Abstract

An optical communication channel is constructed using a heated thermo-electrically pumped, high efficiency infrared light-emitting diode (LED). In these devices, electro-luminescent cooling is observed, resulting in greater than unity (> 100%) efficiency in converting electrical power to optical power. The average amount of electrical energy required to generate a photon (4.3 meV) is much less than the optical energy in that photon (520 meV). Such a light source can serve as a test-bed for fundamental studies of energy-efficient bosonic communication channels. In this low energy consumption mode, we demonstrate data transmission at 3 kilobits per second (kbps) with only 120 picowatts of input electric power. Although the channel employs a mid-infrared source with limited quantum efficiency, a binary digit can be communicated using 40 femtojoules with a bit error rate of 3 x 10−3.

© 2014 Optical Society of America

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References

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2014 (2)

C. Antonelli, A. Mecozzi, M. Shtaif, and P. Winzer, “Quantum Limits on the Energy Consumption of Optical Transmission Systems,” J. Lightwave Technol. 32(10), 1853–1860 (2014).
[Crossref]

M. Notomi, K. Nozaki, A. Shinya, S. Matsuo, and E. Kuramochi, “Toward fJ/bit optical communication in a chip,” Opt. Commun. 314, 3–17 (2014).
[Crossref]

2013 (3)

P. Moser, J. A. Lott, and D. Bimberg, “Energy Efficiency of Directly Modulated Oxide-Confined High Bit Rate 850-nm VCSELs for Optical Interconnects,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1702212 (2013).
[Crossref]

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

P. Santhanam, D. Huang, R. J. Ram, M. A. Remennyi, and B. A. Matveev, “Room temperature thermo-electric pumping in mid-infrared light-emitting diodes,” Appl. Phys. Lett. 103(18), 183513 (2013).
[Crossref]

2012 (1)

P. Santhanam, D. J. Gray, and R. J. Ram, “Thermoelectrically pumped light-emitting diodes operating above unity efficiency,” Phys. Rev. Lett. 108(9), 097403 (2012).
[Crossref] [PubMed]

2011 (1)

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

2008 (1)

J. Lekki, Q. V. Nguyen, T. Bizon, B. Nguyen, J. Kojima, and M. Hizlan, “Extremely low power quantum optical communication link for miniature planetary sensor stations,” J. Aerospace Comput. Inf. Commun. 5(10), 396–408 (2008).
[Crossref]

2006 (1)

1996 (1)

R. Landauer, “Minimal energy requirements in communication,” Science 272(5270), 1914–1918 (1996).
[Crossref] [PubMed]

1987 (1)

R. Landauer, “Energy requirements in communication,” Appl. Phys. Lett. 51(24), 2056–2058 (1987).
[Crossref]

1985 (1)

1984 (1)

1961 (1)

R. Landauer, “Irreversibility and heat generation in the computing process,” IBM J. Res. Develop. 5(3), 183–191 (1961).
[Crossref]

1960 (1)

1957 (1)

J. Tauc, “The share of thermal energy taken from the surroundings in the electro-luminescent energy radiated from ap-n junction,” Cechoslovackij Fiziceskij Zurnal 7, 275–276 (1957).

1948 (1)

C. E. Shannon and W. Weaver, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

Antonelli, C.

Bimberg, D.

P. Moser, J. A. Lott, and D. Bimberg, “Energy Efficiency of Directly Modulated Oxide-Confined High Bit Rate 850-nm VCSELs for Optical Interconnects,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1702212 (2013).
[Crossref]

Bizon, T.

J. Lekki, Q. V. Nguyen, T. Bizon, B. Nguyen, J. Kojima, and M. Hizlan, “Extremely low power quantum optical communication link for miniature planetary sensor stations,” J. Aerospace Comput. Inf. Commun. 5(10), 396–408 (2008).
[Crossref]

Ellis, B.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Friberg, S. R.

Gray, D. J.

P. Santhanam, D. J. Gray, and R. J. Ram, “Thermoelectrically pumped light-emitting diodes operating above unity efficiency,” Phys. Rev. Lett. 108(9), 097403 (2012).
[Crossref] [PubMed]

Haller, E. E.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Harris, J.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Hasebe, K.

