Abstract

We present a rigorous analysis defining the fundamental performance limits of duobinary line-coding for optical communications. First, we compare the back-to-back performance of duobinary and intensity modulation systems in an AWGN channel satisfying the Nyquist criterion, with both linear and quadratic receiver. We show that, also for duobinary and quadratic receiver, matched filtering is the best achievable condition. Then, we derive a detailed performance analysis of duobinary in an ASE-noise-limited direct-detection optical system considering noise on the entire space of polarizations. We show that for duobinary line-coding the expression of the bit error rate depends both on the shape of the transmitted pulse and on the receiver optical filter. Comparing duobinary coded and uncoded intensity modulation systems, we show the intrinsic advantages of using the duobinary line-coding in a system based on quadratic detection. Finally, some results for realistic setups are obtained through simulation and compared to the fundamental limits in order to show how close to those limits state-of-the-art systems can operate.

© 2008 Optical Society of America

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References

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  1. J. G. Proakis, Digital communications, 2nd edition (New York, McGraw-Hill), 1989.
  2. A. Lender, "The duobinary technique for high-speed data transmission," IEEE Trans. Commun. and Elecron. 82, 214-218 (1963).
  3. X. Gu and L.C. Blank, "10 Gbit/s unrepeated three-level optical transmission over 100 km of standard fibre," Electron. Lett. 29, 2209-2211 (1993).
    [CrossRef]
  4. K. Yonenaga, S. Kuwano, and S. Norimatsu, "Optical duobinary transmission system with no receiver sensitivity degradation," Electron. Lett. 31, 302-304 (1995).
    [CrossRef]
  5. D. Penninckx, M. Chbat, L. Pierre, and J.-P. Thierry, "The Phase-Shaped Binary Transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," in Proceedings of ECOC 1996 (Oslo, Norway, Sep. 1996) 2, pp. 173-176.
  6. H. Bissesur, G. Charlet, C. Simonneau, S. Borne, L. Pierre, C. De Barros, P. Train, W. Idler, and R. Dischler, "3.2 Tb/s 80×40 Gb/s C-band transmission over 3×100 km with 0.8 bit/s/Hz efficiency," in Proceedings of ECOC 2001 (Amsterdam, The Netherlands, Sep. 2001) 6, pp. 22-29.
  7. T. Ono, Y. Yano, K. Fukuchi, T. Ito, H. Yamazaki, M. Yamaguchi, and K. Emura, "Characteristics of optical duobinary signals in terabit/s capacity, high-spectral efficiency WDM systems," J. Lightwave Technol. 16, 788-797 (1998).
    [CrossRef]
  8. K. Yonenaga and S. Kuwano, "Dispersion tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530-1537 (1997).
    [CrossRef]
  9. D. Penninckx, M. Chbat, L. Pierre, and J.-P. Thierry, "The Phase-Shaped Binary Transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," IEEE Photon. Technol. Lett. 9, 259-261 (1997).
    [CrossRef]
  10. W. Kaiser, M. Wichers, T. Wuth, W. Rosenkranz, C. Scheerer, C. Glingener, A. Färbert, J.-P. Elbers, and G. Fischer, "SPM limit of Duobinary transmission," Proceedings of ECOC 2000 (Munich, Germany, Sep. 2000), 3, pp. 31-32.
  11. X. Zheng, F. Liu, and P. Jeppesen, "Receiver optimization for 40-Gb/s optical duobinary signal," IEEE Photon. Technol. Lett.,  13, 744-746 (2001).
    [CrossRef]
  12. I. Lyubomirsky and B. Pitchumani, "Impact of Optical Filtering on Duobinary Transmission," IEEE Photon. Technol. Lett. 16, 1969-1971 (2004).
    [CrossRef]
  13. G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Modulation formats suitable for ultrahigh spectral efficient WDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 321-328 (2004).
    [CrossRef]
  14. N. B. Pavloviæ and A. V. T. Cartaxo, "Optimized Bandwidth Limited Duobinary Coding Format for Ultra Dense WDM Systems," in Proc. of ICTON 2005, paper We P.3.
  15. G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct detection optically preamplified receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102-104 (2003).
    [CrossRef]
  16. G. Bosco and R. Gaudino, "On BER estimation in optical system simulation: Monte-Carlo vs. semi-analytical techniques", ECOC 2000 Proceedings, Workshop on Modelling and design of optical networks and systems, (Munich, Germany, Sep.2000), paper 3.3.
  17. E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and post-detection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
    [CrossRef]
  18. J. S. Lee and C.S. Shim, "Bit error rate analysis of optically preamplified receiver using an eigenfunction expansion method in optical frequency domain," J. Lightwave Technol. 12, 1224-1229 (1994).
    [CrossRef]
  19. G. Bosco et al., "A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
    [CrossRef]
  20. D. Penninckx, H. Bissessur, P. Brindel, E. Gohin, and F. Bakhtu, "Optical Differential Phase Shift Keying (DPSK) direct detection considered as a duobinary signal," Proceedings of ECOC 2001 (Amsterdam, Sep. 2001), paper We.P.40.

