## A family of Nyquist pulses for coherent optical communications |

Optics Express, Vol. 20, Issue 8, pp. 8397-8416 (2012)

http://dx.doi.org/10.1364/OE.20.008397

Acrobat PDF (2538 KB)

### Abstract

A new family of Nyquist pulses for coherent optical single carrier systems is introduced and is shown to increase the nonlinearity tolerance of dual-polarization (DP)-QPSK and DP-16-QAM systems. Numerical investigations for a single-channel 28 Gbaud DP-16-QAM long-haul system without optical dispersion compensation indicate that the proposed pulse can increase the reach distance by 26% and 19%, for roll-off factors of 1 and 2, respectively. In multi-channel transmissions and for a roll-off factor of 1, a reach distance increase of 20% is reported. Experimental results for DP-QPSK and DP-16-QAM systems at 10 Gbaud confirm the superior nonlinearity tolerance of the proposed pulse.

© 2012 OSA

## 1. Introduction

## 2. Background

1. E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. **22**(23), 1714–1716 (2010). [CrossRef]

3. C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express **19**(14), 12879–12884 (2011). [CrossRef] [PubMed]

1. E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. **22**(23), 1714–1716 (2010). [CrossRef]

3. C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express **19**(14), 12879–12884 (2011). [CrossRef] [PubMed]

1. E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. **22**(23), 1714–1716 (2010). [CrossRef]

**22**(23), 1714–1716 (2010). [CrossRef]

4. M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256-QAM (64 Gb/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” IEEE Photon. Technol. Lett. **22**(3), 185–187 (2010). [CrossRef]

7. K. Roberts, A. Borowiec, and C. Laperle, “Technologies for optical systems beyond 100G,” Opt. Fiber Technol. **17**(5), 387–394 (2011). [CrossRef]

8. B. Farhang-Boroujeny, “A square-root Nyquist (M) filter design for digital communication systems,” IEEE Trans. Signal Process. **56**(5), 2127–2132 (2008). [CrossRef]

10. B. Farhang-Boroujeny and G. Mathew, “Nyquist filters with robust performance against timing jitter,” IEEE Trans. Signal Process. **46**(12), 3427–3431 (1998). [CrossRef]

11. A. Assalini and A. M. Tonello, “Improved Nyquist pulses,” IEEE Commun. Lett. **8**(2), 87–89 (2004). [CrossRef]

12. K. Roberts, M. O’Sullivan, K.-T. Wu, H. Sun, A. Awadalla, D. J. Krause, and C. Laperle, “Performance of dual-polarization QPSK for optical transport systems,” J. Lightwave Technol. **27**(16), 3546–3559 (2009). [CrossRef]

**22**(23), 1714–1716 (2010). [CrossRef]

3. C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express **19**(14), 12879–12884 (2011). [CrossRef] [PubMed]

## 3. Nyquist pulses for optical communication

*T*, 2

*T*, 3

*T*, …) of the received waveform must be zero. To obtain zero ISI, the impulse response of a Nyquist pulse shaping filter should meet the following condition:

*k*/

*T*equals to

*T*, or more generally, to a constant level.

*W*is 2

*W*independent symbols per second. The pulse shaping filter having minimum bandwidth while satisfying the ISI condition expressed by Eq. (2) is thus given by:

*α*) of zero. Its frequency response is represented in Fig. 1 for

*T*= 1. It can be seen that it satisfies the ISI criterion expressed by Eq. (2), here shown for

*k*ranging between –1 and 1. Figure 1 also shows that a RC filter with

*α*= 1 satisfies the ISI criterion, since adding all the delayed spectra would produce a constant level of one. Furthermore, it can be observed that the bandwidth of a RC filter with

*α*= 1 is twice the bandwidth of the minimum-bandwidth Nyquist pulse (

*α*= 0), hence corresponding to an excess bandwidth of 100%.

