Coherent detection in optical fiber

Coherent detection in optical fiber systems

Ezra Ip*, Alan Pak Tao Lau, Daniel J. F. Barros, Joseph M. Kahn

Stanford University, 366 Packard Building, 350 Serra Mall, Stanford, CA 94305-9515, USA.

*Corresponding Author: wavelet@https://www.360docs.net/doc/8d14556680.html,

Abstract: The drive for higher performance in optical fiber systems has

renewed interest in coherent detection. We review detection methods,

including noncoherent, differentially coherent, and coherent detection, as

well as a hybrid method. We compare modulation methods encoding

information in various degrees of freedom (DOF). Polarization-multiplexed

quadrature-amplitude modulation maximizes spectral efficiency and power

efficiency, by utilizing all four available DOF, the two field quadratures in

the two polarizations. Dual-polarization homodyne or heterodyne

downconversion are linear processes that can fully recover the received

signal field in these four DOF. When downconverted signals are sampled at

the Nyquist rate, compensation of transmission impairments can be

performed using digital signal processing (DSP). Linear impairments,

including chromatic dispersion and polarization-mode dispersion, can be

compensated quasi-exactly using finite impulse response filters. Some

nonlinear impairments, such as intra-channel four-wave mixing and

nonlinear phase noise, can be compensated partially. Carrier phase recovery

can be performed using feedforward methods, even when phase-locked

loops may fail due to delay constraints. DSP-based compensation enables a

receiver to adapt to time-varying impairments, and facilitates use of

advanced forward-error-correction codes. We discuss both single- and

multi-carrier system implementations. For a given modulation format, using

coherent detection, they offer fundamentally the same spectral efficiency

and power efficiency, but may differ in practice, because of different

impairments and implementation details. With anticipated advances in

analog-to-digital converters and integrated circuit technology, DSP-based

coherent receivers at bit rates up to 100 Gbit/s should become practical

within the next few years.

?2008 Optical Society of America

OCIS codes: (060.0060) Fiber optics and optical communications; (060.1660) Coherent

communications; (060.2920) Homodyning; (060.4080) Modulation; (060.5060) Phase

modulation; (060.2840) Heterodyne.

References and links

1. C.E. Shannon, “A mathematical theory of communication,” Bell. Syst. Tech. J. 27, 379?423 (1948).

2.J. M. Geist, “Capacity and cutoff rate for dense M-ary PSK constellations,” in MILCOM 1990, (Monterey,

CA, USA, 1990), pp. 768?770.

3.K.-P. Ho, “Exact evaluation of the capacity for intensity-modulated direct-detection channels with optical

amplifier noises,” IEEE Photon. Technol. Lett. 17, 858?860 (2005).

4.P.P.Mitra and J.B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,”

Nature 411, 1027–1030 (2001).

5.J. M. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM

Systems,” ,” J. Sel. Top. Quantum Electron. 10, 259–271 (2004).

6.V. Jungnickel, A. Forck, T. Haustein, S. Schiffermuller, C. Helmolt, F. Luhn, M. Pollock, C. Juchems, M.

Lampe, S. Eichnger, W. Zirwas, E. Schulz, “1 Gbit/s MIMO-OFDM transmission experiments,” in

Proceedings of IEEE Conference on Vehicular Technol. (Institute of Electrical and Electronics Engineers, Dallas, 2005), pp. 861–866.

7.K. Kikuchi, “Coherent detection of phase-shift keying signals using digital carrier-phase estimation,” in

Proceedings of IEEE Conference on Optical Fiber Communications, (Institute of Electrical and Electronics Engineers, Anaheim, 2006), Paper OTuI4.

8.T. Pfau, S. Hoffmann, R. Peveling, S. Bhandard, S. Ibrahim, O. Adamczyk, M. Porrmann, R. Noé and Y.

Achiam, “First real-time data recovery for synchronous QPSK transmission with standard DFB lasers,”

IEEE Photon. Technol. Lett. 18, 1907–1909 (2006).

9. A. Leven, N. Kaneda, U.-V. Koc and Y.-K. Chen, “Coherent receivers for practical optical communication

systems,” in Proceedings of IEEE Conference on Optical Fiber Communications, (Institute of Electrical

and Electronics Engineers, Anaheim, 2007), Paper OThK4.

10.S. J. Savory, G. Gavioli, R. I. Killey, P. Bayvel, “Transmission of 42.8 Gbit/s polarization multiplexed

NRZ-QPSK over 6400 km of standard fiber with no optical dispersion compensation,” in Proceedings of

IEEE Conference on Optical Fiber Communications, (Institute of Electrical and Electronics Engineers,

Anaheim, 2007), Paper OTuA1.

11.K. Sekine, N. Kikuchi, S. Sasaki, S. Hayase, C. Hasegawa and T. Sugawara, “40 Gbit/s, 16-ary (4

bit/symbol) optical modulation/demodulation scheme,” Electron. Lett. 41, 430–432, (2005).

12.J. Hongo, K. Kasai, M. Yoshida and M. Nakazawa, “1-Gsymbol/s 64-QAM coherent optical transmission

over 150 km,” IEEE Photon. Technol. Lett. 19, 638–640 (2007).

13.T. Foggi, E. Forestieri, G. Colavolpe and G. Prati, “Maximum-likelihood sequence detection with closed-

form metrics in OOK optical systems impaired by GVD and PMD,” J. Lightwave Technol. 24, 3073–3087 (2006).

14.M. Nazarathy and E. Simony, “Multichip differential phase encoded optical transmission,” IEEE Photon.

Technol. Lett. 17, 1133–1135 (2005).

15. D. Divsalar and M. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun. 38,

300–308 (1990).

16.S. Benedetto and P. Poggiolini, “Theory of polarization shift keying modulation,” IEEE Trans. Commun.

40, 708?721 (1992).

17.S. Betti, F. Curti, G. de Marchis and E. Iannone, “Multilevel coherent optical system based on Stokes

parameters modulation,” J. Lightwave Technol. 8, 1127?1136 (1990).

18. E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave

Technol., 25, 2675-2692 (2007).

