## Spectral efficiency limits of pre-filtered modulation formats |

Optics Express, Vol. 18, Issue 19, pp. 20273-20281 (2010)

http://dx.doi.org/10.1364/OE.18.020273

Acrobat PDF (1205 KB)

### Abstract

We investigate the spectral efficiency (SE) limit of pre-filtered modulation in optical fiber communication systems. We show that pre-filtering induced symbol correlation generates a modulation with memory and, thus, a higher SE limit than that of the original memoryless modulation. The SE limits of a series of modulation formats with varying number *L* of correlated symbols producing 50% of the original bandwidth are evaluated, which approach the SE limit of a modulation format with 50% pre-filtering as *L* → ∞. We show that the SE limit of a modulation format with *L* correlated symbols approximates a lower bound for the SE limit of the corresponding pre-filtered format.

© 2010 OSA

## 1. Introduction

5. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express **13**(12), 4568–4579 (2005). [CrossRef] [PubMed]

8. J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010).

8. J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010).

11. J. Tang, “The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission,” J. Lightwave Technol. **19**(8), 1104–1109 (2001). [CrossRef]

18. A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. **28**(4), 423–433 (2010). [CrossRef]

## 2. Theoretical model for SE limit analysis of pre-filtered modulation

*T*-second transmission (

*T*→ ∞) in an AWGN channel of band

*W*can be specified by 2

*WT*numbers, and the SE limit is given bywhere

*C*is channel capacity and

*P*and

*N*are average signal and noise power, respectively. This is the ultimate SE limit for any modulation and coding schemes that can be applied to an AWGN channel. Figure 1 depicts the spectrum (a), in-phase and quadrature components (b), and time-domain pulses (c) of a

*T*-second signal that can approach the SE limit given in Eq. (1) as

*T*→ ∞ and the channel input (

*x*

_{1},

*x*

_{2}, …,

*x*,

_{WT}*q*

_{1},

*q*

_{2}, …,

*q*) are independent and identically distributed (i.i.d.) Gaussian random variables. The limit approaching signal has a rectangular spectrum of bandwidth

_{WT}*W*in the frequency domain, and 1/

*W*spaced sinc pulses in the time domain, i.e.

*C*

_{1D}, then the 1D SE limit is half of the limit given in Eq. (1), i.e.,

*x*

_{1},

*x*

_{2}, …,

*x*), i.e.

_{WT}*x*∈ {–1, 1}, with equal probability of taking –1 or 1. The SE limit of such a binary modulation format is given by [10,13

_{i}13. G. Kramer, A. Ashikhmin, A. J. van Wijngaarden, and X. Wei, “Spectral efficiency of coded phase-shift keying for fiber-optic communication,” J. Lightwave Technol. **21**(10), 2438–2445 (2003). [CrossRef]

*C*

_{B}is the binary modulation capacity,

*y*is the channel output corresponding to the noise-impaired

_{i}*x*, and

_{i}*f*(

_{Y}*y*) is the probability density function (pdf) of

_{i}*y*given by [10]

_{i}*N*→ 0 for a given signal power

*P*), the well-known SE limits

*C*

_{1D}/

*W*→ ∞ and

*C*

_{B}/

*W*→ 1 bit/s/Hz can be derived from Eqs. (3) and (4). The difference between the two SE limits can be understood in terms of the usage of signal power. For a signal with statistical properties that approach that of white noise, the signal power is fully utilized to generate uncertainty of the signal [10]. Signals with higher uncertainty carry more information. On the other hand for BPSK modulation, only a portion of the signal power contributes to the signal uncertainty (or carries information). If we loosen the binary constraint on the channel inputs, we would be able to exceed the BPSK SE limit and achieve a SE somewhere in between the two limits shown in Fig. 2.

*W*and all 6 pulses can be modulated independently in the time domain. Each of the 6 pulses carries 1-bit of information without interfering with each other.

5. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express **13**(12), 4568–4579 (2005). [CrossRef] [PubMed]

*k*is the recursion index, and

**A**

*is the resulting alphabet after the*

_{k}*k-*th recursion with 2

*k*−1 symbols included in the calculation on each side of the symbol under investigation. The corresponding probability mass function (pmf) is given by

*k*= 1 the blue and green pulses are included in the recursive calculations in Eqs. (6) and (7). Figure 4 plots the pmf of the signal value of the 50% pre-filtered BPSK modulation format for

*k*= 4. It shows a much more complex modulation than the original BPSK modulation, which results in a higher SE limit as shown in the following sections.

## 3. SE limit of pre-filtered modulation in linear noise-dominated optical fiber channels

*f*

**(**

_{Y}*y*

_{1},

*y*

_{2}, …,

*y*), we need to know the statistics of the noise vector (

_{L}*n*

_{1},

*n*

_{2}, …,

*n*). In a linear optical fiber channel dominated by ASE noise, the noise samples (i.e.,

_{L}*n*

_{1},

*n*

_{2}, …,

*n*) are Gaussian random variables. For the SE limit evaluation, we assume an ideal rectangular filter shape as shown in Fig. 3(b). Thus, the power spectral density of the ASE noise after the receiver filter (with a bandwidth of

_{L}*W*/2 to match the 50% pre-filter at the transmitter) has a rectangular shape. Correspondingly, the autocorrelation function of the filtered noise is a sinc function with nulls at all the sampling instants [Fig. 3(b)]. Hence, the noise samples

*n*

_{1},

*n*

_{2}, …,

*n*are independent Gaussian random variables and, thus, the joint pdf of

_{L}**Y**is given by [22]where

*p*(

*x*,

_{i1}*x*, …,

_{i2}*x*) is the probability of the chosen alphabet vector (

_{iL}*x*,

_{i1}*x*, …,

_{i2}*x*). Note that if the noise is not sampled at the Nyquist rate as in some practical implementations, the resulting noise samples are no longer independent and the noise correlation must be taken into account in the joint statistics [23

_{iL}23. N. Alic, M. Karlsson, M. Sköld, O. Milenkovic, P. A. Andrekson, and S. Radic, “Joint statistics and MLSD in filtered incoherent high-speed fiber optic communications,” J. Lightwave Technol. **28**(10), 1564–1572 (2010). [CrossRef]

## 4. Approximation of SE limit for 50% pre-filtered BPSK and QPSK modulation

*L*= 4 as a lower bound for the SE limit of the 50% pre-filtered BPSK modulation. The SE limit is also bound by the 1D limit of unconstrained modulation and the 2 bits/s/Hz limit derived from Eqs. (9) and (10) for infinite SNR.

5. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express **13**(12), 4568–4579 (2005). [CrossRef] [PubMed]

23. N. Alic, M. Karlsson, M. Sköld, O. Milenkovic, P. A. Andrekson, and S. Radic, “Joint statistics and MLSD in filtered incoherent high-speed fiber optic communications,” J. Lightwave Technol. **28**(10), 1564–1572 (2010). [CrossRef]

25. I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. **26**(6), 73–83 (2008). [CrossRef]

## 5. Conclusions

## References and links

1. | M. Salsi, H. Mardoyan, P. Tran, C. Koebele, E. Dutisseuil, G. Charlet, and S. Bigo, “155x100Gbit/s coherent PDM-QPSK transmission over 7,200km,” Proc. ECOC’09, paper PD2.5 (2009). |

2. | A. Sano, H. Masuda, T. Kobayashi, M. Fujiwara, K. Horikoshi, E. Yoshida, Y. Miyamoto, M. Matsui, M. Mizoguchi, H. Yamazaki, Y. Sakamaki, and H. Ishii, “69.1-Tb/s (432x171-Gb/s) C- and extended L-band transmission over 240 km using PDM-16-QAM modulation and digital coherent detection,” Proc. OFC/NFOEC’10, paper PDPB7 (2010). |

