Abstract

A novel coherent optical code-division multiple access (OCDMA) scheme is proposed that uses spectral line pairing to generate signals suitable for heterodyne decoding. Both signal and local reference are transmitted via a single optical fiber and a simple balanced receiver performs sourceless heterodyne detection, canceling speckle noise and multiple-access interference (MAI). To validate the idea, a 16 user fully loaded phase encoded system is simulated. Effects of fiber dispersion on system performance are studied as well. Both second and third order dispersion management is achieved by using a spectral phase encoder to adjust phase shifts of spectral components at the optical network unit (ONU).

© 2012 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011

Z. Gao, X. Wang, N. Kataoka, and N. Wada, “Demonstration of a two-user time domain spectral phase enoding OCDMA system with variable bandwidth spectrum shaper based decoder,” Microw. Opt. Technol. Lett.53(8), 1879–1882 (2011).
[CrossRef]

2010

J. Chen, Q. Zhang, C. Yu, X. Xin, Y. Shi, F. Deng, and C. Jin, “40Gbit/s PON over OCDMA uplink using DQPSK/OOK orthogonal re-modulation,” Proc. SPIE7848, 784837, 784837-8 (2010).
[CrossRef]

X. Wang, Z. S. Gao, N. Kataoka, and N. Wada, “Time domain spectral phase encoding/DPSK data modulation using single phase modulator for OCDMA application,” Opt. Express18(10), 9879–9890 (2010).
[CrossRef] [PubMed]

2009

P. Toliver, A. Agarwal, R. Menendez, J. Jackel, and S. Etemad, “Optical code division multiplexing for confidentiality at the photonic layer in metro networks and beyond,” Proc. SPIE7235, 723506, 723506-10 (2009).
[CrossRef]

2007

2006

A. Agarwal, P. Toliver, R. Menendez, T. Banwell, J. Jackel, and S. Etemad, “Spectrally efficient six-user coherent OCDMA system using reconfigurable integrated ring resonator circuits,” IEEE Photon. Technol. Lett.18(18), 1952–1954 (2006).
[CrossRef]

J. Cao, R. G. Broeke, N. K. Fontaine, C. Ji, Y. Du, N. Chubun, K. Aihara, A.-V. Pham, F. Olsson, S. Lourdudoss, and S. J. B. Yoo, “Demonstration of spectral phase O-CDMA encoding and decoding in monolithically integrated arrayed-waveguide-grating-based encoder,” IEEE Photon. Technol. Lett.18(24), 2602–2604 (2006).
[CrossRef]

2004

2001

P. C. Teh, P. Petropoulos, M. Ibsen, and D. J. Richardson, “A comparative study of the performance of seven and 63-chip optical code-division multiple-access encoders and decoders based on superstructured fiber Bragg gratings,” J. Lightwave Technol.19(9), 1352–1365 (2001).
[CrossRef]

T. Mizuno, T. Kitoh, T. Saida, Y. Inoue, M. Itoh, T. Shibata, Y. Hibino, and Y. Hida, “Low-loss 1.5%-∆ arrayed waveguide grating with narrow laterally tapered spotsize converter,” Electron. Lett.37(24), 1452–1454 (2001).
[CrossRef]

1999

S. Shen and A. M. Weiner, “Complete dispersion compensation for 400-fs pulse transmission over 10-km fiber link using dispersion compensating fiber and spectral phase equalizer,” IEEE Photon. Technol. Lett.11(7), 827–829 (1999).
[CrossRef]

1998

C. F. Lam, D. T. K. Tong, M. C. Wu, and E. Yablonovitvh, “Experimental demonstration of bipolar optical CDMA system using a balanced transmitter and complementary spectral encoding,” IEEE Photon. Technol. Lett.10(10), 1504–1506 (1998).
[CrossRef]

C. C. Chang, H. P. Sardesai, and A. M. Weiner, “Code-division multiple-access encoding and decoding of femtosecond optical pulses over a 2.5-km fiber link,” IEEE Photon. Technol. Lett.10(1), 171–173 (1998).
[CrossRef]

C. C. Chang, H. P. Sardesai, and A. M. Weiner, “Dispersion-free fiber transmission for femtosecond pulses by use of a dispersion-compensating fiber and a programmable pulse shaper,” Opt. Lett.23(4), 283–285 (1998).
[CrossRef] [PubMed]