19. E. Ip and J.M. Kahn, “Digital equalization of chromatic dispersion and polarization mode dispersion,” J.

Lightwave Technol. 25, 2033-2043 (2007).

20. E. Ip and J.M. Kahn, “Carrier synchronization for 3- and 4-bit-per-symbol optical transmission,” J.

Lightwave Technol. 23, 4110–4124 (2005).

21.S. Tsukamoto, K. Katoh and K. Kikuchi, “Coherent demodulation of optical multilevel phase-shift-keying

signals using homodyne detection and digital signal processing,” IEEE Photon. Technol. Lett. 18, 1131–

1133 (2006).

22.G. P. Agrawal, Fiber-Optic Communiation Systems, 3rd ed. (Wiley, New York, 2002).

23.J. R. Barry and J.M. Kahn, “Carrier synchronization for homodyne and heterodyne detection of optical

quadriphase-shift keying,” J. Lightwave Technol. 10, 1939–1951 (1992).

24.R. Noé, “Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with

feedforward carrier recovery,” J. Lightwave Technol. 23, 802?808 (2005).

25.J. Rebola and A. Cartaxo, “Optimization of level spacing in quaternary optical communication systems,”

Proc. SPIE 4087, 49?59 (2000).

26.J. J. Bussgang and M. Leiter, “Error rate approximations for differential phase-shift keying.” IEEE Trans.

Commun. Systems 12, 18–27 (1964).

27.J. G. Proakis, “Probabilities of error for adaptive reception of M-phase signals,” IEEE Trans. Commun.

Tech. 16, 71–81 (1968).

28.S. Benedetto and P. Poggiolini, “Multilevel polarization shift keying: optimum receiver structure and

performance evaluation,” IEEE Trans. Commun. 42, 1174–1186 (1994).

29.J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, New York, 2001).

30.M. Suzuki and N. Edagawa, “Dispersion-managed high-capacity ultra-long-haul transmission,” J.

Lightwave Technol. 21, 916?929 (2003).

31. E. Forestieri and G. Prati, “Exact analytical evaluation of second-order PMD impact on the outage

probability for a compensated system,” J. Lightwave Technol. 22, 988?996 (2004).

32. C. D. Poole, R. W. Tkach, A. R. Chraplyvy and D. A.Fishman, “Fading in lightwave systems due to

polarization-mode dispersion,” IEEE Photon. Technol. Lett. 3, 68–70 (1991).

33.H. Bülow, W. Baumert, H. Schmuck, F. Mohr, T. Schulz, F. Küppers and W. Weiershausen, “Measurement

of the maximum speed of PMD fluctuation in installed field fiber,” in Proceedings of IEEE Conference on Optical Fiber Communications, (Institute of Electrical and Electronics Engineers, San Diego, 1999), Paper OWE4.

34. C. D. Poole, “Statistical treatment of polarization dispersion in single-mode fiber,” Opt. Lett. 13, 687–689

(1988).

35.N. Gisin, J.-P. Von der Weid and J.-P. Pellaux, “Polarization mode dispersion of short and long single-

mode fibers,” J. Lightwave Technol. 9, 821–827 (1991).

36.G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single-mode fibers,” J.

Lightwave Technol. 9, 1439–1456 (1991).

37.H. Bülow, “System outage probability due to first- and second-order PMD,” IEEE Photon. Technol. Lett.

10, 696–698 (1998).

38.H. Sunnerud, C. Xie, M. Karlsson, R. Samuelsson and P. Andrekson, “A comparison between different

PMD compensation techniques,” J. Lightwave Technol. 20, 368–378 (2002).

39. F. Buchali and H. Bülow, “Adaptive PMD compensation by electrical and optical techniques,” J.

Lightwave Technol. 22, 1116–1126 (2004).

40.R. Noé, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Sch?pflin, C. Flingener, E. Gottwald, C.

Scheerer, G. Fischer, T. Weyrauch and W. Haase, “Polarization mode dispersion compensation at 10, 20,

and 40 Gb/s with various optical equalizer,” J. Lightwave Technol. 17, 1602–1616 (1999).

41.S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner and J. Feinberg, “Adjustable

compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg

grating,” IEEE Photon. Technol. Lett. 11, 1277–1279 (1999).

42.T. Saida, K. Takiguchi, S. Kuwahara, Y. Kisaka, Y. Miyamoto, Y. Hashizume, T. Shibata and K. Okamoto,

“Planar lightwave circuit polarization-mode dispersion compensator,” IEEE Photon. Technol. Lett. 14,

507–509 (2002).

43.J. Wang and J. M. Kahn, “Performance of electrical equalizers in optical amplified OOK and DPSK

systems,” IEEE Photon. Technol. Lett. 16, 1397–1399 (2004).

44. C. Vinegoni, M. Karlsson, M. Petersson and H. Sunnerud, “The statistics of polarization-dependent loss in

a recirculating loop,” J. Lightwave Technol. 22, 968?976 (2004).

45. A. H. Gnauck, P. J. Winzer and S. Chandrasekhar, “Hybrid 10/40-G transmission on a 50-GHz Grid

through 2800 km of SSMF and seven optical add-drops,” IEEE Photon. Technol. Lett. 17, 2203?2205

(2005).

46.G. Goldfarb and G. Li, “Chromatic dispersion compensation using digital IIR filtering with coherent

detection,” IEEE Photon. Technol. Lett. 19, 969?971 (2007).

47.N. Amitay and J. Salz, “Linear Equalization Theory in Digital Data Transmission over Dually Polarized

Fading Radio Channels,” Bell. Syst. Tech. J. 63, 2215–2259 (1984).

48.J. Salz, “Digital transmission over cross-coupled linear channels,” AT&T Tech. J. 64, 1147–1159 (1985).

49.H. Meyr, M. Moeneclaey and S. Fechtel, Digital Communication Receivers. (John Wiley, New York,

1997).

50.R.D. Gitlin and S. B. Weinstein, “Fractionally spaced equalization: an improved digital transversal

equalizer,” Bell. Syst. Tech. J. 60, 275–296 (1981).