3. | X. Zhou, J. Yu, M. Huang, Y. Shao, T. Wang, L. Nelson, P. Magill, M. Birk, P. I. Borel, D. W. Peckham, and R. Lingle, Jr., “64-Tb/s (640x107-Gb/s) PDM-36QAM transmission over 320km using both pre- and post-transmission digital equalization,” Proc. OFC/NFOEC’10, paper PDPB9 (2010). |

4. | X. Liu, S. Chandrasekhar, B. Chu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “Transmission of a 448-Gb/s reduced-guard-interval CO-OFDM signal with a 60-GHz optical bandwidth over 2000 km of ULAF and five 80-GHz-Grid ROADMs,” Proc. OFC/NFOEC’10, paper PDPC2 (2010). |

5. | N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express |

6. | M. Rubsamen, P. J. Winzer, and R.-J. Essiambre, “MLSE receivers for narrow-band optical filtering,” Proc. OFC/NFOEC‘06, paper OWB6 (2006). |

7. | N. Alic, E. Myslivets, and S. Radic, “1.0 bit/s/Hz spectral efficiency in single polarization at 2000km with narrowly filtered intensity modulated signals,” Proc. IEEE Summer Topical Meeting 2008, paper TuD3.4 (2008). |

8. | J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010). |

9. | Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010). |

10. | C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. |

11. | J. Tang, “The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission,” J. Lightwave Technol. |

12. | Y. Cai, N. Ramanujam, J. M. Morris, T. Adali, G. Lenner, A. B. Puc, and A. Pilipetskii, “Performance limit of forward error correction codes in optical fiber communications,” Proc. OFC/IOOC’01, paper TuF2 (2001). |

13. | G. Kramer, A. Ashikhmin, A. J. van Wijngaarden, and X. Wei, “Spectral efficiency of coded phase-shift keying for fiber-optic communication,” J. Lightwave Technol. |

14. | J. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” J. Sel. Top. Quantum Electron. |

15. | I. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightwave Technol. |

16. | Y. Cai, “Performance limits of FEC and modulation formats in optical fiber communications,” Proc. LEOS’06, paper WH1 (2006). |

17. | R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. |

18. | A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. |

19. | D. Arnold, and H. Loeliger, “On the information rate of binary-input channels with memory,” Proc. ICC’2001, pp. 2692–2695 (2001). |

20. | T. Cover, and J. Thomas, |

21. | R. G. Gallager, |

22. | J. G. Proakis, |

23. | N. Alic, M. Karlsson, M. Sköld, O. Milenkovic, P. A. Andrekson, and S. Radic, “Joint statistics and MLSD in filtered incoherent high-speed fiber optic communications,” J. Lightwave Technol. |

24. | Y. Cai, D. Foursa, C. R. Davidson, J.-X. Cai, O. Sinkin, M. Nissov, and A. Pilipetskii, “Experimental demonstration of coherent MAP detection for nonlinearity mitigation in long-haul transmissions,” Proc. OFC/NFOEC’10, paper OTuE1 (2010). |

25. | I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.2330) Fiber optics and optical communications : Fiber optics communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 28, 2010

Revised Manuscript: August 31, 2010

Manuscript Accepted: September 2, 2010

Published: September 8, 2010

**Citation**

Yi Cai, Jin-Xing Cai, Alexei Pilipetskii, Georg Mohs, and Neal S. Bergano, "Spectral efficiency limits of pre-filtered modulation formats," Opt. Express **18**, 20273-20281 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20273

