu, “On channel capacity per unit cost,” IEEE Trans. Inf. Theor. 36(5), 1019–1030 (1990).
    [Crossref]
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    [Crossref] [PubMed]

2017 (1)

H. W. Chung, S. Guha, and L. Zheng, “On capacity of optical communications over a lossy bosonic channel with a receiver employing the most general coherent electro-optic feedback control,” Phys. Rev. A 96, 012320 (2017).
[Crossref]

2016 (1)

M. Rosati, A. Mari, and V. Giovannetti, “Multiphase Hadamard receivers for classical communication on lossy bosonic channels,” Phys. Rev. A 94(6), 062325 (2016).
[Crossref]

2015 (1)

2014 (2)

V. Giovannetti, R. Garcia-Patron, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Phot. 8, 796–800 (2014).
[Crossref]

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

2013 (2)

2011 (3)

A. Waseda, M. Sasaki, M. Takeoka, M. Fujiwara, M. Toyoshima, and A. Assalini, “Numerical evaluation of PPM for deep-space links,” J. Opt. Commun. Netw. 3(6), 514–521 (2011).
[Crossref]

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3), 257–265 (2011)
[Crossref]

S. Guha, “Structured optical receivers to attain superadditive capacity and the Holevo limit,” Phys. Rev. Lett. 106(24), 240502 (2011).
[Crossref] [PubMed]

2007 (1)

M. Toyoshima, W. R. Leeb, H. Kunimori, and T. Takano, “Comparison of microwave and light wave communication systems in space applications,” Optical Engineering 46(1), 015003 (2007).
[Crossref]

1997 (1)

L. Rizzo, “Effective erasure codes for reliable computer communication protocols,” ACM SIGCOMM Computer Communication Review 27(2), 24–36 (1997).
[Crossref]

1990 (1)

S. Verdu, “On channel capacity per unit cost,” IEEE Trans. Inf. Theor. 36(5), 1019–1030 (1990).
[Crossref]

1964 (1)

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316–A334 (1964).
[Crossref]

Albota, M. A.

Assalini, A.

Banaszek, K.

M. Jarzyna, P. Kuszaj, and K. Banaszek, “Incoherent on-off keying with classical and non-classical light,” Opt. Express 23(3), 3170–3175 (2015).
[Crossref] [PubMed]

K. Banaszek and M. Jachura, “Structured optical receivers for efficient deep-space communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2017), pp. 176–181.

M. Jarzyna, W. Zwoliński, M. Jachura, and K. Banaszek, “Optimizing deep-space optical communication under power constraints,” Proc. SPIE 10524 Free-Space Laser Communication and Atmospheric Propagation XXX, 105240A (15 February 2018).

Birnbaum, K. M.

S. Dolinar, K. M. Birnbaum, B. I. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2011), pp. 269–278.

Biswas, A.

W. D. Williams, M. Collins, D. M. Boronson, J. Lesch, and A. Biswas, “RF and optical communications: a comparison of high data rate returns from deep space in the 2020 timeframe,” NASA/TM-2007-214459, NASA Glenn Research Center, Cleveland, Ohio, 1–16 (2007)

Boronson, D. M.

W. D. Williams, M. Collins, D. M. Boronson, J. Lesch, and A. Biswas, “RF and optical communications: a comparison of high data rate returns from deep space in the 2020 timeframe,” NASA/TM-2007-214459, NASA Glenn Research Center, Cleveland, Ohio, 1–16 (2007)

Boroson, D. M.

Caplan, D. O.

Carney, J. J.

Cerf, N. J.

V. Giovannetti, R. Garcia-Patron, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Phot. 8, 796–800 (2014).
[Crossref]

Chung, H. W.

H. W. Chung, S. Guha, and L. Zheng, “On capacity of optical communications over a lossy bosonic channel with a receiver employing the most general coherent electro-optic feedback control,” Phys. Rev. A 96, 012320 (2017).
[Crossref]

Collins, M.