51.G. Ungerboeck, “Fractional tap-spacing equalizer and consequences for clock recovery in data modems,”

IEEE Trans. Commun. 24, 856–864 (1976).

52.S. Qureshi, “Adaptive equalization,” Proceedings of the IEEE 73, 1349–1387 (1985).

53. B. Widrow and S. D. Stearns, Adaptive Signal Processing, (Prentice Hall, Englewood Cliffs, NJ, 1985).

54. A. Oppenheim and R. Schafer, Discrete-Time Signal Processing, (Prentice Hall, Englewood Cliffs, NJ,

1989).

55. A. Mecozzi, C. B. Clausen and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed

optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392?394 (2000).

56.I. Shake, H. Takara, K. Mori, S. Kawanishi and Y. Yamabayashi, “Influence of inter-bit four-wave mixing

in optical TDM transmission,” Electron. Lett. 34, 1600?1601 (1998).

57.R.-J. Essiambre, B. Mikkelsen and G. Raybon, “Intra-channel cross-phase modulation and four-wave

mixing in high-speed TDM systems,” Electron. Lett. 35, 1576?1578 (1999).

58. A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park and A. H. Gnauck, “Cancellation of timing and amplitude

jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett. 13, 445?447 (2001).

59. A. Striegler and B. Schmauss, “Compensation of intrachannel effects in symmetric dispersion-managed

transmission systems,” J. Lightwave Technol. 22, 1877?1882 (2004).

60.N. Alic and Y. Fainman, “Data-dependent phase coding for suppression of ghost pulses in optical fibers,”

IEEE Photon. Technol. Lett. 16, 1212?1214 (2004).

61.I.B. Djordjevic and B. Vasic, “Constrained coding techniques for suppression of intrachannel nonlinear

effects in high-speed optical transmission,” J. Lightwave Technol. 24, 411?419 (2006).

62.X. Wei and X. Liu, “Analysis of intrachannel four-wave mixing in differential phase-shift keying

transmission with large dispersion,” Opt. Lett.28, 2300?2302 (2003).

63. A. P. T. Lau, S. Rabbani and J. M. Kahn are preparing a manuscript to be called “On the statistics of intra-

channel four-wave-mixing induced phase noise in phase modulated systems.”

64.J. P. Gordon and L.F. Mollenauer, “Phase noise in photonic communications systems using linear

amplifiers,” Opt. Lett. 15, 1351?1353 (1990).

65.K.-P. Ho, Phase-Modulated Optical Communication Systems, (Springer, New York, 2005).

66.K.-P. Ho, “Statistical properties of nonlinear phase noise,” in Advances in Optics and Laser Research3,

(Nova Science Publishers, New York, 2003).

67. A. P. T. Lau and J. M. Kahn, “Design of inline amplifiers gain and spacing to minimize phase noise in

optical transmission systems,” J. Lightwave Technol. 24, 1334?1341 (2006).

68. A.P.T. Lau and J.M. Kahn, “Power profile optimization in phase-modulated systems in presence of

nonlinear phase noise," IEEE Photon. Technol. Lett. 18, 2514?2516 (2006).

69.K.-P. Ho and J. M. Kahn, “Detection technique to mitigate Kerr effect phase noise,” J. Lightwave Technol.

22, 779?783 (2004).

70. D.-S. Ly-Gagnon and K. Kikuchi, “Cancellation of nonlinear phase noise in DPSK transmission,” 2004

Optoelectronics and Communications Conference and International Conference on Optical Internet

(OECC/COIN2004), paper 14C3-3 (2004).

71.X.Liu, X. Wei, R. E. Slusher and C. J. McKinstrie, “Improving transmission performance in differential

phase-shift-keyed systems by use of lumped nonlinear phase-shift compensation," Opt. Lett. 27, 1616?1618 (2002).

72.K. Kikuchi, M. Fukase and S.-Y. Kim, “Electronic post-compensation for nonlinear phase noise in a 1000-

km 20-Gbit/s optical QPSK transmission system using the homodyne receiver with digital signal

processing,” in Proceedings of IEEE Conference on Optical Fiber Communications, (Institute of Electrical and Electronics Engineers, Anaheim, 2007), Paper OTuA2.

73.G. Charlet, N. Maaref, J. Renaudier, H. Mardoyan, P. Tran and S. Bigo, “Transmission of 40 Gb/s QPSK

with coherent detection over ultra-long distance improved by nonlinearity mitigation,” in Proceedings

ECOC 2006, Cannes, France, 2006, Postdeadline paper Th4.3.4.

74.G. Zhu, L. Mollenauer and C. Xu, “Experimental demonstration of post-nonlinearity compensation in a

multispan DPSK transmission,” IEEE Photon. Technol. Lett. 18, 1007?1009 (2006).

75.K.P. Ho, “Mid-span compensation of nonlinear phase noise,” Opt. Comm. 245, 391?398 (2005).

76. A. P. T. Lau and J. M. Kahn, “Signal design and detection in presence of nonlinear phase noise,” J.

Lightwave Technol. 25, 3008?3016 (2007).

77. A.G. Green, P.P. Mitra, L.G. L. Wegener, “Effect of chromatic dispersion on nonlinear phase noise,” Opt.

Lett. 28, 2455?2457 (2003).

78.S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30,

3278?3280 (2005).

79.K.-P. Ho and H.C. Wang, “On the effect of dispersion on nonlinear phase noise,” Opt. Lett. 31, 2109?2111

(2006).

80.S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear

optical transmission systems,” Opt. Express 15, 2166?2177 (2007).

81. D. Boivin, G.-K. Chang, J. R. Barry and M. Hanna, “Reduction of Gordon-Mollenauer phase noise in

dispersion-managed systems using in-line spectral inversion,” J. Opt. Soc. Am. A. B 23, 2019?2023 (2006).

82.P. Serena, A. Orlandini and A. Bononi, “Parametric-Gain approach to the analysis of single-channel

DPSK/DQPSK systems with nonlinear phase noise,” J. Lightwave Technol. 24, 2026?2037 (2006).

83.K.P. Ho and H.C. Wang, “Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-

DPSK signals in dispersive transmission systems,” IEEE Photon. Technol. Lett. 17, 1426?1428 (2005).