1995

M. Kavehrad and D. Zaccarin, “Optical code-division-multiplexed systems based on spectral encoding of noncoherent sources,” J. Lightwave Technol.13(3), 534–545 (1995).
[CrossRef]

1993

D. Zaccarin and M. Kavehrad, “An optical CDMA system based on spectral encoding of LED,” IEEE Photon. Technol. Lett.5(4), 479–482 (1993).
[CrossRef]

1990

W. H. C. de Krom, “Impact of laser phase noise on the performance of a 3 × 3 phase and polarization diversity optical homodyne DPSK receiver,” J. Lightwave Technol.8(11), 1709–1715 (1990).
[CrossRef]

Agarwal, A.

P. Toliver, A. Agarwal, R. Menendez, J. Jackel, and S. Etemad, “Optical code division multiplexing for confidentiality at the photonic layer in metro networks and beyond,” Proc. SPIE7235, 723506, 723506-10 (2009).
[CrossRef]

A. Agarwal, P. Toliver, R. Menendez, T. Banwell, J. Jackel, and S. Etemad, “Spectrally efficient six-user coherent OCDMA system using reconfigurable integrated ring resonator circuits,” IEEE Photon. Technol. Lett.18(18), 1952–1954 (2006).
[CrossRef]

Aihara, K.

Anh-Vu Pham, S.

Baek, J.-H.

Banwell, T.

A. Agarwal, P. Toliver, R. Menendez, T. Banwell, J. Jackel, and S. Etemad, “Spectrally efficient six-user coherent OCDMA system using reconfigurable integrated ring resonator circuits,” IEEE Photon. Technol. Lett.18(18), 1952–1954 (2006).
[CrossRef]

Broeke, R. G.

Cao, J.

Chang, C. C.

C. C. Chang, H. P. Sardesai, and A. M. Weiner, “Dispersion-free fiber transmission for femtosecond pulses by use of a dispersion-compensating fiber and a programmable pulse shaper,” Opt. Lett.23(4), 283–285 (1998).
[CrossRef] [PubMed]

C. C. Chang, H. P. Sardesai, and A. M. Weiner, “Code-division multiple-access encoding and decoding of femtosecond optical pulses over a 2.5-km fiber link,” IEEE Photon. Technol. Lett.10(1), 171–173 (1998).
[CrossRef]

Chen, J.

J. Chen, Q. Zhang, C. Yu, X. Xin, Y. Shi, F. Deng, and C. Jin, “40Gbit/s PON over OCDMA uplink using DQPSK/OOK orthogonal re-modulation,” Proc. SPIE7848, 784837, 784837-8 (2010).
[CrossRef]

Chu, S. N.

Chubun, N.

J. Cao, R. G. Broeke, N. K. Fontaine, C. Ji, Y. Du, N. Chubun, K. Aihara, A.-V. Pham, F. Olsson, S. Lourdudoss, and S. J. B. Yoo, “Demonstration of spectral phase O-CDMA encoding and decoding in monolithically integrated arrayed-waveguide-grating-based encoder,” IEEE Photon. Technol. Lett.18(24), 2602–2604 (2006).
[CrossRef]

Cong, W.

Cooper, A. B.

A. B. Cooper, J. B. Khurgin, S. M. Xu, and J. U. Kang, “Phase and polarization diversity for minimum MAI in OCDMA networks,” IEEE J. Sel. Top. Quantum Electron.13(5), 1386–1395 (2007).
[CrossRef]

de Krom, W. H. C.

W. H. C. de Krom, “Impact of laser phase noise on the performance of a 3 × 3 phase and polarization diversity optical homodyne DPSK receiver,” J. Lightwave Technol.8(11), 1709–1715 (1990).
[CrossRef]

Deng, F.

J. Chen, Q. Zhang, C. Yu, X. Xin, Y. Shi, F. Deng, and C. Jin, “40Gbit/s PON over OCDMA uplink using DQPSK/OOK orthogonal re-modulation,” Proc. SPIE7848, 784837, 784837-8 (2010).
[CrossRef]

Ding, Z.

Du, Y.

Etemad, S.