W. D. Williams, M. Collins, D. M. Boronson, J. Lesch, and A. Biswas, “RF and optical communications: a comparison of high data rate returns from deep space in the 2020 timeframe,” NASA/TM-2007-214459, NASA Glenn Research Center, Cleveland, Ohio, 1–16 (2007)

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 2006), Chap. 8.

Dauler, E. A.

Dolinar, S.

S. Dolinar, K. M. Birnbaum, B. I. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2011), pp. 269–278.

Erkmen, B. I.

S. Dolinar, K. M. Birnbaum, B. I. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2011), pp. 269–278.

Farr, W.

B. Moisson and W. Farr, “Range dependence of the optical communications channel,” IPN Progerss Report 42–199 (2014).

Fujiwara, M.

Garcia-Patron, R.

V. Giovannetti, R. Garcia-Patron, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Phot. 8, 796–800 (2014).
[Crossref]

Giovannetti, V.

M. Rosati, A. Mari, and V. Giovannetti, “Multiphase Hadamard receivers for classical communication on lossy bosonic channels,” Phys. Rev. A 94(6), 062325 (2016).
[Crossref]

V. Giovannetti, R. Garcia-Patron, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Phot. 8, 796–800 (2014).
[Crossref]

Guha, S.

H. W. Chung, S. Guha, and L. Zheng, “On capacity of optical communications over a lossy bosonic channel with a receiver employing the most general coherent electro-optic feedback control,” Phys. Rev. A 96, 012320 (2017).
[Crossref]

S. Guha, “Structured optical receivers to attain superadditive capacity and the Holevo limit,” Phys. Rev. Lett. 106(24), 240502 (2011).
[Crossref] [PubMed]

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3), 257–265 (2011)
[Crossref]

Habif, J. L.

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3), 257–265 (2011)
[Crossref]

Hakimi, F.

Hamilton, S. A.

Hemmati, H.

H. Hemmati, Deep-Space Optical Communication (Wiley, 2005) Chap. 4.

Holevo, A. S.

V. Giovannetti, R. Garcia-Patron, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Phot. 8, 796–800 (2014).
[Crossref]

Jachura, M.

M. Jarzyna, W. Zwoliński, M. Jachura, and K. Banaszek, “Optimizing deep-space optical communication under power constraints,” Proc. SPIE 10524 Free-Space Laser Communication and Atmospheric Propagation XXX, 105240A (15 February 2018).

K. Banaszek and M. Jachura, “Structured optical receivers for efficient deep-space communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2017), pp. 176–181.

Jarzyna, M.

M. Jarzyna, P. Kuszaj, and K. Banaszek, “Incoherent on-off keying with classical and non-classical light,” Opt. Express 23(3), 3170–3175 (2015).
[Crossref] [PubMed]

M. Jarzyna, W. Zwoliński, M. Jachura, and K. Banaszek, “Optimizing deep-space optical communication under power constraints,” Proc. SPIE 10524 Free-Space Laser Communication and Atmospheric Propagation XXX, 105240A (15 February 2018).

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316–A334 (1964).
[Crossref]

Kerman, A. J.

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316–A334 (1964).
[Crossref]

Kochman, Y.

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

Kunimori, H.

M. Toyoshima, W. R. Leeb, H. Kunimori, and T. Takano, “Comparison of microwave and light wave communication systems in space applications,” Optical Engineering 46(1), 015003 (2007).
[Crossref]

Kuszaj, P.

Leeb, W. R.

M. Toyoshima, W. R. Leeb, H. Kunimori, and T. Takano, “Comparison of microwave and light wave communication systems in space applications,” Optical Engineering 46(1), 015003 (2007).
[Crossref]

Lesch, J.

W. D. Williams, M. Collins, D. M. Boronson, J. Lesch, and A. Biswas, “RF and optical communications: a comparison of high data rate returns from deep space in the 2020 timeframe,” NASA/TM-2007-214459, NASA Glenn Research Center, Cleveland, Ohio, 1–16 (2007)

Mari, A.