84. F. Zhang, C. A. Bunge and K. Petermann, “Analysis of nonlinear phase noise in single-channel return-to-

zero differential phase-shift keying transmission systems,” Opt. Lett. 31, 1038?1040 (2006).

85. F. Zhang, C. A. Bunge, K. Petermann and A. Richter, “Optimum dispersion mapping of single-channel 40

Gbit/s return-to-zero differential phase-shift keying transmission systems,” Optics Express 14, 6613?6618 (2006).

86.X. Zhu, S. Kumar and X. Li, “Analysis and comparison of impairments in differential phase-shift keying

and on-off keying transmission systems based on the error probability,” Appl. Opt. 45, 6812?6822 (2006).

87. C. Henry, “Theory of the phase noise and power spectrum of a single mode injection laser,” J. Quantum

Electron. 19, 1391–1397 (1983).

88.M. Tur, B. Moslehi and J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay

lines,” J. Lightwave Technol. 3, 20–31 (1985).

89. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949, (1958).

90. F.M. Gardner, Phaselock Techniques, 3rd ed. (John Wiley, Hoboken, NJ, 2005).

91.M. A. Grant, W. C. Michie, M. J. Fletcher, “The performance of optical phase-locked loops in the presence

of nonnegligible loop propagation delay,” J. Lightwave Technol. 5, 592–597 (1987).

92.K. Kikuchi, “Phase-diversity homodyne detection of multilevel optical modulation with digital carrier

phase estimation,” J. Sel. Top. Quantum Electron. 12, 563?570 (2006).

93.M. G. Taylor, “Accurate digital phase estimation process for coherent detection using a parallel digital

processor,” in Proceedings ECOC 2005, Glasgow, UK, 2005, Paper Tu4.2.6.

94. D.-S. Ly-Gagnon, S. Tsukamoto, K. Katoh and K. Kikuchi, “Coherent detection of optical quadrature

phase-shift keying signals with coherent phase estimation,” J. Lightwave Technol. 24, 12–21, (2006).

95. F. J. Foschini, R. D. Gitlin and S. B. Weinstein, “On the selction of a two-dimensional signal constellation

in the presence of phase jitter and Gaussian noise,” Bell. Syst. Tech. J. 52, 927–965 (1973).

96. A. Bahai, B. Saltzberg and M. Ergen, Multi-carrier Digital Communications: Theory and Applications of

OFDM, 2nd Ed. (Springer, New York, 2004).

97.R. Prasad, “OFDM for wireless communications systems,” (Artech House Publishers, Boston, 2004).

98.W. Shieh, X. Yi, and Y. Tang, “Experimental demonstration of transmission of coherent optical OFDM

Systems,” in Proceedings of IEEE Conference on Optical Fiber Communications, (Institute of Electrical

and Electronics Engineers, Anaheim, 2007), Paper OMP2.

99.W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett.

42, 587?589 (2006).

100.N. Cvijetic, L. Xu and T. Wang, “Adaptive PMD compensation using OFDM in long-haul 10 Gb/s DWDM systems,” in Proceedings of IEEE Conference on Optical Fiber Communications, (Institute of Electrical

and Electronics Engineers, Anaheim, 2007), Paper OTuA5.

101.A. Lowery and J. Armstrong, “Orthogonal-frequency-division multiplexing for optical dispersion compensation,” in Proceedings of IEEE Conference on Optical Fiber Communications, (Institute of

Electrical and Electronics Engineers, Anaheim, 2007), Paper OTuA4.

102.W. Henkel, G. Taubock, P. Odling, P. O. Borjesson and N. Petersson, “The cyclic prefix of OFDM/DMT – an analysis,” IEEE International Seminar on Broadband Communications, (Institute of Electrical and

Electronic Engineers, Zurich, 2002).

103.D. J.F. Barros and J. M. Kahn are preparing a manuscript to be called “Optimized dispersion compensation using orthogonal frequency-division multiplexing.”

104.W. Shieh, X. Yi, Y. Ma and Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Optics Express 15, 9936?9947 (2007).

105.C. Cover and J. Thomas, Elements of Information Theory. (John Wiley, New York, 1991).

106.C. Y. Wong, R. S. Cheng, K. B. Letaief and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” J. Sel. Top. Commun. 17, 1747?1758 (1999).

107.B. S. Krongold, K. Ramchandran and D. L. Jones, “Computationally efficient optimal power allocation algorithms for multicarrier communication systems,” IEEE Trans. Commun. 48, 23?27 (2000).

108.J. Jang, K. B. Lee and Y.-H. Lee, “Transmit power and bit allocations for OFDM systems in a fading channel,” in Proceedings of IEEE GLOBECOM, (Institute of Electrical and Electronics Engineers, San

Francisco, 2003), pp. 858?862.

109.S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,”

IEEE Trans. Commun. 52, 1988?1996 (2004).

110.A. G. Armada and M. Calvo, “Phase noise and sub-carrier spacing effects on the performance of an OFDM communication system,” IEEE Commun. Lett. 2, 11?13 (1998).

111.S. Wu and Y. Bar-Ness, “A phase noise suppression algorithm for OFDM based WLANs,” IEEE Commun.

Lett. 6, 535?537 (2002).

112.H. Ochiai and H. Imai, “On the distribution of the peak-to-average power ratio in OFDM signals,” IEEE Trans. Commun. 49, 282?289 (2001).

113.A. J. Lowery, “Fiber nonlinearity mitigation in optical links that use OFDM for dispersion compensation,”

IEEE Photon Technol. Lett. 19, 1556?1558 (2007).

114.A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,”

Optics Express 15, 12965?12970 (2007).

115.A. J. Lowery, S. Wang and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Optics Express 15, 13282?13287 (2007).

116.D.-S. Ly-Gagnon, “Information recovery using coherent detection and digital signal pocessing for phase-shift-keying modulation formats in optical communication systems,” M.S. Thesis, University of Tokyo

(2004).

