Abstract

QAM modulation utilizing subcarrier frequency lower than the symbol rate is both theoretically and experimentally investigated. High spectral efficiency and concentration of power in low frequencies make sub-cycle QAM signals attractive for optical fiber links with direct modulated light sources. Real-time generated 10-Gbps 4-level QAM signal in a 7.5-GHz bandwidth utilizing subcarrier frequency at a half symbol rate was successfully transmitted over 20-km SMF using an un-cooled 1.5-µm VCSEL. Only 2.5-dB fiber transmission power penalty was observed with no equalization applied.

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  1. W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
    [CrossRef]
  2. P. Moser, W. Hofmann, P. Wolf, G. Fiol, J. A. Lott, N. N. Ledentsov, and D. Bimberg, “83 fJ/bit energy-to-data ratio of 850-nm VCSEL at 17 Gb/s,” in Proceedings of 37th European Conference on Optical Communication (2011), pp. 1–3.
  3. W. Hofmann, M. Görblich, G. Böhm, M. Ortsiefer, L. Xie, and M.-C. Amann, “Long-wavelength 2-D VCSEL arrays for optical interconnects,” in Proceedings of Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (2008), pp.1- 2.
  4. M. C. Y. Huang, K. B. Cheng, Y. Zhou, A. Pisano, and C. Chang-Hasnain, “Monolithic integrated piezoelectric MEMS-tunable VCSEL,” IEEE IEEE J. Sel. Topics Quantum Electron.13(2), 374–380 (2007).
    [CrossRef]
  5. B. Zhang, X. Zhao, L. Christen, D. Parekh, W. Hofmann, M. C. Wu, M. C. Amann, C. J. Chang-Hasnain, and A. E. Willner, “Adjustable chirp injection-locked 1.55-μm VCSELs for enhanced chromatic dispersion compensation at 10-Gbit/s,” in Optical Fiber Communication Conference (Optical Society of America, 2008) paper OWT7.
  6. L. Xu, H. K. Tsang, W. Hofmann, and M.-C. Amann, “10-Gb/s colorless re-modulation of signal from 1550nm vertical cavity surface emitting laser array in WDM PON,” in Proceedings of Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (2009), paper CI3_4.
  7. T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
    [CrossRef]
  8. K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).
  9. R. Rodes, J. Estaran, B. Li, M. Muller, J. B. Jensen, T. Gruendl, M. Ortsiefer, C. Neumeyr, J. Rosskopf, K. J. Larsen, M.-C. Amann, and I. T. Monroy, “100 Gb/s single VCSEL data transmission link,” in Optical Fiber Communication Conference (Optical Society of America, 2012), paper PDP5D.
  10. E. Hugues-Salas, R. P. Giddings, X. Q. Jin, J. L. Wei, X. Zheng, Y. Hong, C. Shu, and J. M. Tang, “Real-time experimental demonstration of low-cost VCSEL intensity-modulated 11.25 Gb/s optical OFDM signal transmission over 25 km PON systems,” Opt. Express19(4), 2979–2988 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-2979 .
    [CrossRef] [PubMed]
  11. S. C. J. Lee, F. Breyer, S. Randel, J. Zeng, F. Huijskens, H. P. van den Boom, A. M. Koonen, and N. Hanik, “24-Gb/s transmission over 730 m of multimode fiber by direct modulation of an 850-nm VCSEL using discrete multi-tone modulation,” in Optical Fiber Communication Conference (Optical Society of America, 2009), paper PDP5.
  12. K. Szczerba, B.-E. Olsson, P. Westbergh, A. Rhodin, J. S. Gustavsson, A. Haglund, M. Karlsson, A. Larsson, and P. A. Andrekson, "37 Gbps transmission over 200 m of MMF using single cycle subcarrier modulation and a VCSEL with 20 GHz modulation bandwidth," in Proceedings of 36th European Conference on Optical Communication (2010), paper We.7.B.2.
  13. J. Proakis and M. Salehi, Digital Communications (McGraw-Hill, 2007).
  14. J. Justesen, “Performance of product codes and related structures with iterated decoding,” IEEE Trans. Commun.59(2), 407–415 (2011).
    [CrossRef]

2011 (5)

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).