M. Rosati, A. Mari, and V. Giovannetti, “Multiphase Hadamard receivers for classical communication on lossy bosonic channels,” Phys. Rev. A 94(6), 062325 (2016).
[Crossref]

Moision, B.

S. Dolinar, K. M. Birnbaum, B. I. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2011), pp. 269–278.

Moisson, B.

B. Moisson and W. Farr, “Range dependence of the optical communications channel,” IPN Progerss Report 42–199 (2014).

Molnar, R. J.

Moores, J. D.

Proakis, J. G.

J. G. Proakis and M. Salehi, Communication systems engineering (Prentice-Hall, Inc., 1994) pp. 11.

Rizzo, L.

L. Rizzo, “Effective erasure codes for reliable computer communication protocols,” ACM SIGCOMM Computer Communication Review 27(2), 24–36 (1997).
[Crossref]

Robinson, B. S.

Rosati, M.

M. Rosati, A. Mari, and V. Giovannetti, “Multiphase Hadamard receivers for classical communication on lossy bosonic channels,” Phys. Rev. A 94(6), 062325 (2016).
[Crossref]

Rosenberg, D.

Salehi, M.

J. G. Proakis and M. Salehi, Communication systems engineering (Prentice-Hall, Inc., 1994) pp. 11.

Sasaki, M.

Savage, S. J.

Takano, T.

M. Toyoshima, W. R. Leeb, H. Kunimori, and T. Takano, “Comparison of microwave and light wave communication systems in space applications,” Optical Engineering 46(1), 015003 (2007).
[Crossref]

Takeoka, M.

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3), 257–265 (2011)
[Crossref]

A. Waseda, M. Sasaki, M. Takeoka, M. Fujiwara, M. Toyoshima, and A. Assalini, “Numerical evaluation of PPM for deep-space links,” J. Opt. Commun. Netw. 3(6), 514–521 (2011).
[Crossref]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 2006), Chap. 8.

Toyoshima, M.

A. Waseda, M. Sasaki, M. Takeoka, M. Fujiwara, M. Toyoshima, and A. Assalini, “Numerical evaluation of PPM for deep-space links,” J. Opt. Commun. Netw. 3(6), 514–521 (2011).
[Crossref]

M. Toyoshima, W. R. Leeb, H. Kunimori, and T. Takano, “Comparison of microwave and light wave communication systems in space applications,” Optical Engineering 46(1), 015003 (2007).
[Crossref]

Verdu, S.

S. Verdu, “On channel capacity per unit cost,” IEEE Trans. Inf. Theor. 36(5), 1019–1030 (1990).
[Crossref]

Wang, L.

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

Waseda, A.

Williams, W. D.

W. D. Williams, M. Collins, D. M. Boronson, J. Lesch, and A. Biswas, “RF and optical communications: a comparison of high data rate returns from deep space in the 2020 timeframe,” NASA/TM-2007-214459, NASA Glenn Research Center, Cleveland, Ohio, 1–16 (2007)

Wornell, G. W.

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

Zheng, L.

H. W. Chung, S. Guha, and L. Zheng, “On capacity of optical communications over a lossy bosonic channel with a receiver employing the most general coherent electro-optic feedback control,” Phys. Rev. A 96, 012320 (2017).
[Crossref]

Zwolinski, W.

M. Jarzyna, W. Zwoliński, M. Jachura, and K. Banaszek, “Optimizing deep-space optical communication under power constraints,” Proc. SPIE 10524 Free-Space Laser Communication and Atmospheric Propagation XXX, 105240A (15 February 2018).

