Abstract

In the context of communications based on light-emitting diodes (LEDs), spectrally efficient modulation has been considered for overcoming their limited bandwidth, and one scheme under investigation is quadrature-amplitude modulation on discrete multitones. The dependence of the output optical power on the driving current of practical LEDs is nonlinear, which distorts the transmitted signal. We investigate the impact of the nonlinear LED transfer function, i.e., the dependence of the emitted optical power on the driving current, on discrete multitone modulation. The effect incurred by this distortion was analyzed by using detailed numerical simulations addressing the impact of clipping, individual subcarriers, signal-to-noise ratio, and bit-error ratio. The approach was generalized to describe the impact of the nonlinearity of arbitrary LEDs and laser diodes, resulting in a powerful tool for assessing the impact of the nonlinearity on the link performance. This approach was applied to three types of LED, showing anything from a minuscule effect to the case in which error-free data transmission is made impossible by the transfer-function nonlinearity.

© 2009 Optical Society of America

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2008 (2)

J. Armstrong, B. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett., vol. 12, no. 5, pp. 343–345, 2008.
[CrossRef]

S. C. J. Lee, F. Breyer, S. Randel, H. P. A. van den Boom, A. M. J. Koonen, “High-speed transmission over multimode fiber using discrete multitone modulation,” J. Opt. Netw., vol. 7, no. 2, pp. 183–196, 2008.
[CrossRef]

2007 (3)

J. M. Tang, K. Shore, “Maximizing the transmission performance of adaptively modulated optical OFDM signals in multimode-fiber links by optimizing analog-to-digital converters,” J. Lightwave Technol., vol. 25, no. 3, pp. 787–798, 2007.
[CrossRef]

J. Grubor, K. Langer, J. Walewski, S. Randel, “High-speed wireless indoor communication via visible light,” ITG Fachbericht, vol. 198, pp. 203–208, 2007.

Y. Tang, W. Shieh, X. Yi, R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett., vol. 19, no. 7, pp. 483–485, 2007.
[CrossRef]

2006 (1)

A. Chorti, M. Brookes, “On the effects of memoryless nonlinearities on M-QAM and DQPSK OFDM signals,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 8, pp. 3301–3315, 2006.
[CrossRef]

2005 (2)

B. Le, T. W. Rondeau, J. H. Reed, C. W. Bostian, “Analog-to-digital converters,” IEEE Signal Process. Mag., vol. 22, no. 6, pp. 69–77, 2005.
[CrossRef]

C. J. Henderson, B. Robertson, D. G. Leyva, T. D. Wilkinson, D. C. O’Brien, G. E. Faulkner, “Control of a free-space adaptive optical interconnect using a liquid-crystal spatial light modulator for beam steering,” Opt. Eng., vol. 44, no. 7, p. 075401, 2005.
[CrossRef]

2004 (1)

T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
[CrossRef]

2001 (1)

C. Tellambura, “Computation of the continuous-time PAR of an OFDM signal with BPSK subcarriers,” IEEE Commun. Lett., vol. 5, no. 5, pp. 185–187, 2001.
[CrossRef]

1997 (1)

C.-L. Liu, “The effect on nonlinearity on a QPSK-OFDM-QAM signal,” IEEE Trans. Consumer Electron., vol. 43, no. 3, pp. 443–447, 1997.
[CrossRef]

1994 (1)

J. E. Mazo, N. F. Frigo, “On a method for calculating the strength of intermodulation distortion and products,” IEEE Trans. Commun., vol. 42, nos. 2/3/4, pp. 1475–1477, 1994.
[CrossRef]

1990 (1)

J. Kragten, “Least-squares polynomial curve-fitting for calibration purposes (statcal-calibra),” Anal. Chim. Acta, vol. 241, pp. 1–13, 1990.
[CrossRef]

1979 (1)

F. G. Gfeller, U. Bapst, “Wireless in-house data communication via diffuse infrared radiation,” Proc. IEEE, vol. 67, no. 11, pp. 1474–1486, 1979.
[CrossRef]

1977 (1)

T. P. Lee, “The nonlinearity of double-heterostructure LED’s for optical communications,” Proc. IEEE, vol. 65, no. 9, pp. 1408–1410, 1977.
[CrossRef]

1966 (1)

J. H. van Vleck, D. Middleton, “The spectrum of clipped noise,” Proc. IEEE, vol. 54, no. 1, pp. 2–19, 1966.
[CrossRef]

Akita, M.