*α*= 0 is very slow. This translates in a waveform with a high PAPR that is also more sensitive to synchronization errors. Nevertheless, recent experiments on optical coherent systems [5–7

7. K. Roberts, A. Borowiec, and C. Laperle, “Technologies for optical systems beyond 100G,” Opt. Fiber Technol. **17**(5), 387–394 (2011). [CrossRef]

17. F. Zhang, “XPM Statistics in 100% Precompensated WDM Transmission for OOK and DPSK Formats,” IEEE Photon. Technol. Lett. **21**(22), 1707–1709 (2009). [CrossRef]

19. C. Behrens, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Nonlinear Distortion in Transmission of Higher Order Modulation Formats,” IEEE Photon. Technol. Lett. **22**(15), 1111–1113 (2010). [CrossRef]

**22**(23), 1714–1716 (2010). [CrossRef]

**19**(14), 12879–12884 (2011). [CrossRef] [PubMed]

*α*> 0, the rectangular shape is the low-pass filter shape that maximizes the most the energy content in the higher portion of the spectrum. The design of the pulse is thus achieved first, by specifying a frequency response that is a rectangular function, and then by manually setting its passband shape, so that the ISI criterion expressed by Eq. (2) is fulfilled. For excess bandwidth ranging from 0% to 100%, the proposed pulse response is given by:

## 4. System model

^{15}, pulse shaping finite impulse response (FIR) filters, 6-bit digital-to-analog converters (DACs), in-phase-quadrature (I-Q) modulators and a polarization beam combiner (PBC). As in [3

**19**(14), 12879–12884 (2011). [CrossRef] [PubMed]

**19**(14), 12879–12884 (2011). [CrossRef] [PubMed]

**19**(14), 12879–12884 (2011). [CrossRef] [PubMed]

^{2}/km, the effective area to 80 µm

^{2}, the nonlinear coefficient to 1.2 W

^{−1}km

^{−1}, the PMD coefficient to 0.1 ps/km

^{0.5}, and the EDFA noise figure to 4.5 dB. A linewidth of 100 kHz is specified for the transmitter and receiver lasers, and the frequency offset between the transmitter and receiver laser is assumed to be negligible. The value

*L*corresponds to the number of spans. At the receiver, the WDM demultiplexer (Demux) is modeled with the same bandpass filters as the transmitter Mux. Following the Demux, a coherent front-end integrates polarization beam splitters, optical hybrids, a local oscillator and the photodetectors [12

12. K. Roberts, M. O’Sullivan, K.-T. Wu, H. Sun, A. Awadalla, D. J. Krause, and C. Laperle, “Performance of dual-polarization QPSK for optical transport systems,” J. Lightwave Technol. **27**(16), 3546–3559 (2009). [CrossRef]

^{−1}) are implemented by FIR filters. The final stages of the receiver digital signal processing functions include a 13-tap fractionally-spaced decision-directed butterfly equalizer (EQ) and a decision-directed second order phase-lock loop (PLL) [12

12. K. Roberts, M. O’Sullivan, K.-T. Wu, H. Sun, A. Awadalla, D. J. Krause, and C. Laperle, “Performance of dual-polarization QPSK for optical transport systems,” J. Lightwave Technol. **27**(16), 3546–3559 (2009). [CrossRef]

## 5. Numerical analysis

### 5.1 System performance in single-channel transmission

*T*, or in this case to 56 GHz. But has can be seen in Fig. 5(a), the out-of-band attenuation of the NRZ pulse is lower than it is for the two other pulses. The RRC and proposed pulses have an out-of-band attenuation of more than 40 dB. The out-of-band attenuation of the RRC and proposed pulse is limited in this system by the finite resolution of the DACs.

**19**(14), 12879–12884 (2011). [CrossRef] [PubMed]

^{−3}is considered. It can be seen that in the linear regime, for launch powers between –7 dBm and –3 dBm, the system performance when using the NRZ, RRC and proposed pulses is practically the same. This is explained by the fact that these pulses are all matched root-Nyquist pulses satisfying the ISI criterion. It can also be seen that the performance of the NRZ and RRC pulses is almost the same in the nonlinear regime, from –3 dBm to 4 dBm, resulting in a maximum reach of around 1530 km. This can be understood by the fact that they have similar frequency characteristics in the passband region, as shown in Fig. 5(a). Therefore, when the NRZ and RRC pulses propagate, they experience similar dispersion, resulting in comparable sensitivity to SPM effects. It can also be seen in Fig. 6(a) that the maximal reach obtained with the proposed pulse is 1963 km, corresponding to a 26% reach increase.