1. Introduction

An important goal of a long-haul optical fiber system is to transmit the highest data throughput over the longest distance without signal regeneration. Given constraints on the bandwidth imposed by optical amplifiers and ultimately by the fiber itself, it is important to maximize spectral efficiency, measured in bit/s/Hz. But given constraints on signal power imposed by fiber nonlinearity, it is also important to maximize power (or SNR) efficiency, i.e., to minimize the required average transmitted energy per bit (or the required signal-to-noise ratio per bit). Most current systems use binary modulation formats, such as on-off keying or differential phase-shift keying, which encode one bit per symbol. Given practical constraints on filters for dense wavelength-division multiplexing (DWDM), these are able to achieve spectral efficiencies of 0.8 bit/s/Hz per polarization. Spectral efficiency limits for various detection and modulation methods have been studied in the linear [1?3] and nonlinear regimes [4,5]. Noncoherent detection and differentially coherent detection offer good power efficiency only at low spectral efficiency, because they limit the degrees of freedom available for encoding of information [5].

The most promising detection technique for achieving high spectral efficiency while maximizing power (or SNR) efficiency, is coherent detection with polarization multiplexing, as symbol decisions are made using the in-phase (I) and quadrature (Q) signals in the two field polarizations, allowing information to be encoded in all the available degrees of freedom. When the outputs of an optoelectronic downconverter are sampled at Nyquist rate, the digitized waveform retains full information of the electric field, which enables compensation of transmission impairments by digital signal processing (DSP). A DSP-based receiver is highly advantageous because adaptive algorithms can be used to compensate time-varying transmission impairments. Advanced forward error-correction coding can also be implemented. Moreover, digitized signals can be delayed, split and amplified without degradation in signal quality. DSP-based receivers are ubiquitous in wireless and digital subscriber line (DSL) systems at lower data rates. In such systems, computationally intensive techniques have been demonstrated, such as orthogonal frequency-division multiplexing (OFDM) with multiple-input-multiple-output (MIMO) transmission in a real-time 1 Gbit/s wireless link [6]. Continued hardware improvements will enable deployment of DSP-based coherent optical systems in the next few years.

Experimental results in coherent optical communication have been promising. Kikuchi demonstrated polarization-multiplexed 4-ary quadrature-amplitude modulation (4-QAM) transmission at 40 Gbit/s with a channel bandwidth of 16 GHz (2.5 bit/s/Hz) [7]. This experiment used a high-speed sampling oscilloscope to record the output of a homodyne downconverter. DSP was performed offline because of the unavailability of sufficiently fast processing hardware. The first demonstration of real-time coherent detection occurred in 2006, when an 800 Mbit/s 4-QAM signal was coherently detected using a receiver with 5-bit analog-to-digital converters (ADC) followed by a field programmable gate array [8]. In 2007, feedforward carrier recovery was demonstrated in real time for 4-QAM at 4.4 Gbit/s [9]. Savory showed the feasibility of digitally compensating the chromatic dispersion (CD) in 6,400 km of SMF without inline dispersion compensating fiber (DCF), with only 1.2 dB OSNR penalty at 42.8 Gbit/s [10]. Coherent detection of large QAM constellations has also been demonstrated. For example, 16-ary transmission at 40 Gbit/s using an amplitude-phase-shift keying (APSK) format was shown by Sekine et al [11]. In 2007, Hongo et al demonstrated 64-QAM transmission over 150 km of dispersion-shifted fiber [12].

This paper provides an overview of detection and modulation methods, with emphasis on coherent detection and digital compensation of channel impairments. The paper is organized as follows: in Section 3, we review signal detection methods, including noncoherent, differentially coherent and coherent detection. We compare these techniques and the modulation formats they enable. In Section 4, we compare the bit-error ratio (BER) performance of different modulation formats in the presence of additive white Gaussian noise (AWGN). In Section 5, we review the major channel impairments in long-haul optical transmission. We show how these can be compensated digitally in single-carrier systems, and we compare digital compensation to traditional compensation methods. In Section 6, we review OFDM, which is a multi-carrier modulation format. Finally, in Section 7, we compare implementation complexities of single- and multi-carrier systems.

2. Notation

In this paper, we represent optical signal and noise electric fields as complex-valued, baseband quantities, which are denoted as ()t

E subscript. In Section 3, photocurrents are represented as real passband signals as ()t

I subscript. After we derive the canonical model for a coherent receiver, in all subsequent sections, we write all signals and noises as complex-valued baseband quantities, with the convention that the output of the optoelectronic downconverter is denoted by y, the transmitted symbol as x, the output of the linear equalizer as x~, and the phase de-rotated output prior to symbol decisions as x?. Signals that occupy one polarization only are represented as scalars and are written in italics as shown; while dually

polarized signals are denoted as vectors and are written in bold face, such as k y . Unless stated otherwise, we employ a linear (x , y ) basis for decomposing dually polarized signals. In Section 5 dealing with impairment compensation, matrix and vector quantities are denoted in bold face.

3. Optical detection methods

3.1 Noncoherent detection

Fig. 1. Noncoherent receivers for (a) amplitude-shift modulation (ASK) and (b) binary

frequency-shift keying (FSK).

In noncoherent detection, a receiver computes decision variables based on a measurement of signal energy . An example of noncoherent detection is direct detection of on-off-keying (OOK) using a simple photodiode (Fig. 1(a)). To encode more than one bit per symbol, multi-level amplitude-shift keying (ASK) – also known as pulse-amplitude modulation – can be used. Another example of noncoherent detection is frequency-shift keying (FSK) with wide frequency separation between the carriers. Fig. 1(b) shows a noncoherent receiver for binary FSK.

The limitations of noncoherent detection are: (a) detection based on energy measurement allows signals to encode only one degree of freedom (DOF) per polarization per carrier, reducing spectral efficiency and power efficiency, (b) the loss of phase information during detection is an irreversible transformation that prevents full equalization of linear channel impairments like CD and PMD by linear filters. Although maximum-likelihood sequence detection (MLSD) can be used to find the best estimate of the transmitted sequence given only a sequence of received intensities, the achievable performance is suboptimal compared with optical or electrical equalization making use of the full electric field [13].