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Hizlan, M.

J. Lekki, Q. V. Nguyen, T. Bizon, B. Nguyen, J. Kojima, and M. Hizlan, “Extremely low power quantum optical communication link for miniature planetary sensor stations,” J. Aerospace Comput. Inf. Commun. 5(10), 396–408 (2008).
[Crossref]

Hong, C. K.

Huang, D.

P. Santhanam, D. Huang, R. J. Ram, M. A. Remennyi, and B. A. Matveev, “Room temperature thermo-electric pumping in mid-infrared light-emitting diodes,” Appl. Phys. Lett. 103(18), 183513 (2013).
[Crossref]

Kakitsuka, T.

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Kobayashi, W.

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Kojima, J.

J. Lekki, Q. V. Nguyen, T. Bizon, B. Nguyen, J. Kojima, and M. Hizlan, “Extremely low power quantum optical communication link for miniature planetary sensor stations,” J. Aerospace Comput. Inf. Commun. 5(10), 396–408 (2008).
[Crossref]

Kuramochi, E.

M. Notomi, K. Nozaki, A. Shinya, S. Matsuo, and E. Kuramochi, “Toward fJ/bit optical communication in a chip,” Opt. Commun. 314, 3–17 (2014).
[Crossref]

Landauer, R.

R. Landauer, “Minimal energy requirements in communication,” Science 272(5270), 1914–1918 (1996).
[Crossref] [PubMed]

R. Landauer, “Energy requirements in communication,” Appl. Phys. Lett. 51(24), 2056–2058 (1987).
[Crossref]

R. Landauer, “Irreversibility and heat generation in the computing process,” IBM J. Res. Develop. 5(3), 183–191 (1961).
[Crossref]

Lekki, J.

J. Lekki, Q. V. Nguyen, T. Bizon, B. Nguyen, J. Kojima, and M. Hizlan, “Extremely low power quantum optical communication link for miniature planetary sensor stations,” J. Aerospace Comput. Inf. Commun. 5(10), 396–408 (2008).
[Crossref]

Lott, J. A.

P. Moser, J. A. Lott, and D. Bimberg, “Energy Efficiency of Directly Modulated Oxide-Confined High Bit Rate 850-nm VCSELs for Optical Interconnects,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1702212 (2013).
[Crossref]

Majumdar, A.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Mandel, L.

Matsuo, S.

M. Notomi, K. Nozaki, A. Shinya, S. Matsuo, and E. Kuramochi, “Toward fJ/bit optical communication in a chip,” Opt. Commun. 314, 3–17 (2014).
[Crossref]

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Matveev, B. A.

P. Santhanam, D. Huang, R. J. Ram, M. A. Remennyi, and B. A. Matveev, “Room temperature thermo-electric pumping in mid-infrared light-emitting diodes,” Appl. Phys. Lett. 103(18), 183513 (2013).
[Crossref]

Mayer, M. A.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Mecozzi, A.

Moser, P.

P. Moser, J. A. Lott, and D. Bimberg, “Energy Efficiency of Directly Modulated Oxide-Confined High Bit Rate 850-nm VCSELs for Optical Interconnects,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1702212 (2013).
[Crossref]

Narimanov, E. E.

Nguyen, B.

J. Lekki, Q. V. Nguyen, T. Bizon, B. Nguyen, J. Kojima, and M. Hizlan, “Extremely low power quantum optical communication link for miniature planetary sensor stations,” J. Aerospace Comput. Inf. Commun. 5(10), 396–408 (2008).
[Crossref]

Nguyen, Q. V.

J. Lekki, Q. V. Nguyen, T. Bizon, B. Nguyen, J. Kojima, and M. Hizlan, “Extremely low power quantum optical communication link for miniature planetary sensor stations,” J. Aerospace Comput. Inf. Commun. 5(10), 396–408 (2008).
[Crossref]

Notomi, M.

M. Notomi, K. Nozaki, A. Shinya, S. Matsuo, and E. Kuramochi, “Toward fJ/bit optical communication in a chip,” Opt. Commun. 314, 3–17 (2014).
[Crossref]

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Nozaki, K.