*α*= 1.4, the null-to-null bandwidth of the 28 Gbaud DP-16-QAM signal is 67.2 GHz. Figure 5(b) indicates that the out-of-band attenuation provided by the RZ pulse is similar to the out-of-band attenuation obtained when using the NRZ pulse, and that better out-of-band rejection can be achieved by the RRC and proposed pulses.

**19**(14), 12879–12884 (2011). [CrossRef] [PubMed]

*α*> 0.3. For lower roll-off factors, the frequency characteristics of the proposed pulse is very similar to the frequency characteristics of the RRC pulse, explaining the absence of significant improvements. As the roll-off factor is increased, the maximum reach of the system for both pulses is increased, due to the larger bandwidth that is occupied and to the increased dispersion. For roll-off factors of 0.5, 1.0, 1.5 and 2.0, the increase in reach distance when using the proposed pulse is respectively 9%, 26%, 26%, and 19%.

*T*) and infinity, to its total energy:

*E*) obtained using Eq. (9) as a function of the roll-off factor is displayed in Fig. 8 for the RRC and proposed pulses. It can be seen that the proposed metric is directly proportional to the system maximum reach for single-channel transmission, presented in Fig. 7. For the RRC pulse, the energy ratio increases almost constantly as the roll-off factor is increased. In the case of the proposed pulse, the energy ratio increase rate is higher for roll-off factors between 0 and 1, than it is for roll-off factors between 1 and 2. This can be explained by considering the proposed pulse transfer function expressed by Eq. (5) and Eq. (6). For 0 <

_{R}*α*≤ 1, the magnitude of the spectrum for frequencies higher than 1/(2

*T*) is

*α*≤ 2, the magnitude varies as a function of the roll-off factor and is only

*T*) to infinity in the case of 1 <

*α*≤ 2 results in a lower energy ratio increase rate. The excellent correspondence between the energy ratio and the maximum reach further indicates that for coherent systems without optical dispersion compensation, the energy content of the pulse spectrum in its higher portion determines its tolerance to single-channel fiber nonlinearity, as previously suggested in section 3.

### 5.2 System performance in multi-channel transmission

*α*= 0.9, the use of the proposed pulse decreases the maximum reach of the system. This is due to the fact that the bandwidth of the signal is then 53.2 GHz, and that it exceeds the 50 GHz Mux/Demux bandwidth. The Mux and Demux alter the spectrum of the signal, breaking both the matched filter and the ISI conditions. Although the same phenomenon occurs with the RRC pulse, it has less energy content at high frequencies and is therefore less affected by the Mux and Demux responses. For a channel spacing of 100 GHz, a similar effect is observed in Fig. 9(b) reducing the maximal reach at a roll-off factor of 2.0.

20. Y. Jiang, X. Tang, J. C. Cartledge, and K. Roberts, “Electronic Pre-Compensation of Narrow Optical Filtering for OOK, DPSK and DQPSK Modulation Formats,” J. Lightwave Technol. **27**(16), 3689–3698 (2009). [CrossRef]

### 5.3 Effect of pulse truncation

*s*(

*n*) is the impulse response of the optimized FIR pulse shaping filter,

*M*is the impulse response length in symbol periods,

*N*corresponds to the oversampling rate,

*n*is the discrete time index (

*n*= –

*MT*/2, ..., –2

*T*/

*N*, –

*T*/

*N*, 0,

*T*/

*N*, 2

*T*/

*N*, ...,

*MT*/2),

*S*(

*f*) is the Fourier transform of

*s*(

*n*), and

*f*is the frequency index. The number of coefficients of the FIR filter

*s*(

*n*) is thus given by

*M*∙

*N*+ 1. By maximizing the pulse energy of the center coefficient

*s*(0), the first term of the objective function seeks to minimize the pulse width, and consequently the energy content in the higher part of its spectrum. By maximizing the energy of

*S*(

*f*) between 0 and

*γ*relative to the optimization weight

*µ*, the stopband attenuation can be set. To assure zero ISI, or equivalently that Nyquist's first criterion is met, the following constraint is formulated:

*r*(

*n*) is a Nyquist pulse given by

*s*(

*n*) corresponds to a root-Nyquist pulse and matched filtering is achieved since

*s*(

*n*) is used both at the transmitter and receiver. A final constraint ensures that the filter impulse response is symmetric:

21. M. J. D. Powell, “A fast algorithm for nonlinearly constrained optimization calculations,” Lect. Notes Math. **630**, 144–157 (1978). [CrossRef]

*M*of 2048 symbol periods. It can be seen that the first-null bandwidth of the optimized responses is exactly the same as of the ideal response, but that as the length of the impulse response is reduced, the amount of energy in the higher part of the passband (from 0.65 Hz to 0.75 Hz) is also reduced. Figure 11 also shows the frequency response of the proposed pulse using a rectangular window and

*M*= 64. In this case, the side-lobe attenuation is reduced to only 24 dB, while a side-lobe attenuation of at least 40 dB is achieved by all the optimized truncated versions of the pulse, thereby showing the effectiveness of the proposed truncation procedure.

*α*= 0.5,

*T*= 1,

*N*= 16 and

*M*= 64. To produce these eye diagrams, 2

^{18}symbols were used and no noise was added to the signal. It can be seen in Fig. 12(a) that truncating the RRC pulse response to 64 symbol periods with a rectangular window results in a 16-QAM signal with zero ISI. On the contrary, when using a rectangular window and truncating the proposed pulse response to 64 symbol periods, Fig. 12(b) shows that the level of ISI is increased, since the vertical eye opening is reduced by 20%, from 2.0 to 1.6. When using the truncation procedure described above for the proposed pulse and

*M*= 64, Fig. 13(c) demonstrates that the vertical eye opening is optimal and that the proposed pulse obtained by numerical optimization meets the zero ISI constraint.

*M*= 16 the reach increase is only 3.7%, while it is 7.6% and 8.9% when

*M*= 32 and

*M*= 64, respectively. This variation of performance can be explained by comparing the frequency responses of the truncated pulses in Fig. 11, that show less energy content in the high frequencies, and therefore less nonlinearity tolerance for reduced pulse lengths. For a roll-off factor of 1.0, the effect of truncation is less important since the truncation to 16 symbol periods does not bring significant penalties compared to pulse lengths of 32 and 64 symbol periods. For a roll-off factor of 1.5, the proposed pulse truncation method has no effect on system performance. Since the RRC pulse has a smoother transition band and a faster decay rate, its truncation does not lead to any performance variations, for the considered roll-off factor values and impulse response lengths.

### 5.4 Effect of DAC resolution

### 5.5 Performance under timing jitter

^{15}, followed by a 16-QAM mapper and by a FIR pulse shaping filter. The channel is modeled by an additive white gaussian noise (AWGN) channel. At the receiver, a sampler acquires the signal. In practical systems, the sampler function is realized by ADCs. Timing detection is accomplished by Gardner's timing error detector (TED) [22

22. F. M. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. **34**(5), 423–429 (1986). [CrossRef]

*T*/1000 is chosen and the length of the FIR pulse shaper, matched filter and prefilter is set to 64 symbol periods. The oversampling rate of the matched filter, prefilter, TED and loop filter is set to

*N*= 2. The model was implemented in Simulink (R2010a).

23. A. N. D’Andrea and M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. **41**(9), 1296–1299 (1993). [CrossRef]

*g*(

*t*) corresponds to the convolution of the channel impulse response

*p*(

*t*) and the prefilter

*h*(

_{p}*t*). It is shown in [23

23. A. N. D’Andrea and M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. **41**(9), 1296–1299 (1993). [CrossRef]

*P*(

*f*) by ± 1/

*T*. In the time domain, this corresponds to:

*α*= 1 are shown in Figs. 15(a) and 15(d), respectively. Some level of ISI is present, since the signal at the output of a root-Nyquist pulse shaping filter is not necessarily ISI free. Figures 15(b) and 15(e) show the eye diagrams obtained after the receiver matched filter for the RRC and proposed pulse, for a channel without noise. As expected, both signals are free from any ISI. Figures 15(c) and 15(f) show the eye diagrams at the output of the prefilter for a sequence filtered by the RRC and the proposed pulse. As it can be seen, the output of the prefilter has regular

*T*-spaced zero-crossings.