ACM SIGCOMM Computer Communication Review (1)

L. Rizzo, “Effective erasure codes for reliable computer communication protocols,” ACM SIGCOMM Computer Communication Review 27(2), 24–36 (1997).
[Crossref]

IEEE Trans. Inf. Theor. (1)

S. Verdu, “On channel capacity per unit cost,” IEEE Trans. Inf. Theor. 36(5), 1019–1030 (1990).
[Crossref]

IEEE Trans. Inf. Theory (1)

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

J. Mod. Opt. (1)

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3), 257–265 (2011)
[Crossref]

J. Opt. Commun. Netw. (1)

Nat. Phot. (1)

V. Giovannetti, R. Garcia-Patron, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Phot. 8, 796–800 (2014).
[Crossref]

Opt. Express (3)

Optical Engineering (1)

M. Toyoshima, W. R. Leeb, H. Kunimori, and T. Takano, “Comparison of microwave and light wave communication systems in space applications,” Optical Engineering 46(1), 015003 (2007).
[Crossref]

Phys. Rev. (1)

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316–A334 (1964).
[Crossref]

Phys. Rev. A (2)

H. W. Chung, S. Guha, and L. Zheng, “On capacity of optical communications over a lossy bosonic channel with a receiver employing the most general coherent electro-optic feedback control,” Phys. Rev. A 96, 012320 (2017).
[Crossref]

M. Rosati, A. Mari, and V. Giovannetti, “Multiphase Hadamard receivers for classical communication on lossy bosonic channels,” Phys. Rev. A 94(6), 062325 (2016).
[Crossref]

Phys. Rev. Lett. (1)

S. Guha, “Structured optical receivers to attain superadditive capacity and the Holevo limit,” Phys. Rev. Lett. 106(24), 240502 (2011).
[Crossref] [PubMed]

Other (8)

K. Banaszek and M. Jachura, “Structured optical receivers for efficient deep-space communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2017), pp. 176–181.

B. Moisson and W. Farr, “Range dependence of the optical communications channel,” IPN Progerss Report 42–199 (2014).

J. G. Proakis and M. Salehi, Communication systems engineering (Prentice-Hall, Inc., 1994) pp. 11.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 2006), Chap. 8.

M. Jarzyna, W. Zwoliński, M. Jachura, and K. Banaszek, “Optimizing deep-space optical communication under power constraints,” Proc. SPIE 10524 Free-Space Laser Communication and Atmospheric Propagation XXX, 105240A (15 February 2018).

S. Dolinar, K. M. Birnbaum, B. I. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in Proceedings of the IEEE International Conference on Satellite Optical Systems and Applications (ICSOS) (IEEE2011), pp. 269–278.

W. D. Williams, M. Collins, D. M. Boronson, J. Lesch, and A. Biswas, “RF and optical communications: a comparison of high data rate returns from deep space in the 2020 timeframe,” NASA/TM-2007-214459, NASA Glenn Research Center, Cleveland, Ohio, 1–16 (2007)

H. Hemmati, Deep-Space Optical Communication (Wiley, 2005) Chap. 4.

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Figures (5)