P. Toliver, A. Agarwal, R. Menendez, J. Jackel, and S. Etemad, “Optical code division multiplexing for confidentiality at the photonic layer in metro networks and beyond,” Proc. SPIE7235, 723506, 723506-10 (2009).
[CrossRef]

A. Agarwal, P. Toliver, R. Menendez, T. Banwell, J. Jackel, and S. Etemad, “Spectrally efficient six-user coherent OCDMA system using reconfigurable integrated ring resonator circuits,” IEEE Photon. Technol. Lett.18(18), 1952–1954 (2006).
[CrossRef]

Fontaine, N. K.

Gao, Z.

Z. Gao, X. Wang, N. Kataoka, and N. Wada, “Demonstration of a two-user time domain spectral phase enoding OCDMA system with variable bandwidth spectrum shaper based decoder,” Microw. Opt. Technol. Lett.53(8), 1879–1882 (2011).
[CrossRef]

Gao, Z. S.

Gruezke, L. A.

Hamm, R. A.

Heritage, J. P.

Hernandez, V. J.

Hibino, Y.

T. Mizuno, T. Kitoh, T. Saida, Y. Inoue, M. Itoh, T. Shibata, Y. Hibino, and Y. Hida, “Low-loss 1.5%-∆ arrayed waveguide grating with narrow laterally tapered spotsize converter,” Electron. Lett.37(24), 1452–1454 (2001).
[CrossRef]

Hida, Y.

T. Mizuno, T. Kitoh, T. Saida, Y. Inoue, M. Itoh, T. Shibata, Y. Hibino, and Y. Hida, “Low-loss 1.5%-∆ arrayed waveguide grating with narrow laterally tapered spotsize converter,” Electron. Lett.37(24), 1452–1454 (2001).
[CrossRef]

Hobson, W. S.

Ibsen, M.

Inoue, Y.

T. Mizuno, T. Kitoh, T. Saida, Y. Inoue, M. Itoh, T. Shibata, Y. Hibino, and Y. Hida, “Low-loss 1.5%-∆ arrayed waveguide grating with narrow laterally tapered spotsize converter,” Electron. Lett.37(24), 1452–1454 (2001).
[CrossRef]

Itoh, M.

T. Mizuno, T. Kitoh, T. Saida, Y. Inoue, M. Itoh, T. Shibata, Y. Hibino, and Y. Hida, “Low-loss 1.5%-∆ arrayed waveguide grating with narrow laterally tapered spotsize converter,” Electron. Lett.37(24), 1452–1454 (2001).
[CrossRef]

Jackel, J.

P. Toliver, A. Agarwal, R. Menendez, J. Jackel, and S. Etemad, “Optical code division multiplexing for confidentiality at the photonic layer in metro networks and beyond,” Proc. SPIE7235, 723506, 723506-10 (2009).
[CrossRef]

A. Agarwal, P. Toliver, R. Menendez, T. Banwell, J. Jackel, and S. Etemad, “Spectrally efficient six-user coherent OCDMA system using reconfigurable integrated ring resonator circuits,” IEEE Photon. Technol. Lett.18(18), 1952–1954 (2006).
[CrossRef]

Ji, C.

J. Cao, R. G. Broeke, N. K. Fontaine, C. Ji, Y. Du, N. Chubun, K. Aihara, A.-V. Pham, F. Olsson, S. Lourdudoss, and S. J. B. Yoo, “Demonstration of spectral phase O-CDMA encoding and decoding in monolithically integrated arrayed-waveguide-grating-based encoder,” IEEE Photon. Technol. Lett.18(24), 2602–2604 (2006).
[CrossRef]

Jiang, W.

Jin, C.

J. Chen, Q. Zhang, C. Yu, X. Xin, Y. Shi, F. Deng, and C. Jin, “40Gbit/s PON over OCDMA uplink using DQPSK/OOK orthogonal re-modulation,” Proc. SPIE7848, 784837, 784837-8 (2010).
[CrossRef]

Kang, J. U.

A. B. Cooper, J. B. Khurgin, S. M. Xu, and J. U. Kang, “Phase and polarization diversity for minimum MAI in OCDMA networks,” IEEE J. Sel. Top. Quantum Electron.13(5), 1386–1395 (2007).
[CrossRef]

Kataoka, N.

Z. Gao, X. Wang, N. Kataoka, and N. Wada, “Demonstration of a two-user time domain spectral phase enoding OCDMA system with variable bandwidth spectrum shaper based decoder,” Microw. Opt. Technol. Lett.53(8), 1879–1882 (2011).
[CrossRef]

X. Wang, Z. S. Gao, N. Kataoka, and N. Wada, “Time domain spectral phase encoding/DPSK data modulation using single phase modulator for OCDMA application,” Opt. Express18(10), 9879–9890 (2010).
[CrossRef] [PubMed]

Kavehrad, M.