24. A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. **42**(234), 1391–1399 (1994). [CrossRef]

*B*is the noise loop bandwidth, E

_{L}_{b}/N

_{0}is the energy per bit and

*M*is the order of the QAM modulation. Figures 16(a) and 16(b) show the normalized jitter variance against the E

_{QAM}_{b}/N

_{0}, for roll-off factors of 0.5 and 1.0. The MCRB of the proposed pulse is lower than the MCRB of the RRC pulse, suggesting that the former has better synchronization properties. Figures 16(a) and 16(b) confirm this expectation, and show that when the input of the TED is the matched filter (i.e. no prefilter is used), a lower jitter floor is reached. For a roll-off factor of 0.5, the minimum jitter variance is 5.5 × 10

^{−6}and 1.8 × 10

^{−5}for the proposed and RRC pulses, respectively. For a roll-off factor of 1, the minimum jitter variance is 3.5 × 10

^{−7}and 3.5 × 10

^{−6}for the proposed and RRC pulses, respectively. When using a prefilter, Figs. 16(a) and 16(b) show that the effect of self-noise is practically removed and that the proposed pulse has better jitter tolerance. For a jitter variance of 10

^{−5}, performance gains of 4.0 dB and 3.9 dB are obtained for roll-off factors of 0.5 and 1.0, respectively.

25. X. Zhou, X. Chen, W. Zhou, Y. Fan, H. Zhu, and Z. Li, “All-digital timing recovery and adaptive equalization for 112 Gbit/s POLMUX-NRZ-DQPSK optical coherent receivers,” J. Opt. Commun. Netw. **2**(11), 984–990 (2010). [CrossRef]

## 6. Experimental results

^{14}, but two symbol pattern periods, totaling 2

^{15}symbols for each polarization, were used to produce BER statistics. Prior to launching the signal in the fiber, a polarization scrambler (PS) was added to continuously vary the polarization state. The optical link consisted of

*L*G.652 fiber spans of 80 km. Erbium doped fiber amplifiers (EDFAs) were used to amplify the signal. At the receiver a broadband noise source was inserted in order to adjust the OSNR and an optical spectrum analyzer (OSA) was used to determine the OSNR. Prior to demodulation, a 50 GHz Demux was inserted. The receiver architecture is also similar to the one described by Fig. 4. The received signal first went through a coherent front end. An 8-bit high speed real-time oscilloscope operated at 50 GSamples/s and having a 3 dB bandwidth of 16 GHz was then used to digitize the signal. The same clock source was driving the DACs and the oscilloscope. Off-line processing of the remaining digital signal processing functions was done on a personal computer. A quadrature-imbalance compensation (QIC) circuit [26

26. A. Leven, N. Kaneda, and S. Corteselli, “Real-time implementation of digital signal processing for coherent optical digital communication systems,” IEEE J. Sel. Top. Quantum Eelectron. **16**(5), 1227–1234 (2010). [CrossRef]

*T*/2 fractional equalizer (EQ) was then used to recover the signal from the two polarizations. For the DP-QPSK experiment, a feedforward Viterbi-Viterbi [27

27. A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with applications to burst digital transmission,” IEEE Trans. Inf. Theory **29**(4), 543–551 (1983). [CrossRef]

28. D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. **28**(11), 1867–1875 (1980). [CrossRef]

*T*/2 equalizer, while carrier recovery was achieved by a second order PLL. The dashed lines in Fig. 17 between the decision circuit, the CR and the

*T*/2 EQ represent the feedback paths for the DP-16-QAM implementation.

*α*= 1, as transmit pulse shaping and receive matched filters. Note that the proposed pulse response was in this case obtained by the optimization problem defined in [13], with

*M*= 16. For a roll-off factor of 1, the formulation in [13] practically gives the same result as the formulation expressed by Eq. (10), Eq. (11), and Eq. (12).

^{−3}and for a launch power of –2 dBm, an increased tolerance to fiber nonlinearity of 1.2 dB is provided by the proposed pulse. At this power level, simulation results indicate a 0.9 dB of added tolerance to fiber nonlinearity. At 1200 km and for a launch power of 0 dBm, Fig. 19(c) shows that the system with the proposed pulse almost reaches the BER threshold, while the system using RRC pulses is limited to a BER of 1.8 × 10

^{−2}. At this BER level, the optimized pulse outperforms the RRC pulse by 4.3 dB. Figures 19(a), 19(b) and 19(c) show good general agreement between the experimental results and the simulation model. It should be noted that the introduction of soft-decision forward error correction (FEC) [29] would be required to be able to operate at a FEC threshold of 8 × 10

^{−3}.