T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
[CrossRef]

Armstrong, J.

J. Armstrong, B. Schmidt, “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,” IEEE Commun. Lett., vol. 12, no. 5, pp. 343–345, 2008.
[CrossRef]

Audeh, M. D.

R. Narasimhan, M. D. Audeh, J. Kahn, “Effect of electronic-ballast fluorescent lighting on wireless infrared links,” in IEEE Int. Conf. on Communications, 1996. ICC 96, Conf. Rec., Converging Technologies for Tomorrow's Applications, Dallas, TX, June 23–27, 1996, vol. 2, pp. 1213–1219.

Bapst, U.

F. G. Gfeller, U. Bapst, “Wireless in-house data communication via diffuse infrared radiation,” Proc. IEEE, vol. 67, no. 11, pp. 1474–1486, 1979.
[CrossRef]

Barth, U.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

Biran, B.

D. J. G. Mestdagh, P. Spruyt, B. Biran, “Analysis of clipping effect in DMT-based ADSL systems,” in IEEE Int. Conf. on Communications, 1994. ICC '94, SUPERCOMM/ICC '94, Conf. Rec., “Serving Humanity Through Communications”, New Orleans, LA, May 1–5, 1994, vol. 1, pp. 293–300, 1994.

Borges, J.

D. O’Brien, L. Zeng, H. Le-Minh, G. Faulkner, S. Randel, J. Walewski, J. Borges, K. Langer, J. Grubor, K. Lee, E. Won, “Visible-light communications,” in Short-Range Wireless Communications: Emerging Technologies and Applications. R. Kraemer and M. Katz, eds. New York, NY: Wiley, 2009.
[CrossRef]

Bostian, C. W.

B. Le, T. W. Rondeau, J. H. Reed, C. W. Bostian, “Analog-to-digital converters,” IEEE Signal Process. Mag., vol. 22, no. 6, pp. 69–77, 2005.
[CrossRef]

Bouchet, O.

O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

Breyer, F.

S. C. J. Lee, F. Breyer, S. Randel, H. P. A. van den Boom, A. M. J. Koonen, “High-speed transmission over multimode fiber using discrete multitone modulation,” J. Opt. Netw., vol. 7, no. 2, pp. 183–196, 2008.
[CrossRef]

S. C. J. Lee, J. W. Walewski, S. Randel, F. Breyer, H. P. A. van den Boom, A. M. J. Koonen, “Discrete multitone for novel application areas of optical communications,” in 2008 Dig. IEEE/LEOS Summer Topical Meetings, Acapulco, Mexico, July 21–23, 2008, pp. 163–164.
[CrossRef]

S. Lee, F. Breyer, S. Randel, O. Ziemann, H. van den Boom, A. Koonen, “Low-cost and robust 1-Gbit∕s plastic optical fiber link based on light-emitting diode technology,” in Optical Fiber Communication Conf. and Expo. and The Nat. Fiber Optic Engineers Conf., San Diego, California, OSA Technical Digest (CD), Washington, DC: Optical Society of America, Feb. 24, 2008, paper OWB3.

Brookes, M.

A. Chorti, M. Brookes, “On the effects of memoryless nonlinearities on M-QAM and DQPSK OFDM signals,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 8, pp. 3301–3315, 2006.
[CrossRef]

Chorti, A.

A. Chorti, M. Brookes, “On the effects of memoryless nonlinearities on M-QAM and DQPSK OFDM signals,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 8, pp. 3301–3315, 2006.
[CrossRef]

Egner, P.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

Evans, R.

Y. Tang, W. Shieh, X. Yi, R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett., vol. 19, no. 7, pp. 483–485, 2007.
[CrossRef]

Faulkner, G.

O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

D. O’Brien, L. Zeng, H. Le-Minh, G. Faulkner, S. Randel, J. Walewski, J. Borges, K. Langer, J. Grubor, K. Lee, E. Won, “Visible-light communications,” in Short-Range Wireless Communications: Emerging Technologies and Applications. R. Kraemer and M. Katz, eds. New York, NY: Wiley, 2009.
[CrossRef]

Faulkner, G. E.