Sort: Year | Journal | Reset

### References

- M. Salsi, H. Mardoyan, P. Tran, C. Koebele, E. Dutisseuil, G. Charlet, and S. Bigo, “155x100Gbit/s coherent PDM-QPSK transmission over 7,200km,” Proc. ECOC’09, paper PD2.5 (2009).
- A. Sano, H. Masuda, T. Kobayashi, M. Fujiwara, K. Horikoshi, E. Yoshida, Y. Miyamoto, M. Matsui, M. Mizoguchi, H. Yamazaki, Y. Sakamaki, and H. Ishii, “69.1-Tb/s (432x171-Gb/s) C- and extended L-band transmission over 240 km using PDM-16-QAM modulation and digital coherent detection,” Proc. OFC/NFOEC’10, paper PDPB7 (2010).
- X. Zhou, J. Yu, M. Huang, Y. Shao, T. Wang, L. Nelson, P. Magill, M. Birk, P. I. Borel, D. W. Peckham, and R. Lingle, Jr., “64-Tb/s (640x107-Gb/s) PDM-36QAM transmission over 320km using both pre- and post-transmission digital equalization,” Proc. OFC/NFOEC’10, paper PDPB9 (2010).
- X. Liu, S. Chandrasekhar, B. Chu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “Transmission of a 448-Gb/s reduced-guard-interval CO-OFDM signal with a 60-GHz optical bandwidth over 2000 km of ULAF and five 80-GHz-Grid ROADMs,” Proc. OFC/NFOEC’10, paper PDPC2 (2010).
- N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]
- M. Rubsamen, P. J. Winzer, and R.-J. Essiambre, “MLSE receivers for narrow-band optical filtering,” Proc. OFC/NFOEC‘06, paper OWB6 (2006).
- N. Alic, E. Myslivets, and S. Radic, “1.0 bit/s/Hz spectral efficiency in single polarization at 2000km with narrowly filtered intensity modulated signals,” Proc. IEEE Summer Topical Meeting 2008, paper TuD3.4 (2008).
- J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010).
- Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).
- C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
- J. Tang, “The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission,” J. Lightwave Technol. 19(8), 1104–1109 (2001). [CrossRef]
- Y. Cai, N. Ramanujam, J. M. Morris, T. Adali, G. Lenner, A. B. Puc, and A. Pilipetskii, “Performance limit of forward error correction codes in optical fiber communications,” Proc. OFC/IOOC’01, paper TuF2 (2001).
- G. Kramer, A. Ashikhmin, A. J. van Wijngaarden, and X. Wei, “Spectral efficiency of coded phase-shift keying for fiber-optic communication,” J. Lightwave Technol. 21(10), 2438–2445 (2003). [CrossRef]
- J. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” J. Sel. Top. Quantum Electron. 10(2), 259–272 (2004). [CrossRef]
- I. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightwave Technol. 23(11), 3755–3763 (2005). [CrossRef]
- Y. Cai, “Performance limits of FEC and modulation formats in optical fiber communications,” Proc. LEOS’06, paper WH1 (2006).
- R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]
- A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. 28(4), 423–433 (2010). [CrossRef]
- D. Arnold, and H. Loeliger, “On the information rate of binary-input channels with memory,” Proc. ICC’2001, pp. 2692–2695 (2001).
- T. Cover, and J. Thomas, Elements of Information Theory, John Wiley & Sons, New York (1991).
- R. G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, New York (1968).
- J. G. Proakis, Digital Communications, 4th Edition, McGraw-Hill, New York (2001).
- N. Alic, M. Karlsson, M. Sköld, O. Milenkovic, P. A. Andrekson, and S. Radic, “Joint statistics and MLSD in filtered incoherent high-speed fiber optic communications,” J. Lightwave Technol. 28(10), 1564–1572 (2010). [CrossRef]
- Y. Cai, D. Foursa, C. R. Davidson, J.-X. Cai, O. Sinkin, M. Nissov, and A. Pilipetskii, “Experimental demonstration of coherent MAP detection for nonlinearity mitigation in long-haul transmissions,” Proc. OFC/NFOEC’10, paper OTuE1 (2010).
- I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.