## 7. Conclusion

## Acknowledgments

## References and links

1. | E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. |

2. | S. Makovejs, E. Torrengo, D. S. Millar, R. I. Killey, S. J. Savory, and P. Bayvel, “Comparison of pulse shapes in a 224 Gbit/s (28 Gbaud) PDM-QAM16 long-haul transmission experiment,” in |

3. | C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express |

4. | M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256-QAM (64 Gb/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” IEEE Photon. Technol. Lett. |

5. | X. Zhou, L. Nelson, P. Magill, R. Isaac, B. Zhu, D. W. Peckham, P. Borel, and K. Carlson, “8x450-Gb/s, 50-GHz-spaced, PDM-32QAM transmission over 400km and one 50GHz-grid ROADM,” in |

6. | R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, B. Nebendahl, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse modulation transmitter generating rectangular shaped spectra of 112 Gbit/s 16QAM signals,” in |

7. | K. Roberts, A. Borowiec, and C. Laperle, “Technologies for optical systems beyond 100G,” Opt. Fiber Technol. |

8. | B. Farhang-Boroujeny, “A square-root Nyquist (M) filter design for digital communication systems,” IEEE Trans. Signal Process. |

9. | P. S. Rha and S. Hsu, “Peak-to-average ratio (PAR) reduction by pulse shaping using a new family of generalized raised cosine filters,” in |

10. | B. Farhang-Boroujeny and G. Mathew, “Nyquist filters with robust performance against timing jitter,” IEEE Trans. Signal Process. |

11. | A. Assalini and A. M. Tonello, “Improved Nyquist pulses,” IEEE Commun. Lett. |

12. | K. Roberts, M. O’Sullivan, K.-T. Wu, H. Sun, A. Awadalla, D. J. Krause, and C. Laperle, “Performance of dual-polarization QPSK for optical transport systems,” J. Lightwave Technol. |

13. | B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. |

14. | B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “Optimized pulse shaping for intra-channel nonlinearities mitigation in a 10 Gbaud dual-polarization 16-QAM system,” in |

15. | H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans. |

16. | J. G. Proakis, |

17. | F. Zhang, “XPM Statistics in 100% Precompensated WDM Transmission for OOK and DPSK Formats,” IEEE Photon. Technol. Lett. |

18. | R. J. Essiambre and P. J. Winzer, “Impact of fiber nonlinearities on advanced modulation formats using electronic pre-distortion,” in |

19. | C. Behrens, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Nonlinear Distortion in Transmission of Higher Order Modulation Formats,” IEEE Photon. Technol. Lett. |

20. | Y. Jiang, X. Tang, J. C. Cartledge, and K. Roberts, “Electronic Pre-Compensation of Narrow Optical Filtering for OOK, DPSK and DQPSK Modulation Formats,” J. Lightwave Technol. |

21. | M. J. D. Powell, “A fast algorithm for nonlinearly constrained optimization calculations,” Lect. Notes Math. |

22. | F. M. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. |

23. | A. N. D’Andrea and M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. |

24. | A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. |

25. | X. Zhou, X. Chen, W. Zhou, Y. Fan, H. Zhu, and Z. Li, “All-digital timing recovery and adaptive equalization for 112 Gbit/s POLMUX-NRZ-DQPSK optical coherent receivers,” J. Opt. Commun. Netw. |

26. | A. Leven, N. Kaneda, and S. Corteselli, “Real-time implementation of digital signal processing for coherent optical digital communication systems,” IEEE J. Sel. Top. Quantum Eelectron. |

27. | A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with applications to burst digital transmission,” IEEE Trans. Inf. Theory |

28. | D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. |

29. | K. Onohara, Y. Miyata, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuochi, “Soft decision FEC for 100G transport systems,” in |

**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: November 9, 2011

Revised Manuscript: December 20, 2011

Manuscript Accepted: December 23, 2011

Published: March 27, 2012

**Citation**

Benoît Châtelain, Charles Laperle, Kim Roberts, Mathieu Chagnon, Xian Xu, Andrzej Borowiec, François Gagnon, and David. V. Plant, "A family of Nyquist pulses for coherent optical communications," Opt. Express **20**, 8397-8416 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8397

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