C. J. Henderson, B. Robertson, D. G. Leyva, T. D. Wilkinson, D. C. O’Brien, G. E. Faulkner, “Control of a free-space adaptive optical interconnect using a liquid-crystal spatial light modulator for beam steering,” Opt. Eng., vol. 44, no. 7, p. 075401, 2005.
[CrossRef]

Fettweis, G.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

Franke, M.

O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

Frigo, N. F.

J. E. Mazo, N. F. Frigo, “On a method for calculating the strength of intermodulation distortion and products,” IEEE Trans. Commun., vol. 42, nos. 2/3/4, pp. 1475–1477, 1994.
[CrossRef]

Fujita, H.

T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
[CrossRef]

Gfeller, F. G.

F. G. Gfeller, U. Bapst, “Wireless in-house data communication via diffuse infrared radiation,” Proc. IEEE, vol. 67, no. 11, pp. 1474–1486, 1979.
[CrossRef]

Grass, E.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

Grubor, J.

J. Grubor, K. Langer, J. Walewski, S. Randel, “High-speed wireless indoor communication via visible light,” ITG Fachbericht, vol. 198, pp. 203–208, 2007.

D. O’Brien, L. Zeng, H. Le-Minh, G. Faulkner, S. Randel, J. Walewski, J. Borges, K. Langer, J. Grubor, K. Lee, E. Won, “Visible-light communications,” in Short-Range Wireless Communications: Emerging Technologies and Applications. R. Kraemer and M. Katz, eds. New York, NY: Wiley, 2009.
[CrossRef]

O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

J. Grubor, J. Lee, K. D. Langer, A. M. J. Koonen, J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in 33rd European Conf. and Exhibition on Optical Communication, Berlin, Germany, Sept. 16–20, 2007, postdeadline paper no. PD3.6.

Henderson, C. J.

C. J. Henderson, B. Robertson, D. G. Leyva, T. D. Wilkinson, D. C. O’Brien, G. E. Faulkner, “Control of a free-space adaptive optical interconnect using a liquid-crystal spatial light modulator for beam steering,” Opt. Eng., vol. 44, no. 7, p. 075401, 2005.
[CrossRef]

Hiertz, G.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

Idrus, S.

R. Ramirez-Iniguez, S. Idrus, Z. Sun, Optical Wireless Communications—IR for Wireless Connectivity, 1st ed. Boca Raton, FL: Auerbach, 2008, vol. 6.
[CrossRef]

Inan, B.

I. Neokosmidis, T. Kamalakis, J. W. Walewski, B. Inan, T. Sphicopoulos, “Impact of nonlinear LED transfer function on discrete multi-tone modulation: analytical approach,” J. Lightwave Technol., to be published.

B. Inan, S. C. J. Lee, S. Randel, I. Neokosmidis, A. M. J. Koonen, J. W. Walewski, “The impact of LED transfer function nonlinearity on high-speed optical wireless communications based on discrete-multitone modulation,” in Nat. Fiber Optic Engineers Conf., San Diego, CA, OSA Technical Digest (CD), Washington, DC: Optical Society of America, March 22, 2009, paper JThA52.

Irmer, R.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

Kahn, J.

R. Narasimhan, M. D. Audeh, J. Kahn, “Effect of electronic-ballast fluorescent lighting on wireless infrared links,” in IEEE Int. Conf. on Communications, 1996. ICC 96, Conf. Rec., Converging Technologies for Tomorrow's Applications, Dallas, TX, June 23–27, 1996, vol. 2, pp. 1213–1219.

Kamalakis, T.

O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

I. Neokosmidis, T. Kamalakis, J. W. Walewski, B. Inan, T. Sphicopoulos, “Impact of nonlinear LED transfer function on discrete multi-tone modulation: analytical approach,” J. Lightwave Technol., to be published.