2004

I. Lyubomirsky and B. Pitchumani, "Impact of Optical Filtering on Duobinary Transmission," IEEE Photon. Technol. Lett. 16, 1969-1971 (2004).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Modulation formats suitable for ultrahigh spectral efficient WDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 321-328 (2004).
[CrossRef]

2003

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct detection optically preamplified receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102-104 (2003).
[CrossRef]

2001

X. Zheng, F. Liu, and P. Jeppesen, "Receiver optimization for 40-Gb/s optical duobinary signal," IEEE Photon. Technol. Lett.,  13, 744-746 (2001).
[CrossRef]

G. Bosco et al., "A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

2000

1998

1997

K. Yonenaga and S. Kuwano, "Dispersion tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530-1537 (1997).
[CrossRef]

D. Penninckx, M. Chbat, L. Pierre, and J.-P. Thierry, "The Phase-Shaped Binary Transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," IEEE Photon. Technol. Lett. 9, 259-261 (1997).
[CrossRef]

1995

K. Yonenaga, S. Kuwano, and S. Norimatsu, "Optical duobinary transmission system with no receiver sensitivity degradation," Electron. Lett. 31, 302-304 (1995).
[CrossRef]

1994

J. S. Lee and C.S. Shim, "Bit error rate analysis of optically preamplified receiver using an eigenfunction expansion method in optical frequency domain," J. Lightwave Technol. 12, 1224-1229 (1994).
[CrossRef]

1993

X. Gu and L.C. Blank, "10 Gbit/s unrepeated three-level optical transmission over 100 km of standard fibre," Electron. Lett. 29, 2209-2211 (1993).
[CrossRef]

1963

A. Lender, "The duobinary technique for high-speed data transmission," IEEE Trans. Commun. and Elecron. 82, 214-218 (1963).

Blank, L.C.

X. Gu and L.C. Blank, "10 Gbit/s unrepeated three-level optical transmission over 100 km of standard fibre," Electron. Lett. 29, 2209-2211 (1993).
[CrossRef]

Bosco, G.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Modulation formats suitable for ultrahigh spectral efficient WDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 321-328 (2004).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct detection optically preamplified receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102-104 (2003).
[CrossRef]

G. Bosco et al., "A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

Carena, A.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Modulation formats suitable for ultrahigh spectral efficient WDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 321-328 (2004).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct detection optically preamplified receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102-104 (2003).
[CrossRef]

Chbat, M.

D. Penninckx, M. Chbat, L. Pierre, and J.-P. Thierry, "The Phase-Shaped Binary Transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," IEEE Photon. Technol. Lett. 9, 259-261 (1997).
[CrossRef]

Curri, V.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Modulation formats suitable for ultrahigh spectral efficient WDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 321-328 (2004).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct detection optically preamplified receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102-104 (2003).
[CrossRef]

Emura, K.

Forestieri, E.

Fukuchi, K.

Gaudino, R.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Modulation formats suitable for ultrahigh spectral efficient WDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 321-328 (2004).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct detection optically preamplified receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102-104 (2003).
[CrossRef]

Gu, X.

X. Gu and L.C. Blank, "10 Gbit/s unrepeated three-level optical transmission over 100 km of standard fibre," Electron. Lett. 29, 2209-2211 (1993).
[CrossRef]

Ito, T.

Jeppesen, P.