3.2 Differentially coherent detection

Fig. 2. Differentially coherent phase detection of (a) 2-DPSK (b) M -DPSK, M > 2.

In differentially coherent detection, a receiver computes decision variables based on a measurement of differential phase between the symbol of interest and one or more reference symbol(s). In differential phase-shift keying (DPSK), the phase reference is provided by the previous symbol; in polarization-shift keying (PolSK), the phase reference is provided by the signal in the adjacent polarization. A binary DPSK receiver is shown in Fig. 2(a). Its output photocurrent is:

()()(){}s s s DPSK T t E t E R t I ?=*Re , where ()t E s is the received signal, R is the responsivity of each photodiode, and T s is the symbol period. This receiver can also be used to detect continuous-phase frequency-shift

(t E s ()t I i DPSK ,()

t I q DPSK ,

(1)

(a) (b) (t E s ()t DPSK

keying (CPFSK), as the delay interferometer is equivalent to a delay-and-multiply demodulator. A receiver for M -ary DPSK, M > 2, can similarly be constructed as shown in Fig. 2(b). Its output photocurrents are:

()()(){}s s s i DPSK T t E t E R t I ?=*21,Re , and ()()(){}

s s s q DPSK T t E t E R t I ?=*21,Im . A key motivation for employing differentially coherent detection is that binary DPSK has

2.8 dB higher sensitivity than noncoherent OOK at a BER of 10?9 [5]. However, the constraint that signal points can only differ in phase allows only one DOF per polarization per carrier, the same as noncoherent detection. As the photocurrents in Eq. (1) to (3) are not linear functions of the E -field, linear impairments, such as CD and PMD, also cannot be compensated fully in the electrical domain after photodetection.

A more advanced detector for M -ary DPSK is the multichip DPSK receiver, which has multiple DPSK receivers arranged in parallel, each with a different interferometer delay that is an integer multiple of T s [14,15]. Since a multichip receiver compares the phase of the current symbol to a multiplicity of previous symbols, the extra information available to the detector enables higher sensitivity. In the limit that the number of parallel DPSK receivers is infinite, the performance of multi-chip DPSK approaches coherent PSK [15]. In practice, the number of parallel DPSK receivers required for good performance needs to be equal to the impulse duration of the channel divided by T s . Although multi-chip DPSK does not require a local oscillator (LO) laser, carrier synchronization and polarization control, the hardware complexity can be a significant disadvantage.

3.3 Hybrid of noncoherent and differentially coherent detection

A hybrid of noncoherent and differentially coherent detection can be used to recover information from both amplitude and differential phase. One such format is polarization-shift keying (PolSK), which encodes information in the Stokes parameter. If we let ()()()t j x x x e t a t E φ= and ()()()t j y y y e t a t E φ= be the E -fields in the two polarizations, the

Stokes parameters are 221y x a a S ?=, ()δcos 22y x a a S = and ()δsin 23y x a a S =, where

()()()t t t y x φφδ?= [16]. A PolSK receiver is shown in Fig. 3. The phase noise tolerance of PolSK is evident by examining S 1 to S 3. Firstly, S 1 is independent of phase. Secondly, S 2 and S 3 depend on the phase difference ()()t t y x φφ?. As ()t x φ and ()t y φ are both corrupted by the same phase noise of the transmitter (TX) laser, their arithmetic difference ()t δ is free of phase noise. In practice, the phase noise immunity of PolSK is limited by the bandwidth of the photodetectors [16]. It has been shown that 8-PolSK can tolerate laser linewidths as large as 01.0≈Δb T ν [17], which is about 100 times greater than the phase noise tolerance of coherent 8-QAM (Section 5.3.3). This was a significant advantage in the early 1990s, when symbol rates were only in the low GHz range. In modern systems, symbol rates of tens of GHz, in conjunction with tunable laser having linewidths <100 kHz, has diminished the advantages of PolSK. Recent results have shown that feedforward carrier synchronization enables coherent detection of 16-QAM at b T νΔ ~10?5 [18], which is within the limits of current technology. As systems are increasingly driven by the need for high spectral efficiency, polarization-multiplexed QAM is likely to be more attractive because of its higher sensitivity (Section 4).

(2)

()()()??????=t E t E t y x s E ()t 1()

t 2()

t 3 Fig. 3. Polarization-shift keying (PolSK) receiver.

3.4 Coherent detection

The most advanced detection method is coherent detection, where the receiver computes decision variables based on the recovery of the full electric field , which contains both amplitude and phase information. Coherent detection thus allows the greatest flexibility in modulation formats, as information can be encoded in amplitude and phase, or alternatively in both in-phase (I) and quadrature (Q) components of a carrier. Coherent detection requires the receiver to have knowledge of the carrier phase, as the received signal is demodulated by a LO that serves as an absolute phase reference. Traditionally, carrier synchronization has been performed by a phase-locked loop (PLL). Optical systems can use (i) an optical PLL (OPLL) that synchronizes the frequency and phase of the LO laser with the TX laser, or (ii) an electrical PLL where downconversion using a free-running LO laser is followed by a second-stage demodulation by an analog or digital electrical VCO whose frequency and phase are synchronized. Use of an electrical PLL can be advantageous in duplex systems, as the transceiver may use one laser as both TX and LO. PLLs are sensitive to propagation delay in the feedback path, and the delay requirement can be difficult to satisfy (Section 5.3.1). Feedforward (FF) carrier synchronization overcomes this problem. In addition, as a FF synchronizer uses both past and future symbols to estimate the carrier phase, it can achieve better performance than a PLL which, as a feedback system, can only employ past symbols. Recently, DSP has enabled polarization alignment and carrier synchronization to be performed in software.

Fig. 4. Coherent transmission system (a) implementation, (b) system model.