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G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

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S. C. J. Lee, F. Breyer, S. Randel, H. P. A. van den Boom, A. M. J. Koonen, “High-speed transmission over multimode fiber using discrete multitone modulation,” J. Opt. Netw., vol. 7, no. 2, pp. 183–196, 2008.
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S. C. J. Lee, J. W. Walewski, S. Randel, F. Breyer, H. P. A. van den Boom, A. M. J. Koonen, “Discrete multitone for novel application areas of optical communications,” in 2008 Dig. IEEE/LEOS Summer Topical Meetings, Acapulco, Mexico, July 21–23, 2008, pp. 163–164.
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K. D. Langer, J. Vucic, C. Kottke, L. F. Rosal, S. Nerreter, J. W. Walewski, “Advances and prospects in high-speed information broadcast using phosphorescent white-light LEDs,” in 11th Int. Conf. on Transparent Optical Networks, 2009. ICTON '09, Azores, Portugal, June 28–July 2, 2009, pp. 1–6.
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T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
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J. Grubor, J. Lee, K. D. Langer, A. M. J. Koonen, J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in 33rd European Conf. and Exhibition on Optical Communication, Berlin, Germany, Sept. 16–20, 2007, postdeadline paper no. PD3.6.

K. D. Langer, J. Vucic, C. Kottke, L. F. Rosal, S. Nerreter, J. W. Walewski, “Advances and prospects in high-speed information broadcast using phosphorescent white-light LEDs,” in 11th Int. Conf. on Transparent Optical Networks, 2009. ICTON '09, Azores, Portugal, June 28–July 2, 2009, pp. 1–6.
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J. Grubor, J. Lee, K. D. Langer, A. M. J. Koonen, J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in 33rd European Conf. and Exhibition on Optical Communication, Berlin, Germany, Sept. 16–20, 2007, postdeadline paper no. PD3.6.

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D. O’Brien, L. Zeng, H. Le-Minh, G. Faulkner, S. Randel, J. Walewski, J. Borges, K. Langer, J. Grubor, K. Lee, E. Won, “Visible-light communications,” in Short-Range Wireless Communications: Emerging Technologies and Applications. R. Kraemer and M. Katz, eds. New York, NY: Wiley, 2009.
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S. C. J. Lee, F. Breyer, S. Randel, H. P. A. van den Boom, A. M. J. Koonen, “High-speed transmission over multimode fiber using discrete multitone modulation,” J. Opt. Netw., vol. 7, no. 2, pp. 183–196, 2008.
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T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
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T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
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R. Narasimhan, M. D. Audeh, J. Kahn, “Effect of electronic-ballast fluorescent lighting on wireless infrared links,” in IEEE Int. Conf. on Communications, 1996. ICC 96, Conf. Rec., Converging Technologies for Tomorrow's Applications, Dallas, TX, June 23–27, 1996, vol. 2, pp. 1213–1219.

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I. Neokosmidis, T. Kamalakis, J. W. Walewski, B. Inan, T. Sphicopoulos, “Impact of nonlinear LED transfer function on discrete multi-tone modulation: analytical approach,” J. Lightwave Technol., to be published.

B. Inan, S. C. J. Lee, S. Randel, I. Neokosmidis, A. M. J. Koonen, J. W. Walewski, “The impact of LED transfer function nonlinearity on high-speed optical wireless communications based on discrete-multitone modulation,” in Nat. Fiber Optic Engineers Conf., San Diego, CA, OSA Technical Digest (CD), Washington, DC: Optical Society of America, March 22, 2009, paper JThA52.

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K. D. Langer, J. Vucic, C. Kottke, L. F. Rosal, S. Nerreter, J. W. Walewski, “Advances and prospects in high-speed information broadcast using phosphorescent white-light LEDs,” in 11th Int. Conf. on Transparent Optical Networks, 2009. ICTON '09, Azores, Portugal, June 28–July 2, 2009, pp. 1–6.
[CrossRef]

O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

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O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

D. O’Brien, L. Zeng, H. Le-Minh, G. Faulkner, S. Randel, J. Walewski, J. Borges, K. Langer, J. Grubor, K. Lee, E. Won, “Visible-light communications,” in Short-Range Wireless Communications: Emerging Technologies and Applications. R. Kraemer and M. Katz, eds. New York, NY: Wiley, 2009.
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C. J. Henderson, B. Robertson, D. G. Leyva, T. D. Wilkinson, D. C. O’Brien, G. E. Faulkner, “Control of a free-space adaptive optical interconnect using a liquid-crystal spatial light modulator for beam steering,” Opt. Eng., vol. 44, no. 7, p. 075401, 2005.
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T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
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S. C. J. Lee, F. Breyer, S. Randel, H. P. A. van den Boom, A. M. J. Koonen, “High-speed transmission over multimode fiber using discrete multitone modulation,” J. Opt. Netw., vol. 7, no. 2, pp. 183–196, 2008.
[CrossRef]

J. Grubor, K. Langer, J. Walewski, S. Randel, “High-speed wireless indoor communication via visible light,” ITG Fachbericht, vol. 198, pp. 203–208, 2007.