X. Zheng, F. Liu, and P. Jeppesen, "Receiver optimization for 40-Gb/s optical duobinary signal," IEEE Photon. Technol. Lett.,  13, 744-746 (2001).
[CrossRef]

Kuwano, S.

K. Yonenaga and S. Kuwano, "Dispersion tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530-1537 (1997).
[CrossRef]

K. Yonenaga, S. Kuwano, and S. Norimatsu, "Optical duobinary transmission system with no receiver sensitivity degradation," Electron. Lett. 31, 302-304 (1995).
[CrossRef]

Lee, J. S.

J. S. Lee and C.S. Shim, "Bit error rate analysis of optically preamplified receiver using an eigenfunction expansion method in optical frequency domain," J. Lightwave Technol. 12, 1224-1229 (1994).
[CrossRef]

Lender, A.

A. Lender, "The duobinary technique for high-speed data transmission," IEEE Trans. Commun. and Elecron. 82, 214-218 (1963).

Liu, F.

X. Zheng, F. Liu, and P. Jeppesen, "Receiver optimization for 40-Gb/s optical duobinary signal," IEEE Photon. Technol. Lett.,  13, 744-746 (2001).
[CrossRef]

Lyubomirsky, I.

I. Lyubomirsky and B. Pitchumani, "Impact of Optical Filtering on Duobinary Transmission," IEEE Photon. Technol. Lett. 16, 1969-1971 (2004).
[CrossRef]

Norimatsu, S.

K. Yonenaga, S. Kuwano, and S. Norimatsu, "Optical duobinary transmission system with no receiver sensitivity degradation," Electron. Lett. 31, 302-304 (1995).
[CrossRef]

Ono, T.

Penninckx, D.

D. Penninckx, M. Chbat, L. Pierre, and J.-P. Thierry, "The Phase-Shaped Binary Transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," IEEE Photon. Technol. Lett. 9, 259-261 (1997).
[CrossRef]

Pierre, L.

D. Penninckx, M. Chbat, L. Pierre, and J.-P. Thierry, "The Phase-Shaped Binary Transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," IEEE Photon. Technol. Lett. 9, 259-261 (1997).
[CrossRef]

Pitchumani, B.

I. Lyubomirsky and B. Pitchumani, "Impact of Optical Filtering on Duobinary Transmission," IEEE Photon. Technol. Lett. 16, 1969-1971 (2004).
[CrossRef]

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Modulation formats suitable for ultrahigh spectral efficient WDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 321-328 (2004).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct detection optically preamplified receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102-104 (2003).
[CrossRef]

Shim, C.S.

J. S. Lee and C.S. Shim, "Bit error rate analysis of optically preamplified receiver using an eigenfunction expansion method in optical frequency domain," J. Lightwave Technol. 12, 1224-1229 (1994).
[CrossRef]

Thierry, J.-P.

D. Penninckx, M. Chbat, L. Pierre, and J.-P. Thierry, "The Phase-Shaped Binary Transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," IEEE Photon. Technol. Lett. 9, 259-261 (1997).
[CrossRef]

Yamaguchi, M.

Yamazaki, H.

Yano, Y.

Yonenaga, K.

K. Yonenaga and S. Kuwano, "Dispersion tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530-1537 (1997).
[CrossRef]

K. Yonenaga, S. Kuwano, and S. Norimatsu, "Optical duobinary transmission system with no receiver sensitivity degradation," Electron. Lett. 31, 302-304 (1995).
[CrossRef]

Zheng, X.

X. Zheng, F. Liu, and P. Jeppesen, "Receiver optimization for 40-Gb/s optical duobinary signal," IEEE Photon. Technol. Lett.,  13, 744-746 (2001).
[CrossRef]

Electron. Lett.