J. Justesen, “Performance of product codes and related structures with iterated decoding,” IEEE Trans. Commun.59(2), 407–415 (2011).
[CrossRef]

E. Hugues-Salas, R. P. Giddings, X. Q. Jin, J. L. Wei, X. Zheng, Y. Hong, C. Shu, and J. M. Tang, “Real-time experimental demonstration of low-cost VCSEL intensity-modulated 11.25 Gb/s optical OFDM signal transmission over 25 km PON systems,” Opt. Express19(4), 2979–2988 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-2979 .
[CrossRef] [PubMed]

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

2007 (1)

M. C. Y. Huang, K. B. Cheng, Y. Zhou, A. Pisano, and C. Chang-Hasnain, “Monolithic integrated piezoelectric MEMS-tunable VCSEL,” IEEE IEEE J. Sel. Topics Quantum Electron.13(2), 374–380 (2007).
[CrossRef]

Amann, M.-C.

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

Bimberg, D.

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

Böhm, G.

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

Chang-Hasnain, C.

M. C. Y. Huang, K. B. Cheng, Y. Zhou, A. Pisano, and C. Chang-Hasnain, “Monolithic integrated piezoelectric MEMS-tunable VCSEL,” IEEE IEEE J. Sel. Topics Quantum Electron.13(2), 374–380 (2007).
[CrossRef]

Cheng, K. B.

M. C. Y. Huang, K. B. Cheng, Y. Zhou, A. Pisano, and C. Chang-Hasnain, “Monolithic integrated piezoelectric MEMS-tunable VCSEL,” IEEE IEEE J. Sel. Topics Quantum Electron.13(2), 374–380 (2007).
[CrossRef]

Gibbon, T. B.

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).

Giddings, R. P.

Gründl, T.

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

Hofmann, W.

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

Hong, Y.

Huang, M. C. Y.

M. C. Y. Huang, K. B. Cheng, Y. Zhou, A. Pisano, and C. Chang-Hasnain, “Monolithic integrated piezoelectric MEMS-tunable VCSEL,” IEEE IEEE J. Sel. Topics Quantum Electron.13(2), 374–380 (2007).
[CrossRef]

Hugues-Salas, E.

Jin, X. Q.

Justesen, J.

J. Justesen, “Performance of product codes and related structures with iterated decoding,” IEEE Trans. Commun.59(2), 407–415 (2011).
[CrossRef]

Ma, M.

K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).

Monroy, I. T.

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

Müller, M.

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

Mutig, A.

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

Neumeyr, C.

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).

Ortsiefer, M.

K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

Pham, T. T.

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

Pisano, A.

M. C. Y. Huang, K. B. Cheng, Y. Zhou, A. Pisano, and C. Chang-Hasnain, “Monolithic integrated piezoelectric MEMS-tunable VCSEL,” IEEE IEEE J. Sel. Topics Quantum Electron.13(2), 374–380 (2007).
[CrossRef]

Prince, K.

K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

Rönneberg, E.

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).

Shu, C.

Tafur Monroy, I.

K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).

Tang, J. M.

Tatarczak, A.

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

Wei, J. L.

Wolf, P.

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

Zheng, X.

Zhou, Y.

M. C. Y. Huang, K. B. Cheng, Y. Zhou, A. Pisano, and C. Chang-Hasnain, “Monolithic integrated piezoelectric MEMS-tunable VCSEL,” IEEE IEEE J. Sel. Topics Quantum Electron.13(2), 374–380 (2007).
[CrossRef]

Electron. Lett. (1)

W. Hofmann, M. Müller, P. Wolf, A. Mutig, T. Gründl, G. Böhm, D. Bimberg, and M.-C. Amann, “40 Gbit/s modulation of 1550 nm VCSEL,” Electron. Lett.47(4), 270–271 (2011).
[CrossRef]