D. O’Brien, L. Zeng, H. Le-Minh, G. Faulkner, S. Randel, J. Walewski, J. Borges, K. Langer, J. Grubor, K. Lee, E. Won, “Visible-light communications,” in Short-Range Wireless Communications: Emerging Technologies and Applications. R. Kraemer and M. Katz, eds. New York, NY: Wiley, 2009.
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S. C. J. Lee, J. W. Walewski, S. Randel, F. Breyer, H. P. A. van den Boom, A. M. J. Koonen, “Discrete multitone for novel application areas of optical communications,” in 2008 Dig. IEEE/LEOS Summer Topical Meetings, Acapulco, Mexico, July 21–23, 2008, pp. 163–164.
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B. Inan, S. C. J. Lee, S. Randel, I. Neokosmidis, A. M. J. Koonen, J. W. Walewski, “The impact of LED transfer function nonlinearity on high-speed optical wireless communications based on discrete-multitone modulation,” in Nat. Fiber Optic Engineers Conf., San Diego, CA, OSA Technical Digest (CD), Washington, DC: Optical Society of America, March 22, 2009, paper JThA52.

O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

S. Lee, F. Breyer, S. Randel, O. Ziemann, H. van den Boom, A. Koonen, “Low-cost and robust 1-Gbit∕s plastic optical fiber link based on light-emitting diode technology,” in Optical Fiber Communication Conf. and Expo. and The Nat. Fiber Optic Engineers Conf., San Diego, California, OSA Technical Digest (CD), Washington, DC: Optical Society of America, Feb. 24, 2008, paper OWB3.

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B. Le, T. W. Rondeau, J. H. Reed, C. W. Bostian, “Analog-to-digital converters,” IEEE Signal Process. Mag., vol. 22, no. 6, pp. 69–77, 2005.
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G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

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C. J. Henderson, B. Robertson, D. G. Leyva, T. D. Wilkinson, D. C. O’Brien, G. E. Faulkner, “Control of a free-space adaptive optical interconnect using a liquid-crystal spatial light modulator for beam steering,” Opt. Eng., vol. 44, no. 7, p. 075401, 2005.
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B. Le, T. W. Rondeau, J. H. Reed, C. W. Bostian, “Analog-to-digital converters,” IEEE Signal Process. Mag., vol. 22, no. 6, pp. 69–77, 2005.
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K. D. Langer, J. Vucic, C. Kottke, L. F. Rosal, S. Nerreter, J. W. Walewski, “Advances and prospects in high-speed information broadcast using phosphorescent white-light LEDs,” in 11th Int. Conf. on Transparent Optical Networks, 2009. ICTON '09, Azores, Portugal, June 28–July 2, 2009, pp. 1–6.
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G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

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Y. Tang, W. Shieh, X. Yi, R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett., vol. 19, no. 7, pp. 483–485, 2007.
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T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
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Sphicopoulos, T.

I. Neokosmidis, T. Kamalakis, J. W. Walewski, B. Inan, T. Sphicopoulos, “Impact of nonlinear LED transfer function on discrete multi-tone modulation: analytical approach,” J. Lightwave Technol., to be published.

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D. J. G. Mestdagh, P. Spruyt, B. Biran, “Analysis of clipping effect in DMT-based ADSL systems,” in IEEE Int. Conf. on Communications, 1994. ICC '94, SUPERCOMM/ICC '94, Conf. Rec., “Serving Humanity Through Communications”, New Orleans, LA, May 1–5, 1994, vol. 1, pp. 293–300, 1994.

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G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

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T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
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Tang, Y.

Y. Tang, W. Shieh, X. Yi, R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett., vol. 19, no. 7, pp. 483–485, 2007.
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G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

Trautwein, U.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

van den Boom, H.