X. Gu and L.C. Blank, "10 Gbit/s unrepeated three-level optical transmission over 100 km of standard fibre," Electron. Lett. 29, 2209-2211 (1993).
[CrossRef]

K. Yonenaga, S. Kuwano, and S. Norimatsu, "Optical duobinary transmission system with no receiver sensitivity degradation," Electron. Lett. 31, 302-304 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Modulation formats suitable for ultrahigh spectral efficient WDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 321-328 (2004).
[CrossRef]

IEEE Photon. Technol. Lett.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct detection optically preamplified receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102-104 (2003).
[CrossRef]

D. Penninckx, M. Chbat, L. Pierre, and J.-P. Thierry, "The Phase-Shaped Binary Transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," IEEE Photon. Technol. Lett. 9, 259-261 (1997).
[CrossRef]

X. Zheng, F. Liu, and P. Jeppesen, "Receiver optimization for 40-Gb/s optical duobinary signal," IEEE Photon. Technol. Lett.,  13, 744-746 (2001).
[CrossRef]

I. Lyubomirsky and B. Pitchumani, "Impact of Optical Filtering on Duobinary Transmission," IEEE Photon. Technol. Lett. 16, 1969-1971 (2004).
[CrossRef]

IEEE Trans. Commun.

G. Bosco et al., "A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

IEEE Trans. Commun. and Elecron.

A. Lender, "The duobinary technique for high-speed data transmission," IEEE Trans. Commun. and Elecron. 82, 214-218 (1963).

J. Lightwave Technol.

K. Yonenaga and S. Kuwano, "Dispersion tolerant optical transmission system using duobinary transmitter and binary receiver," J. Lightwave Technol. 15, 1530-1537 (1997).
[CrossRef]

E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and post-detection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
[CrossRef]

T. Ono, Y. Yano, K. Fukuchi, T. Ito, H. Yamazaki, M. Yamaguchi, and K. Emura, "Characteristics of optical duobinary signals in terabit/s capacity, high-spectral efficiency WDM systems," J. Lightwave Technol. 16, 788-797 (1998).
[CrossRef]

J. S. Lee and C.S. Shim, "Bit error rate analysis of optically preamplified receiver using an eigenfunction expansion method in optical frequency domain," J. Lightwave Technol. 12, 1224-1229 (1994).
[CrossRef]

Other

D. Penninckx, H. Bissessur, P. Brindel, E. Gohin, and F. Bakhtu, "Optical Differential Phase Shift Keying (DPSK) direct detection considered as a duobinary signal," Proceedings of ECOC 2001 (Amsterdam, Sep. 2001), paper We.P.40.

D. Penninckx, M. Chbat, L. Pierre, and J.-P. Thierry, "The Phase-Shaped Binary Transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," in Proceedings of ECOC 1996 (Oslo, Norway, Sep. 1996) 2, pp. 173-176.

H. Bissesur, G. Charlet, C. Simonneau, S. Borne, L. Pierre, C. De Barros, P. Train, W. Idler, and R. Dischler, "3.2 Tb/s 80×40 Gb/s C-band transmission over 3×100 km with 0.8 bit/s/Hz efficiency," in Proceedings of ECOC 2001 (Amsterdam, The Netherlands, Sep. 2001) 6, pp. 22-29.

W. Kaiser, M. Wichers, T. Wuth, W. Rosenkranz, C. Scheerer, C. Glingener, A. Färbert, J.-P. Elbers, and G. Fischer, "SPM limit of Duobinary transmission," Proceedings of ECOC 2000 (Munich, Germany, Sep. 2000), 3, pp. 31-32.

G. Bosco and R. Gaudino, "On BER estimation in optical system simulation: Monte-Carlo vs. semi-analytical techniques", ECOC 2000 Proceedings, Workshop on Modelling and design of optical networks and systems, (Munich, Germany, Sep.2000), paper 3.3.

J. G. Proakis, Digital communications, 2nd edition (New York, McGraw-Hill), 1989.

N. B. Pavloviæ and A. V. T. Cartaxo, "Optimized Bandwidth Limited Duobinary Coding Format for Ultra Dense WDM Systems," in Proc. of ICTON 2005, paper We P.3.

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Figures (6)

Fig. 1.
Fig. 1.

The transmission channel.

Fig. 2.
Fig. 2.

Schematics of the duobinary system

Fig. 3.
Fig. 3.

Noisy eye-diagram of the duobinary signal before photo-detection.

Fig. 4.
Fig. 4.

Example of duobinary pulse shape.

Fig. 5.
Fig. 5.

Schematic of the PSBT duobinary transmitter

Fig. 6.
Fig. 6.

BER vs. OSNR for the three configurations of duobinary systems reported in Tab. 2. The theoretical limit curve defined by Eq. (36) and the performance curve of a system employing a realistic PIN receiver are also shown for comparison.