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

M. C. Y. Huang, K. B. Cheng, Y. Zhou, A. Pisano, and C. Chang-Hasnain, “Monolithic integrated piezoelectric MEMS-tunable VCSEL,” IEEE IEEE J. Sel. Topics Quantum Electron.13(2), 374–380 (2007).
[CrossRef]

IEEE Trans. Commun. (1)

J. Justesen, “Performance of product codes and related structures with iterated decoding,” IEEE Trans. Commun.59(2), 407–415 (2011).
[CrossRef]

IEEE/OSA JOCN. (1)

K. Prince, M. Ma, T. B. Gibbon, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. Tafur Monroy, “Free-running 1550 nm VCSEL for 10.7 Gb/s transmission in 99.7 km PON,” IEEE/OSA JOCN.3, 399–403 (2011).

Opt. Express (1)

Opt. Fiber Technol. (1)

T. B. Gibbon, K. Prince, T. T. Pham, A. Tatarczak, C. Neumeyr, E. Rönneberg, M. Ortsiefer, and I. T. Monroy, “VCSEL transmission at 10Gb/s for 20km single mode fiber WDM-PON without dispersion compensation or injection locking,” Opt. Fiber Technol.17(1), 41–45 (2011).
[CrossRef]

Other (8)

R. Rodes, J. Estaran, B. Li, M. Muller, J. B. Jensen, T. Gruendl, M. Ortsiefer, C. Neumeyr, J. Rosskopf, K. J. Larsen, M.-C. Amann, and I. T. Monroy, “100 Gb/s single VCSEL data transmission link,” in Optical Fiber Communication Conference (Optical Society of America, 2012), paper PDP5D.

S. C. J. Lee, F. Breyer, S. Randel, J. Zeng, F. Huijskens, H. P. van den Boom, A. M. Koonen, and N. Hanik, “24-Gb/s transmission over 730 m of multimode fiber by direct modulation of an 850-nm VCSEL using discrete multi-tone modulation,” in Optical Fiber Communication Conference (Optical Society of America, 2009), paper PDP5.

K. Szczerba, B.-E. Olsson, P. Westbergh, A. Rhodin, J. S. Gustavsson, A. Haglund, M. Karlsson, A. Larsson, and P. A. Andrekson, "37 Gbps transmission over 200 m of MMF using single cycle subcarrier modulation and a VCSEL with 20 GHz modulation bandwidth," in Proceedings of 36th European Conference on Optical Communication (2010), paper We.7.B.2.

J. Proakis and M. Salehi, Digital Communications (McGraw-Hill, 2007).

B. Zhang, X. Zhao, L. Christen, D. Parekh, W. Hofmann, M. C. Wu, M. C. Amann, C. J. Chang-Hasnain, and A. E. Willner, “Adjustable chirp injection-locked 1.55-μm VCSELs for enhanced chromatic dispersion compensation at 10-Gbit/s,” in Optical Fiber Communication Conference (Optical Society of America, 2008) paper OWT7.

L. Xu, H. K. Tsang, W. Hofmann, and M.-C. Amann, “10-Gb/s colorless re-modulation of signal from 1550nm vertical cavity surface emitting laser array in WDM PON,” in Proceedings of Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (2009), paper CI3_4.

P. Moser, W. Hofmann, P. Wolf, G. Fiol, J. A. Lott, N. N. Ledentsov, and D. Bimberg, “83 fJ/bit energy-to-data ratio of 850-nm VCSEL at 17 Gb/s,” in Proceedings of 37th European Conference on Optical Communication (2011), pp. 1–3.

W. Hofmann, M. Görblich, G. Böhm, M. Ortsiefer, L. Xie, and M.-C. Amann, “Long-wavelength 2-D VCSEL arrays for optical interconnects,” in Proceedings of Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (2008), pp.1- 2.

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Figures (13)

Fig. 1
Fig. 1

Simulated waveforms of BPSK signals from I and Q channels and half-cycle 4-QAM signal. There is a half-period offset between the two signal components.