S. Lee, F. Breyer, S. Randel, O. Ziemann, H. van den Boom, A. Koonen, “Low-cost and robust 1-Gbit∕s plastic optical fiber link based on light-emitting diode technology,” in Optical Fiber Communication Conf. and Expo. and The Nat. Fiber Optic Engineers Conf., San Diego, California, OSA Technical Digest (CD), Washington, DC: Optical Society of America, Feb. 24, 2008, paper OWB3.

van den Boom, H. P. A.

S. C. J. Lee, F. Breyer, S. Randel, H. P. A. van den Boom, A. M. J. Koonen, “High-speed transmission over multimode fiber using discrete multitone modulation,” J. Opt. Netw., vol. 7, no. 2, pp. 183–196, 2008.
[CrossRef]

S. C. J. Lee, J. W. Walewski, S. Randel, F. Breyer, H. P. A. van den Boom, A. M. J. Koonen, “Discrete multitone for novel application areas of optical communications,” in 2008 Dig. IEEE/LEOS Summer Topical Meetings, Acapulco, Mexico, July 21–23, 2008, pp. 163–164.
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J. H. van Vleck, D. Middleton, “The spectrum of clipped noise,” Proc. IEEE, vol. 54, no. 1, pp. 2–19, 1966.
[CrossRef]

Vucic, J.

K. D. Langer, J. Vucic, C. Kottke, L. F. Rosal, S. Nerreter, J. W. Walewski, “Advances and prospects in high-speed information broadcast using phosphorescent white-light LEDs,” in 11th Int. Conf. on Transparent Optical Networks, 2009. ICTON '09, Azores, Portugal, June 28–July 2, 2009, pp. 1–6.
[CrossRef]

Walewski, J.

J. Grubor, K. Langer, J. Walewski, S. Randel, “High-speed wireless indoor communication via visible light,” ITG Fachbericht, vol. 198, pp. 203–208, 2007.

D. O’Brien, L. Zeng, H. Le-Minh, G. Faulkner, S. Randel, J. Walewski, J. Borges, K. Langer, J. Grubor, K. Lee, E. Won, “Visible-light communications,” in Short-Range Wireless Communications: Emerging Technologies and Applications. R. Kraemer and M. Katz, eds. New York, NY: Wiley, 2009.
[CrossRef]

O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

Walewski, J. W.

I. Neokosmidis, T. Kamalakis, J. W. Walewski, B. Inan, T. Sphicopoulos, “Impact of nonlinear LED transfer function on discrete multi-tone modulation: analytical approach,” J. Lightwave Technol., to be published.

J. Grubor, J. Lee, K. D. Langer, A. M. J. Koonen, J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in 33rd European Conf. and Exhibition on Optical Communication, Berlin, Germany, Sept. 16–20, 2007, postdeadline paper no. PD3.6.

J. W. Walewski is preparing a manuscript to be called “Inference of nonlinear transfer function calibration curves by use of orthogonal polynomials.”

K. D. Langer, J. Vucic, C. Kottke, L. F. Rosal, S. Nerreter, J. W. Walewski, “Advances and prospects in high-speed information broadcast using phosphorescent white-light LEDs,” in 11th Int. Conf. on Transparent Optical Networks, 2009. ICTON '09, Azores, Portugal, June 28–July 2, 2009, pp. 1–6.
[CrossRef]

S. C. J. Lee, J. W. Walewski, S. Randel, F. Breyer, H. P. A. van den Boom, A. M. J. Koonen, “Discrete multitone for novel application areas of optical communications,” in 2008 Dig. IEEE/LEOS Summer Topical Meetings, Acapulco, Mexico, July 21–23, 2008, pp. 163–164.
[CrossRef]

B. Inan, S. C. J. Lee, S. Randel, I. Neokosmidis, A. M. J. Koonen, J. W. Walewski, “The impact of LED transfer function nonlinearity on high-speed optical wireless communications based on discrete-multitone modulation,” in Nat. Fiber Optic Engineers Conf., San Diego, CA, OSA Technical Digest (CD), Washington, DC: Optical Society of America, March 22, 2009, paper JThA52.

Weckerle, M.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

Wilkinson, T. D.