Tables (2)

Tables Icon

Table 1. Comparison between binary and duobinary modulations with quadratic receiver

Tables Icon

Table 2. Optimized filter bandwidths

Equations (37)

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

A m a m δ ( t m T )
H E ( f ) = H R ( f ) = N Y ( f )
P E = 1 2 erfc A σ 2 = 1 2 erfc A N 0 =
= 1 2 erfc P m T N 0 = 1 2 erfc E N 0
P E = 1 2 erfc A σ 2 = 1 2 erfc P m T λ 2 N 0 λ = N Y ( f ) d f
N Y 1 ( f ) = { T ( cos π f T 2 ) ̅ exp { i π f T 2 } f < 1 T 0 elsewhere
C ( f ) = 1 + exp ( 2 i π f T ) = 2 cos π f T · e i π f T
H E ( f ) = H R ( f ) = N Y ( f ) · C ( f )
P E = 3 4 erfc A σ 2
σ 2 = N 0 2 N Y ( f ) · C ( f ) d f = N 0 2 μ
P E = 3 4 erfc A σ 2 = 3 4 erfc A μ N 0 = 1 2 erfc P m T μ 2 N 0
μ = N Y ( f ) · C ( f ) d f = 2 T 1 2 T 1 2 T cos π f T d f = 4 π
Δ = 20 log 4 π = 2.1 dB
P m , duobinary = μ 2 2 P m , binary = 8 π 2 P m , binary
10 log 8 π 2 = 0.91 dB
Δ Q = 10 log 16 π 2 10 log 2 = 10 log 8 π 2 = 0.91 dB
{ P m , binary = A 2 2 T rect 2 ( t ) d t = A 2 2 P m , duobinary = A 2 T rect ( t ) + rect ( t T ) 2 d t = 2 A 2
X ( f ) = T ( sin c π f T ) 2 · exp ( i π f T )
p n = a ̅ n p n 1
b n = 2 p n 1
S TX ( t ) = [ P ̅ S n b n u ( t n T ) ] e j ω 0 t v ̂ ,
s RX ( t ) = { [ P ̅ s n b n u ( t n T ) + n ( t ) ] v ̂ + n ( t ) v ̂ } e j ω 0 t ,
{ n ( t ) = n i ( t ) + j n q ( t ) n ( t ) = n i ( t ) + j n q ( t )
x ( 0 ) = x ( T ) 0 , x ( n T ) = 0 , n 0 , 1
s RX ( t opt ) = [ P ̅ S c n x ( 0 ) + n F ( t opt ) ] v ̂ + n F ( t opt ) v ̂ ,
c n = { 0 if b n b n 1 2 if b n = b n 1 = + 1 2 if b n = b n 1 = 1
{ n F ( t opt ) = n Fi + jn Fq n F ( t opt ) = n Fi + jn Fq
σ 2 = N 0 2 + H RX ( f ) 2 d f = N 0 2 B eq, RX
i D ( t opt ) = ( P ̅ S c n x ( 0 ) + n Fi ) 2 + n Fq 2 + n Fi 2 + n Fq 2
s 2 = { 0 if b n b n 1 , i. e . a n = 0 4 P ̅ S [ x ( 0 ) ] 2 if b n = b n 1 , i. e . a n = 1
P E = 1 2 { e ϕ ( 1 + ϕ ) + 1 Q 2 ( 4 P ̅ S x 2 ( 0 ) σ 2 , 2 ϕ ) }
P E = 1 2 { e ϕ ( 1 + ϕ ) + 1 Q 2 ( x ( 0 ) x ( T 2 ) 16 OSNR , 2 ϕ ) }
OSNR = P ̅ S 2 N 0 R B
x ( t ) = sin ( π t T ) 4 t T ( 1 t T ) = π 4 [ sin ( π t T ) π t T + sin ( π t T T ) π ( t T T ) ]
X ( f ) = { π T 2 cos ( π f T ) e j π f T , f [ 1 2 T , 1 2 T ] 0 , f [ 1 2 T , 1 2 T ]
u ( t ) = F 1 { π T 2 cos ( π f T ) }
P E = 1 2 { e ϕ ( 1 + ϕ ) + 1 Q 2 ( π 4 16 OSNR , 2 ϕ ) }

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