M. Kavehrad and D. Zaccarin, “Optical code-division-multiplexed systems based on spectral encoding of noncoherent sources,” J. Lightwave Technol.13(3), 534–545 (1995).
[CrossRef]

D. Zaccarin and M. Kavehrad, “An optical CDMA system based on spectral encoding of LED,” IEEE Photon. Technol. Lett.5(4), 479–482 (1993).
[CrossRef]

Khurgin, J. B.

A. B. Cooper, J. B. Khurgin, S. M. Xu, and J. U. Kang, “Phase and polarization diversity for minimum MAI in OCDMA networks,” IEEE J. Sel. Top. Quantum Electron.13(5), 1386–1395 (2007).
[CrossRef]

Kitayama, K.

Kitoh, T.

T. Mizuno, T. Kitoh, T. Saida, Y. Inoue, M. Itoh, T. Shibata, Y. Hibino, and Y. Hida, “Low-loss 1.5%-∆ arrayed waveguide grating with narrow laterally tapered spotsize converter,” Electron. Lett.37(24), 1452–1454 (2001).
[CrossRef]

Kolner, B. H.

Lam, C. F.

C. F. Lam, D. T. K. Tong, M. C. Wu, and E. Yablonovitvh, “Experimental demonstration of bipolar optical CDMA system using a balanced transmitter and complementary spectral encoding,” IEEE Photon. Technol. Lett.10(10), 1504–1506 (1998).
[CrossRef]

Lin, S.

Liou, K. Y.

Lothian, J. R.

Lourdudoss, S.

Matsushima, K.

Menendez, R.

P. Toliver, A. Agarwal, R. Menendez, J. Jackel, and S. Etemad, “Optical code division multiplexing for confidentiality at the photonic layer in metro networks and beyond,” Proc. SPIE7235, 723506, 723506-10 (2009).
[CrossRef]

A. Agarwal, P. Toliver, R. Menendez, T. Banwell, J. Jackel, and S. Etemad, “Spectrally efficient six-user coherent OCDMA system using reconfigurable integrated ring resonator circuits,” IEEE Photon. Technol. Lett.18(18), 1952–1954 (2006).
[CrossRef]

Mizuno, T.

T. Mizuno, T. Kitoh, T. Saida, Y. Inoue, M. Itoh, T. Shibata, Y. Hibino, and Y. Hida, “Low-loss 1.5%-∆ arrayed waveguide grating with narrow laterally tapered spotsize converter,” Electron. Lett.37(24), 1452–1454 (2001).
[CrossRef]

Nishiki, A.

Olsson, F.

Patel, B.

Petropoulos, P.

Pham, A.-V.

J. Cao, R. G. Broeke, N. K. Fontaine, C. Ji, Y. Du, N. Chubun, K. Aihara, A.-V. Pham, F. Olsson, S. Lourdudoss, and S. J. B. Yoo, “Demonstration of spectral phase O-CDMA encoding and decoding in monolithically integrated arrayed-waveguide-grating-based encoder,” IEEE Photon. Technol. Lett.18(24), 2602–2604 (2006).
[CrossRef]

Richardson, D. J.

Saida, T.

T. Mizuno, T. Kitoh, T. Saida, Y. Inoue, M. Itoh, T. Shibata, Y. Hibino, and Y. Hida, “Low-loss 1.5%-∆ arrayed waveguide grating with narrow laterally tapered spotsize converter,” Electron. Lett.37(24), 1452–1454 (2001).
[CrossRef]

Sardesai, H. P.

C. C. Chang, H. P. Sardesai, and A. M. Weiner, “Dispersion-free fiber transmission for femtosecond pulses by use of a dispersion-compensating fiber and a programmable pulse shaper,” Opt. Lett.23(4), 283–285 (1998).
[CrossRef] [PubMed]

C. C. Chang, H. P. Sardesai, and A. M. Weiner, “Code-division multiple-access encoding and decoding of femtosecond optical pulses over a 2.5-km fiber link,” IEEE Photon. Technol. Lett.10(1), 171–173 (1998).
[CrossRef]

Scherer, A.

Scott, R. P.

Seo, S.-W.

Shearn, M.