C. J. Henderson, B. Robertson, D. G. Leyva, T. D. Wilkinson, D. C. O’Brien, G. E. Faulkner, “Control of a free-space adaptive optical interconnect using a liquid-crystal spatial light modulator for beam steering,” Opt. Eng., vol. 44, no. 7, p. 075401, 2005.
[CrossRef]

Wirnitzer, W.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

Wolf, M.

O. Bouchet, M. Tabach, M. Wolf, D. O’Brien, G. Faulkner, J. Walewski, S. Randel, M. Franke, S. Nerreter, K. Langer, J. Grubor, T. Kamalakis, “Hybrid wireless optics (HWO): building the next-generation home network,” in 6th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, 2008. CNSDSP 2008, Gratz, Austria, July 25–25, 2008, pp. 283–287.

Won, E.

D. O’Brien, L. Zeng, H. Le-Minh, G. Faulkner, S. Randel, J. Walewski, J. Borges, K. Langer, J. Grubor, K. Lee, E. Won, “Visible-light communications,” in Short-Range Wireless Communications: Emerging Technologies and Applications. R. Kraemer and M. Katz, eds. New York, NY: Wiley, 2009.
[CrossRef]

Won, E. T.

E. T. Won, “Request for a VLC interest group,” presented at 51st Meeting of the IEEE 802.15 Working Group, Atlanta, GA, Nov. 11–16, 2007, https://mentor.ieee.org/802.15/file/07/15-07-0911-00-wng0-request-vlc-interest-group.ppt.

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Y. Tang, W. Shieh, X. Yi, R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett., vol. 19, no. 7, pp. 483–485, 2007.
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T. Mizuochi, Y. Miyata, T. Koayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, K. Motshima, “Forward error correction based on block turbo code with 3- bit soft decision for 10-gb∕s optic communication systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 376–386, 2004.
[CrossRef]

Zeng, L.

D. O’Brien, L. Zeng, H. Le-Minh, G. Faulkner, S. Randel, J. Walewski, J. Borges, K. Langer, J. Grubor, K. Lee, E. Won, “Visible-light communications,” in Short-Range Wireless Communications: Emerging Technologies and Applications. R. Kraemer and M. Katz, eds. New York, NY: Wiley, 2009.
[CrossRef]

Ziemann, O.

S. Lee, F. Breyer, S. Randel, O. Ziemann, H. van den Boom, A. Koonen, “Low-cost and robust 1-Gbit∕s plastic optical fiber link based on light-emitting diode technology,” in Optical Fiber Communication Conf. and Expo. and The Nat. Fiber Optic Engineers Conf., San Diego, California, OSA Technical Digest (CD), Washington, DC: Optical Society of America, Feb. 24, 2008, paper OWB3.

Zillmann, P.

G. Fettweis, R. Irmer, P. Zillmann, U. Barth, E. Kühn, K. Strohm, R. Kraemer, E. Grass, G. Tränkle, U. Trautwein, W. Wirnitzer, A. Kaufmann, L. Schwoerer, H.-J. Reumerman, M. Weckerle, E. Schulz, P. Egner, G. Hiertz, “System concept for 1 Gbit∕s and beyond,”ieee802.org/802_tutorials/nov05/Fettweis_WIGWAM_IEEE_Tutorial_Nov_2005_final.pdf, Nov. 2005.

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S. C. J. Lee, J. W. Walewski, S. Randel, F. Breyer, H. P. A. van den Boom, A. M. J. Koonen, “Discrete multitone for novel application areas of optical communications,” in 2008 Dig. IEEE/LEOS Summer Topical Meetings, Acapulco, Mexico, July 21–23, 2008, pp. 163–164.
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J. W. Walewski is preparing a manuscript to be called “Inference of nonlinear transfer function calibration curves by use of orthogonal polynomials.”

I. Neokosmidis, T. Kamalakis, J. W. Walewski, B. Inan, T. Sphicopoulos, “Impact of nonlinear LED transfer function on discrete multi-tone modulation: analytical approach,” J. Lightwave Technol., to be published.

B. Inan, S. C. J. Lee, S. Randel, I. Neokosmidis, A. M. J. Koonen, J. W. Walewski, “The impact of LED transfer function nonlinearity on high-speed optical wireless communications based on discrete-multitone modulation,” in Nat. Fiber Optic Engineers Conf., San Diego, CA, OSA Technical Digest (CD), Washington, DC: Optical Society of America, March 22, 2009, paper JThA52.