Fig. 2
Fig. 2

Simulated eye diagram of 5-Gbaud BPSK signal from I channel and 4-QAM signal in two-baud duration (a,b): data bits start at ± kπ phase of subcarrier, (c,d): data bits start at a different phase.

Fig. 3
Fig. 3

Simulated eye diagrams of BPSK signal from a) I channel, b) Q channel and c) 4-QAM signal after multiplication with sine signal for detection. Dash lines indicate center of bits - the optimal sampling instant.

Fig. 4
Fig. 4

Theoretical and simulated BER of half-cycle 4-QAM signal in AWGN channel.

Fig. 5
Fig. 5

Simulated waveforms of BPSK signals from I and Q channels and 4-QAM quarter-cycle signal.

Fig. 6
Fig. 6

Simulated eye diagram of 5-Gbaud quarter-cycle modulation signal in 4-baud duration. (a) BPSK signal, (b) 4-QAM signal, (c) 4-QAM signal after multiplication with sine signal. Dash lines indicate center of bits.

Fig. 7
Fig. 7

Experimental setup: Pulse pattern generator (PPG), photodetector (PD), variable optical attenuator (VOA), single mode fiber (SMF), digital storage oscilloscope (DSO).

Fig. 8
Fig. 8

Spectrum of (a) 5-Gbaud half-cycle 4-QAM signal and (b) 5-Gbaud single-cycle 4-QAM signal

Fig. 9
Fig. 9

Eye-diagram of electrical 10-Gbps 4-QAM signal: (a) after XOR gate and after photodetection (b) at B2B and (c) after 20-km SSMF transmission.

Fig. 10
Fig. 10

Performance of 4-QAM signals at B2B and after fiber transmission: (a) 5 Gbaud (10Gbps) and (b) 8 Gbaud (16 Gbps).

Fig. 11
Fig. 11

Constellation of 5-Gbaud 4-QAM signal: (a) generated electrical signal and (b) at −12.0 dBm B2B.

Fig. 12
Fig. 12

Constellation of 8-Gbaud 4-QAM signal: (a) generated electrical signal and (b) at −6.5 dBm B2B.

Fig. 13
Fig. 13

Performance of 5-Gpbs and 10-Gbps NRZ signals at B2B and after fiber transmission in comparison with 10-Gbps half-cycle 4-QAM signal.

Equations (10)

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

ϕ 1 ( t )= 2 Ε g g( t )cos2π f c t
ϕ 2 ( t )= 2 Ε g g( t )sin2π f c t
s m ( t )= A mI E g 2 ϕ 1 ( t )+ A mQ E g 2 ϕ 2 ( t ), m=1, 2,..., M = A mI g( t )cos2π f c t A mQ g( t )sin2π f c t
ϕ 1 ( t ), ϕ 1 ( t ) = 0 T ( 2 Ε g g( t )cos( 2π f c t+φ ) ) 2 = 0 T 2 Ε g g 2 ( t ) cos 2 ( 2π f c t+φ )
ϕ 2 ( t ), ϕ 2 ( t ) = 0 T ( 2 Ε g g( t )sin( 2π f c t+φ ) ) 2 = 0 T 2 Ε g g 2 ( t ) sin 2 ( 2π f c t+φ )
ϕ 1 ( t ), ϕ 2 ( t ) = 0 T 2 Ε g g( t )cos( 2π f c t+φ ) 2 Ε g g( t )sin( 2π f c t+φ ) = 0 T 2 Ε g g 2 ( t )sin2( 2π f c t+φ )
ϕ 1 ( t )=± 2 Ε g g( t )cos( 2π 1 2T t+φ ) t=[0, T]
ϕ 2 ( t )= 2 Ε g g( t )sin( 2π 1 2T t+φ ) t=[0, T]
ϕ 1 ( t )=± 2 Ε g g( t )cos( 2π 1 4T t+ k 4 π ) t=[0, T], k=1,3
ϕ 2 ( t )= 2 Ε g g( t )sin( 2π 1 4T t+ k 4 π ) t=[0, T], k=1,3

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