A coherent transmission system and its canonical model are shown in Fig. 4. At the transmitter, Mach-Zehnder (MZ) modulators encode data symbols onto an optical carrier and perform pulse shaping. If polarization multiplexing is used, the TX laser output is split into

x ,1x ,2k ,1k ,2(a) (b) Transmitter

two orthogonal polarization components, which are modulated separately and combined in a polarization beam splitter (PBS). We can write the transmitted signal as:

()()()()()()+?=??????=k t t j s k t tx tx tx s s e kT t b P t E t E t φωx E 2,1,, where s T is the symbol period, t P is the average transmitted power, ()t b is the pulse shape (e.g., non-return-to-zero (NRZ) or return-to-zero (RZ)) with the normalization ()∫=s T dt t b 2

, s ω and ()t s φ are the frequency and phase noise of the TX laser, and []T k k k x x ,2,1,=x is a 2×1 complex vector representing the k -th transmitted symbol. We

assume that symbols have normalized energy: 12=??

????k E x . For single-polarization transmission, we can set the unused polarization component k x ,2 to zero.

The channel consists of N A spans of fiber, with inline amplification and DCF after each span. In the absence of nonlinear effects, we can model the channel as a 2×2 matrix:

()()()()()()()()()()ωωωωωH h =?????????????=2221121122211211H H H H t h t h t h t h t F , where ()t h ij denote the response of the i -th output polarization due to an impulse applied at the j -th input polarization of the fiber. The choice of reference polarizations at the transmitter and receiver is arbitrary. Eq. (5) can describe CD, all orders of PMD, polarization-dependent loss (PDL), optical filtering effects and sampling time error [19]. In addition, a coherent optical system is corrupted by AWGN, which includes (i) amplified spontaneous emission (ASE) from inline amplifiers, (ii) receiver LO shot noise, and (iii) receiver thermal noise. In the canonical transmission model, we model the cumulative effect of these noises by an equivalent noise source ()()()[]T t n t n t 21,=n referred to the input of the receiver.

The E -field at the output of the fiber is ()()()[]T s s s t E t E t 2,1,,=E , where:

()()()()()t E e kT t c x P t E l sp t t j k m s lm k m r l s s s ,21

,,+?=+=φω. Under the assumption of Fig. 4 where inline amplification completely compensates propagation loss, t r P P = is the average received power, ()()()t h t b t c lm lm ?= is a normalized pulse shape, and ()t E l sp , is ASE noise in the l -th polarization. Assuming the N A fiber spans are identical and all inline amplifiers have gain G and spontaneous emission factor n sp , the two-sided power spectral density (psd) of ()t E l sp , is ()()G n N f S s sp A Esp 1?=ω W/Hz [20].

The first stage of a coherent receiver is a dual-polarization optoelectronic downconverter that recovers the baseband modulated signal. In a digital implementation, the analog outputs are lowpass filtered and sampled at s KT M T =1, where K M is a rational oversampling ratio. Channel impairments can then be compensated digitally before symbol detection.

(5)

(4) (6)

3.4.1 Single-polarization downconverter

Fig. 5. Single-polarization downconverter employing a (a) heterodyne and (b) homodyne design.

We first consider a single-polarization downconverter, where the LO laser is aligned in the l -th polarization. Downconversion from optical passband to electrical baseband can be achieved in two ways: in a homodyne receiver, the frequency of the LO laser is tuned to that of the TX laser so the photoreceiver output is at baseband. In a heterodyne receiver, the LO and TX lasers differ by an intermediate frequency (IF), and an electrical LO is used to downconvert the IF signal to baseband. Both implementations are shown in Fig. 5. Although we show the optical hybrids as 3-dB fiber couplers, the same networks can be implemented in free-space optics using 50/50 beamsplitters; this was the approach taken by Tsukamoto [21]. Since a beamsplitter has the same transfer function as a fiber coupler, there is no difference in their performances.

In the heterodyne downconverter of Fig. 5(a), the optical frequency bands around IF LO ωω+ and IF LO ωω? both map to the same IF. In order to avoid DWDM crosstalk and to avoid excess ASE from the unwanted image band, optical filtering is required before the downconverter. The output current of the balanced photodetector in Fig. 5(a) is:

()()()()(){}()t I t E t E R t E t E R t I l sh l LO l s l het ,,,2221,Im 2+=???????=?, where ()()()t t j l LO l LO LO LO e P t E φω+=,, is the E -field of the LO laser, and l LO P ,, LO ω and ()t LO φ are its power, frequency and phase noise. ()t I l sh , is the LO shot noise. Assuming s LO P P >>, ()t I l sh , has a two-sided psd of ()LO Ish qRP f S = A 2/Hz.. Substituting Eq. (6) into Eq. (7), we get:

()()()()())()()()t I t E t t y t t y P P R t I l sh l sp IF lq IF li r l LO l het ,',,,cos sin 2+++=ωω, where LO s IF ωωω?= is the IF, ()()()t t t LO s φφφ?= is the carrier phase, and ()t y li and ()t y lq are the real and imaginary parts of:

()()()∑∑=?=k m t j s lm k m l e

kT t c x t y 21,0φ. The term ()t E P R l sp r LO ',,2 in Eq. (8) is sometimes called LO-spontaneous beat noise , and

()()()(){}

t t j l sp l sp LO LO e t E t E φω+?=,',Im has a two-sided psd of ()f S Esp

21. It can similarly be shown that the currents at the outputs of the balanced photodetectors in the homodyne downconverter (Fig. 5(b)) are: (7)

(8)

(9)

(a) (b)

l het ,,()t i ()t q l het ,,

()t I i l hom ,,()

()()()()()()()t I t E t y P P R t E t E R t I li sh li sp li r l LO i l hom ,',,2221,,++=???????=, and ()()()()()()()t I t E t y

P P R t E t E R t I lq sh lq sp lq r l LO q l hom ,',,2423,,++=???????=, where ',li sp E and ',lq sp E are white noises with two-sided psd ()f S Esp 21; and li sh I , and lq sh I , are white noises with two-sided psd ()f S Ish 21. Since it can be shown that thermal

noise is always negligible compared to shot noise and ASE noise [22], we have neglected this term in Eq. (10) and (11). In long-haul systems, the psd of LO-spontaneous beat noise is typically much larger than that of LO shot noise; such systems are thus ASE-limited. Conversely, unamplified systems do not have ASE, and are therefore LO shot-noise-limited.