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Figures (13)

Fig. 1
Fig. 1

Block diagram of the DMT model in our simulation code.

Fig. 2
Fig. 2

Normalized practical and linear static transfer functions.

Fig. 3
Fig. 3

Top, normalized measured transfer function of the white LED, and polynomial fits (second and fifth order). Bottom, residuals of fitting the polynomial models data to the normalized measurement.

Fig. 4
Fig. 4

Constellation diagrams of the first and last subcarrier with the nonlinear LED model. The total number of subcarriers is 3, and γ = 1 . (a) First subcarrier, (b) last subcarrier

Fig. 5
Fig. 5

Number of nonlinear distortions that coincide with a certain subcarrier.

Fig. 6
Fig. 6

Constellation diagrams of first and last subcarrier with linear and nonlinear LED. The total number of subcarriers is 255 and γ = 1 . (a) First subcarrier, linear LED; (b) last subcarrier, linear LED; (c) first subcarrier, nonlinear LED; (d) last subcarrier, nonlinear LED.

Fig. 7
Fig. 7

E b N 0 per subcarriers for different numbers of total subcarriers and γ. (a) 3 carriers, (b) 7 carriers, and (c) 255 carriers.

Fig. 8
Fig. 8

BER as a function of E b N 0 : (a) 3 subcarriers, (b) 255 subcarriers.

Fig. 9
Fig. 9

Required E b N 0 to obtain a BER of 10 3 , displayed as a function of QAM level, for various total numbers of subcarriers and both linear and nonlinear LED transfer functions.

Fig. 10
Fig. 10

Transfer functions of varying nonlinearity and their corresponding ζ values, where y is the normalized optical power and x the normalized driving current.

Fig. 11
Fig. 11

Left, a portion of a normalized quadratic transfer function is chosen according to a preset modulation index γ 1 and bias current I DC 0.5 . Right, the chosen section is mapped to the intervals [0,1] on both axes. Notice that the resulting value of ζ; i.e., ζ is larger, i.e., closer to 0.5, than the original one.

Fig. 12
Fig. 12

E b N 0 as a function of ζ for 255 subcarriers, 64 QAM, and no receiver noise.

Fig. 13
Fig. 13

E b N 0 penalty for a BER of 10 3 as a function of ζ.

Tables (1)

Tables Icon

Table 1 LEDs Modulated for γ = 1 , 255 Carriers, and 64 QAM Modulation a

Equations (13)

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

s k = 1 2 N n = 0 2 N 1 C n e j 2 π k ( n 2 N ) ,
P out ( t ) = n = 0 b n [ I ( t ) I DC ] n ,
I ( t ) = n = 1 N 1 I ̂ n sin ( 2 π f n t + φ n ) + I DC
P out = b 0 + b 2 2 n = 1 N 1 I ̂ n 2 + b 1 n = 0 N 1 I ̂ n sin ( 2 π f n t + φ n ) + b 2 2 n = 1 N 1 I ̂ n 2 cos ( 2 π 2 f n t + 2 φ n ) + b 2 2 n , m = 1 , n m N 1 I ̂ n I ̂ m cos { 2 π ( f n + f m ) t + φ n + φ m } + b 2 2 n , m = 1 , n m N 1 I ̂ n I ̂ m cos { 2 π ( f n f m ) t + φ n φ m } .
γ = Δ I I DC .
SNR Q = 6.02 B + 4.77 + 20 log 10 ( V rms V peak ) + 10 log 10 ( f s 2 B sig ) ,
b 0 = ζ ,
b 1 = 1 ,
b 2 = 4 ζ + 2 .
P out = P out P out , l P out , u P out , l .
ζ = ζ I DC ( 4 ζ 1 ) ( 1 γ ) I DC 2 ( 2 4 ζ ) ( 1 γ ) 2 2 I DC γ ( 4 ζ 1 ) + 4 I DC 2 γ ( 2 4 ζ ) .
ζ = 0.5 [ 1 γ ( 1 2 ζ ) ] .
SFDR dBc = 10 log ( | 1 ζ 1 2 | ) .