Shen, S.

S. Shen and A. M. Weiner, “Complete dispersion compensation for 400-fs pulse transmission over 10-km fiber link using dispersion compensating fiber and spectral phase equalizer,” IEEE Photon. Technol. Lett.11(7), 827–829 (1999).
[CrossRef]

Shi, Y.

J. Chen, Q. Zhang, C. Yu, X. Xin, Y. Shi, F. Deng, and C. Jin, “40Gbit/s PON over OCDMA uplink using DQPSK/OOK orthogonal re-modulation,” Proc. SPIE7848, 784837, 784837-8 (2010).
[CrossRef]

Shibata, T.

T. Mizuno, T. Kitoh, T. Saida, Y. Inoue, M. Itoh, T. Shibata, Y. Hibino, and Y. Hida, “Low-loss 1.5%-∆ arrayed waveguide grating with narrow laterally tapered spotsize converter,” Electron. Lett.37(24), 1452–1454 (2001).
[CrossRef]

Soares, F. M.

Teh, P. C.

Toliver, P.

P. Toliver, A. Agarwal, R. Menendez, J. Jackel, and S. Etemad, “Optical code division multiplexing for confidentiality at the photonic layer in metro networks and beyond,” Proc. SPIE7235, 723506, 723506-10 (2009).
[CrossRef]

A. Agarwal, P. Toliver, R. Menendez, T. Banwell, J. Jackel, and S. Etemad, “Spectrally efficient six-user coherent OCDMA system using reconfigurable integrated ring resonator circuits,” IEEE Photon. Technol. Lett.18(18), 1952–1954 (2006).
[CrossRef]

Tong, D. T. K.

C. F. Lam, D. T. K. Tong, M. C. Wu, and E. Yablonovitvh, “Experimental demonstration of bipolar optical CDMA system using a balanced transmitter and complementary spectral encoding,” IEEE Photon. Technol. Lett.10(10), 1504–1506 (1998).
[CrossRef]

Tsang, W. T.

Vatanapradit, S.

Wada, N.

Wang, X.

Weiner, A. M.

S. Shen and A. M. Weiner, “Complete dispersion compensation for 400-fs pulse transmission over 10-km fiber link using dispersion compensating fiber and spectral phase equalizer,” IEEE Photon. Technol. Lett.11(7), 827–829 (1999).
[CrossRef]

C. C. Chang, H. P. Sardesai, and A. M. Weiner, “Code-division multiple-access encoding and decoding of femtosecond optical pulses over a 2.5-km fiber link,” IEEE Photon. Technol. Lett.10(1), 171–173 (1998).
[CrossRef]

C. C. Chang, H. P. Sardesai, and A. M. Weiner, “Dispersion-free fiber transmission for femtosecond pulses by use of a dispersion-compensating fiber and a programmable pulse shaper,” Opt. Lett.23(4), 283–285 (1998).
[CrossRef] [PubMed]

Wu, M. C.

C. F. Lam, D. T. K. Tong, M. C. Wu, and E. Yablonovitvh, “Experimental demonstration of bipolar optical CDMA system using a balanced transmitter and complementary spectral encoding,” IEEE Photon. Technol. Lett.10(10), 1504–1506 (1998).
[CrossRef]

Xin, X.

J. Chen, Q. Zhang, C. Yu, X. Xin, Y. Shi, F. Deng, and C. Jin, “40Gbit/s PON over OCDMA uplink using DQPSK/OOK orthogonal re-modulation,” Proc. SPIE7848, 784837, 784837-8 (2010).
[CrossRef]

Xu, S. M.

A. B. Cooper, J. B. Khurgin, S. M. Xu, and J. U. Kang, “Phase and polarization diversity for minimum MAI in OCDMA networks,” IEEE J. Sel. Top. Quantum Electron.13(5), 1386–1395 (2007).
[CrossRef]

Yablonovitvh, E.

C. F. Lam, D. T. K. Tong, M. C. Wu, and E. Yablonovitvh, “Experimental demonstration of bipolar optical CDMA system using a balanced transmitter and complementary spectral encoding,” IEEE Photon. Technol. Lett.10(10), 1504–1506 (1998).
[CrossRef]

Yang, C.

Yoo, S. J.

Yoo, S. J. B.