M. Notomi, K. Nozaki, A. Shinya, S. Matsuo, and E. Kuramochi, “Toward fJ/bit optical communication in a chip,” Opt. Commun. 314, 3–17 (2014).
[Crossref]

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Petykiewicz, J.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Ram, R. J.

P. Santhanam, D. Huang, R. J. Ram, M. A. Remennyi, and B. A. Matveev, “Room temperature thermo-electric pumping in mid-infrared light-emitting diodes,” Appl. Phys. Lett. 103(18), 183513 (2013).
[Crossref]

P. Santhanam, D. J. Gray, and R. J. Ram, “Thermoelectrically pumped light-emitting diodes operating above unity efficiency,” Phys. Rev. Lett. 108(9), 097403 (2012).
[Crossref] [PubMed]

P. Santhanam and R. J. Ram, “Light-Emitting Diodes Operating Above Unity Efficiency for Infrared Absorption Spectroscopy,” Proc. of the 2012 International Photonics Conference. (2012).
[Crossref]

Remennyi, M. A.

P. Santhanam, D. Huang, R. J. Ram, M. A. Remennyi, and B. A. Matveev, “Room temperature thermo-electric pumping in mid-infrared light-emitting diodes,” Appl. Phys. Lett. 103(18), 183513 (2013).
[Crossref]

Santhanam, P.

P. Santhanam, D. Huang, R. J. Ram, M. A. Remennyi, and B. A. Matveev, “Room temperature thermo-electric pumping in mid-infrared light-emitting diodes,” Appl. Phys. Lett. 103(18), 183513 (2013).
[Crossref]

P. Santhanam, D. J. Gray, and R. J. Ram, “Thermoelectrically pumped light-emitting diodes operating above unity efficiency,” Phys. Rev. Lett. 108(9), 097403 (2012).
[Crossref] [PubMed]

P. Santhanam and R. J. Ram, “Light-Emitting Diodes Operating Above Unity Efficiency for Infrared Absorption Spectroscopy,” Proc. of the 2012 International Photonics Conference. (2012).
[Crossref]

Sarmiento, T.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Sato, T.

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Shambat, G.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Shannon, C. E.

C. E. Shannon and W. Weaver, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

Shinya, A.

M. Notomi, K. Nozaki, A. Shinya, S. Matsuo, and E. Kuramochi, “Toward fJ/bit optical communication in a chip,” Opt. Commun. 314, 3–17 (2014).
[Crossref]

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Shtaif, M.

Takeda, K.

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Taniyama, H.

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Tauc, J.

J. Tauc, “The share of thermal energy taken from the surroundings in the electro-luminescent energy radiated from ap-n junction,” Cechoslovackij Fiziceskij Zurnal 7, 275–276 (1957).

Vuckovic, J.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Weaver, W.

C. E. Shannon and W. Weaver, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

Weinstein, M. A.

Winzer, P.

Wu, B. B.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

P. Santhanam, D. Huang, R. J. Ram, M. A. Remennyi, and B. A. Matveev, “Room temperature thermo-electric pumping in mid-infrared light-emitting diodes,” Appl. Phys. Lett. 103(18), 183513 (2013).
[Crossref]

R. Landauer, “Energy requirements in communication,” Appl. Phys. Lett. 51(24), 2056–2058 (1987).
[Crossref]

Bell Syst. Tech. J. (1)

C. E. Shannon and W. Weaver, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

Cechoslovackij Fiziceskij Zurnal (1)

J. Tauc, “The share of thermal energy taken from the surroundings in the electro-luminescent energy radiated from ap-n junction,” Cechoslovackij Fiziceskij Zurnal 7, 275–276 (1957).