Fig. 1
Fig. 1 (a) The PPM format uses M equiprobable symbols defined by the location of a light pulse in a frame of otherwise empty bins. In the noise-free scenario the input symbol is either identified unambiguously by the timing of the detector click, or erased. (b) In the presence of noise both a light pulse and an empty time bin can generate a detector click with respective probabilities pc and pb.
Fig. 2
Fig. 2 Photon information efficiency I(K)/na as a function of the average detected signal power na and the PPM order M for a fixed background noise strength nb = 10−3. Results are shown for decoding restricted to sequences containing up to (a) K = 1, (b) K = 2, or (c) K = 5 clicks, as well as for (d) complete decoding with K = M. The dashed curves indicate the optimal PPM order M as a function of na.
Fig. 3
Fig. 3 (a) Photon information efficiency optimized over the PPM order for complete decoding PIE* = maxM(I(M)/na) (solid lines) and simple decoding maxM(I(1)/na) (dashed lines), shown as a function of the average detected signal power na for several values of the background noise strength nb. (b) The corresponding optimal pulse detected optical energy for complete decoding n s * = M * n a (solid lines) and simple decoding (dashed lines). Arrows in the panel (a) and in the inset in the panel (b) indicate asymptotic values for the complete decoding scenario calculated using Eq. (9).
Fig. 4
Fig. 4 (a) The asymptotic photon information efficiency PIEas for a completely decoded PPM link in the limit of the vanishing average detected signal power na → 0 as a function of the background noise strength nb. (b) The corresponding optimal detected pulse energy n s as. The depicted values have been obtained using Eq. (9).
Fig. 5
Fig. 5 (a) The maximum information rate R of a PPM link as function of the distance r (bottom scale) and the equivalent average detected signal power na (top scale) for complete decoding (solid lines) and simple decoding (dashed lines) and several values of the background noise power nb color-coded according to the legend. The system parameters are: carrier frequency fc = 2 · 105 GHz; bandwidth B = 2 GHz; transmitter power Ptx = 4 W; transmitter and receiver antenna diameters Dtx = 0.22 m and Drx = 11.8 m respectively; detector efficiency ηdet = 0.025. The distance r is specified in AU units, 1 AU ≈ 1.5·108 km. (b) The peak signal power (left scale) and the corresponding PPM order (right scale) required for the optimal operation of the PPM link.

Equations (21)

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

n a = η tot P tx B h f c , η tot = η det ( π f c D tx D rx 4 c r ) 2 .
p b = 1 exp ( n b ) , p c = 1 exp ( n s n b ) .
p c ( k ) = p c p b k 1 ( 1 p b ) M k , p e ( k ) = ( 1 p c ) p b k ( 1 p b ) M k 1
p ( k ) = k M p c ( k ) + ( 1 k M ) p e ( k )
I ( K ) = 1 M k = 1 K [ ( M 1 k 1 ) p c ( k ) log 2 p c ( k ) + ( M 1 k ) p e ( k ) log 2 p e ( k ) ( M k ) p ( k ) log 2 p ( k ) ] .
PIE * = max M ( I ( M ) / n a ) = I ( M * ) / n a , n s * = M * n a
I ( M ) I OOK = ( 1 1 M ) D ( p b | | ( 1 M 1 ) p b + M 1 p c ) + 1 M D ( p c | | ( 1 M 1 ) p b + M 1 p c )
D ( p q ) = p log 2 p q + ( 1 p ) log 2 1 p 1 q
PIE as = max n s 0 D ( p c | | p b ) n s ,
PIE * PIE OOK * PIE as .
I ( 1 ) log 2 e ( M 1 ) e ( M 1 ) n b 2 M 2 ( 1 e n b ) n s 2 .
PIE ( 1 ) log 2 e 2 e n b ( 1 e n b ) n a .
n s ( 1 ) ( 1 + n b 1 ) n a
R * = B n a PIE * ( n a , n b ) ,
P peak * = B η tot 1 h f c n s * ( n a , n b ) ,
p ( 0 | 0 ) = 1 p b , p ( 1 | 0 ) = p b
p ( 0 | 1 ) = 1 p c , p ( 1 | 1 ) = p c
I ( M ) = 1 M y 1 , y 2 , , y M = 0 , 1 p ( y 1 | 1 ) p ( y 2 | 0 ) p ( y M | 0 ) log 2 [ 1 M ( 1 + p ( y 1 | 0 ) p ( y 1 | 1 ) j = 2 M p ( y j | 1 ) p ( y j | 0 ) ) ] .
I ( M ) 1 M y = 0 , 1 p ( y | 1 ) log 2 [ 1 M ( 1 + p ( y | 0 ) p ( y | 1 ) ( M 1 ) ) ] .
I ( M ) 1 M D ( p c | | ( 1 M 1 ) p b + M 1 p c )
PIE * 1 n s as D ( p c as | | ( 1 n a / n a as ) p b + n a p c as / n s as ) .

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