J. Cao, R. G. Broeke, N. K. Fontaine, C. Ji, Y. Du, N. Chubun, K. Aihara, A.-V. Pham, F. Olsson, S. Lourdudoss, and S. J. B. Yoo, “Demonstration of spectral phase O-CDMA encoding and decoding in monolithically integrated arrayed-waveguide-grating-based encoder,” IEEE Photon. Technol. Lett.18(24), 2602–2604 (2006).
[CrossRef]

Yu, C.

J. Chen, Q. Zhang, C. Yu, X. Xin, Y. Shi, F. Deng, and C. Jin, “40Gbit/s PON over OCDMA uplink using DQPSK/OOK orthogonal re-modulation,” Proc. SPIE7848, 784837, 784837-8 (2010).
[CrossRef]

Zaccarin, D.

M. Kavehrad and D. Zaccarin, “Optical code-division-multiplexed systems based on spectral encoding of noncoherent sources,” J. Lightwave Technol.13(3), 534–545 (1995).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a). Frequency comb for reference and signal. (b) Unencoded signal or reference pulse at 160 GHz. (c) Encoded pulse shape for Hadamard sequence 9,10, and 15.

Fig. 2
Fig. 2

Proposed system and spectral phase encoding (SPE) using AWGs and phase shifters.

Fig. 3
Fig. 3

Simulated signals for the proposed OCDMA system. The OLT is represented in (a) through (d); (e) to (j) are the simulated signals in the ONU. Index letters correspond to points in the diagram of Fig. 2. Note that, following the subtraction at (h), the speckle is gone.

Fig. 4
Fig. 4

Transmitting experiment. (a) Eye diagram for BER = 10−9; (b) BER for four system load levels with integrate and dump and full loading with LPF. Received power is that which enters the ONU.

Fig. 5
Fig. 5

Eye diagrams at the receiver for different system loads irrespective of whether the intended user is transmitting.

Fig. 6
Fig. 6

Receiver eye diagram for the 9th and 15th encoding sequence.

Fig. 7
Fig. 7

(a) BER vs. Total Dispersion. (b) Added phase to spectral components. For the reference comb, the 10th Hadamard encoding sequence and dispersion compensation are both added, whereas only dispersion compensation phase is added for the signal comb. (c) Pulse shape without compensation and (d) with compensation.

Tables (2)

Tables Icon

Table 1 Encoder phase shifts corresponding to Hadamard sequences 9, 10, and 15.

Tables Icon

Table 2 Received optical power of proposed OCDMA schemes for 10−9 BER

Equations (14)

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

E S (t)= P S   m=1 M A m e j ω o t i=N/2 N/21 C m (i) e j4iΔωt
E l (t)= P LO   e j ω o t i=N/2 N/21 C l (i) e j(4i+1)Δωt ,
E 1 = E l 2 +j E S 2
E 2 =j E l 2 + E S 2
P 1 =| E 1 | 2 = | E l 2 +j E S 2 | 2 = 1 4 ( | E l | 2 + | E S | 2 +j E S E l * j E l E S * )
P 2 =| E 2 | 2 = | j E l 2 + E S 2 | 2 = 1 4 ( | E l | 2 + | E S | 2 +j E l E S * j E S E l * ).
| E l | 2 = P LO 4 i=N/2 N/21 k=N/2 N/21 F(4kΔω) C l (i) C l (i+k) e jk4Δωt ,
| E S | 2 = P S 4 i=N/2 N/21 k=N/2 N/21 [ F(4kΔω) m=1 M n=1 M A m A n * C m (i) C n (i+k) e jk4Δωt ].
P= P 1 P 2 = 1 2 ( j E S E l * +c.c. )
I(t)=η j 2 P LO P S [ m=1 M A m i=N/2 N/21 k=N/2 N/21 F[( 4k±1 )Δω] C m (i) C l (i+k) e j(4k±1)Δωt ]+cc
I(t)~η P LO P S m=1 M A m i=N/2 N/21 C m (i) C l (i) e jΔωt +cc
i=N/2 N/21 C m i C i e j(Δω)t = e j(Δω)t i=N/2 N/21 C m i C l i = { N e j(Δω)t ,(m=l) 0,(ml)
| I(t) | 2 ~ η 2 P LO P S A m 2 N 2
ϕ(Δω)= DL λ 2 4πc (Δω) 2 + LS λ 4 24 π 2 c 2 (Δω) 3 + DL λ 3 12 π 2 c 2 (Δω) 3

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