IBM J. Res. Develop. (1)

R. Landauer, “Irreversibility and heat generation in the computing process,” IBM J. Res. Develop. 5(3), 183–191 (1961).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

P. Moser, J. A. Lott, and D. Bimberg, “Energy Efficiency of Directly Modulated Oxide-Confined High Bit Rate 850-nm VCSELs for Optical Interconnects,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1702212 (2013).
[Crossref]

J. Aerospace Comput. Inf. Commun. (1)

J. Lekki, Q. V. Nguyen, T. Bizon, B. Nguyen, J. Kojima, and M. Hizlan, “Extremely low power quantum optical communication link for miniature planetary sensor stations,” J. Aerospace Comput. Inf. Commun. 5(10), 396–408 (2008).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Nat. Commun. (1)

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[Crossref] [PubMed]

Nat. Photonics (1)

K. Takeda, T. Sato, A. Shinya, K. Nozaki, W. Kobayashi, H. Taniyama, M. Notomi, K. Hasebe, T. Kakitsuka, and S. Matsuo, “Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers,” Nat. Photonics 7(7), 569–575 (2013).
[Crossref]

Opt. Commun. (1)

M. Notomi, K. Nozaki, A. Shinya, S. Matsuo, and E. Kuramochi, “Toward fJ/bit optical communication in a chip,” Opt. Commun. 314, 3–17 (2014).
[Crossref]

Opt. Express (1)

Phys. Rev. Lett. (1)

P. Santhanam, D. J. Gray, and R. J. Ram, “Thermoelectrically pumped light-emitting diodes operating above unity efficiency,” Phys. Rev. Lett. 108(9), 097403 (2012).
[Crossref] [PubMed]

Science (1)

R. Landauer, “Minimal energy requirements in communication,” Science 272(5270), 1914–1918 (1996).
[Crossref] [PubMed]

Other (4)

P. Santhanam and R. J. Ram, “Light-Emitting Diodes Operating Above Unity Efficiency for Infrared Absorption Spectroscopy,” Proc. of the 2012 International Photonics Conference. (2012).
[Crossref]

M. Verhelst and W. Dehaene, “System design of an ultra-low power, low data rate, pulsed UWB receiver in the 0-960 MHz band,” Communications, 2005. ICC 2005. IEEE International Conference on. Vol. 4. IEEE, (2005).

Ltd. Ioffe LED. Led21sr data sheet: Optically immersed 2.15 μm LED in heat-sink optimized housing. Product catalog., May 2013.

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Figures (4)

Fig. 1
Fig. 1 (a) An illustration of the free-space optical channel used in this experiment. Light from an In0.15Ga0.85As0.13Sb0.87 LED emitting at 2.5μm at 167°C is collected using a cooled InGaAs photodiode operating at zero bias. (b) Wall-plug efficiency versus emitted optical power for the LED. This data was acquired using two different receiver circuits: a digital lock-in amplifier and a 16-bit analog-to-digital converter. Both measurements indicate an inverse relationship between efficiency and light power. The inset shows L-I and I-V curves for the LED, using a combination of the ADC and the lock-in for low bias conditions, and a DC measurement at high bias conditions. We note that in modeling the LED, the collector efficiency was a fitting parameter [5]. Therefore, any uncertainty in converting photocurrent at the detector to optical power would appear equally in both the data and the model, but would not affect the 1/L power-law relationship between efficiency and power.
Fig. 2
Fig. 2 (a) Block diagram of the LED communication channel. (b) 8-symbol phase shift keyed coding with seven symbols equally spaced in phase, and the last symbol at zero magnitude. The centroids of each symbol are also labeled. (c) Received signal from an orthogonal frequency division multiplexed (OFDM) channel depicting orthogonal channels with 1Hz spacing; channels at 1000, 1002, 1004, and 1008Hz are transmitting nonzero symbols.
Fig. 3
Fig. 3 (a) Bit error rate (BER) versus energy per bit at various data rates. The curves show the theoretical bit error rate assuming a flat noise power spectrum and varying the magnitude of the current signal driving the LED. Input power can be related to energy per bit by Eq. (2). (b) The inset containing the noise spectrum of the channel indicates more noise at higher frequencies. Higher data rate experiments showed higher bit error rates at a given energy per bit as a result.
Fig. 4
Fig. 4 Bit error rate (BER) versus the energy per bit for the experimental channel, as well as extrapolations for an idealized LED-detector pair. All three curves assume 8-symbol phase shift keying.

Equations (2)

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I LED ( t )= i=1 M [ | B i | I 0 cos( 2π f i t+arg( B i ) ) ]
E= 0 t s | I LED ( t ) | 2 Rdt= I 0 2 R 2 t s i=1 M | B i | 2

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