If one were to demodulate Eq. (8) by an electrical LO at IF ω, as shown in Fig. 5(a), the resulting baseband signals ()t I i l het ,, and ()t I q l het ,, will be the same as Eqs. (10) and (11) for the homodyne downconverter in Fig. 5(b), with all noises having the same psd’s. Hence, the heterodyne and the two-quadrature homodyne downconverters have the same performance

[23]. A difference between heterodyne and homodyne downconversion only occurs when the transmitted signal occupies one quadrature (e.g. 2-PSK) and the system is LO shot-noise-limited, as this enables the use of a single-quadrature homodyne downconverter that has the optical front-end of Fig. 5(a), but has LO s ωω=. Its output photocurrent is

()()()

()t I t E t y P P R l sh l sp lq r l LO ,',,2++. Compared to Eq. (11), the signal term is doubled

(four times the power), while the shot noise power is only increased by two, thus yielding a sensitivity improvement of 3 dB compared to heterodyne or two-quadrature homodyne downconversion. This case is not of practical interest in this paper, however, as long haul systems are ASE-limited, not LO shot-noise-limited. Also, for good spectral and power efficiencies, modulation formats that encode information in both I and Q are preferred. Hence, there is no performance difference between a homodyne and a heterodyne downconverter provided optical filtering is used to reject image-band ASE for the heterodyne downconverter. Since the two downconverters in Fig. 5 ultimately recover the same baseband signals, we can combine Eqs. (10) and (11) and write a normalized, canonical equation for their outputs as: ()()()()t n e kT t c x t y l k m t j s lm k m l +?=∑∑=21,φ,

where ()t n l is complex white noise with a two-sided psd of:

()s s nn T N f S ==0. s γ is the signal-to-noise ratio (SNR) per symbol. The values of s γ for homodyne and heterodyne downconverters in different noise regimes are shown in Table 1. For the shot-noise limited regime using a heterodyne or two-quadrature homodyne downconverter, s γ is simply the number of detected photons per symbol. We note that Eq. (12) is complex-valued, and its real and imaginary parts are the two baseband signals recovered in Fig. 5. For the remainder of this paper, it is understood that all complex arithmetic operations are ultimately implemented using these real and imaginary signals.

The advantages of heterodyne downconversion are that it uses only one balanced photodetector and has a simpler optical hybrid. However, the photocurrent in Eq. (8) has a bandwidth of BW IF +ω, where BW is the signal bandwidth (Fig. 6(a)). To avoid signal distortion caused by overlapping side lobes, ω

IF needs to be sufficiently large. Typically,

(10) (11) (12)

(13)

BW IF ≈ω, thus a heterodyne downconverter requires a balanced photodetector with at least twice the bandwidth of a homodyne downconverter, whose output photocurrents in Eqs. (10) and (11) only have bandwidths of BW (Fig. 6(b)). This extra bandwidth requirement is a major disadvantage. In addition, it is also difficult to obtain electrical mixers with baseband bandwidth as large as the IF. A summary of homodyne and heterodyne receivers is given in Table 2. A comparison of carrier synchronization options is given in Table 3. Table 1. SNR per symbol for various receiver configurations. For the ASE-limited cases, s n is

the average number of photons received per symbol, N A is the number of fiber spans, and n sp is

the spontaneous emission noise factor of the inline amplifiers. For the LO shot-noise-limited cases, s r n n η= is the number of detected photons per symbol, where η is the quantum

efficiency of the photodiodes.

. Fig. 6. Spectrum of a (a) heterodyne and (b) homodyne downconverter measured at the output

of the balanced photodetector.

Table 2. Comparison between homodyne and heterodyne downconverters .

Homodyne Heterodyne No. of balanced photodetectors per

polarization required for QAM

2 1 Minimum photodetector bandwidth BW 2BW

Table 3. Comparison of carrier synchronization options. All three can be used with either homodyne or

heterodyne downconversion.

Optical PLL Electrical PLL FF Carrier Recovery Can the transceiver use

same laser for TX and LO?

No Yes

Yes Does propagation delay

degrade performance?

Yes Yes

No Carrier phase estimate

depends on past or future

symbols?

Past only Past only Past and future Implementation Analog Analog or digital Analog [24] or digital

3.4.2 Dual-polarization downconverter

In the single-polarization downconverter, the LO needed to be in the same polarization as the received signal. One way to align the LO polarization with the received signal is with a polarization controller (PC). There are several drawbacks with this: first, the received polarization can be time-varying due to random birefringence, so polarization tracking is required. Secondly, PMD causes the received Stokes vector to be frequency-dependent. When

(a) (b)

a single-polarization receiver is used, frequency-selective fading occurs unless PMD is first compensated in the optical domain. Thirdly, a single-polarization receiver precludes the use of polarization multiplexing to double the spectral efficiency.

A dual-polarization downconverter is shown inside the receiver of Fig. 4(a). The LO laser is polarized at 45° relative to the PBS, and the received signal is separately demodulated by each LO component using two single-polarization downconverters in parallel, each of which can be heterodyne or homodyne. The four outputs are the I and Q of the two polarizations, which has the full information of ()t s E . CD and PMD are linear distortions that can be compensated quasi-exactly by a linear filter.

Fig. 7. Emulating (a) direct detection, (b) 4-DPSK detection and (c) PolSK detection with optoelectronic downconversion followed by non-linear signal processing in the electronic domain. The signals E x (t ) and E y (t ) are the complex-valued analog outputs described by Eq. (12) for each polarization. We note that in the case of the heterodyne downconverter, the non-linear operations shown can be performed at the IF output(s) of the balanced photoreceiver.

The lossless transformation from the optical to the electrical domain also allows the receiver to emulate noncoherent and differentially coherent detection by nonlinear signal processing in the electrical domain (Fig. 7). In long-haul transmission where ASE is the dominant noise source, these receivers have the same performance as those in Fig. 1?3. A summary of the detection methods discussed in this section is shown in Table 4.

Table 4. Comparison between noncoherent, differentially coherent and coherent detection. For the first two detection methods, direct detection refers to the receiver implementations shown in Figs. 1?3, while homodyne/heterodyne refers to the equivalent implementations shown in Fig. 7. Noncoherent Detection Differentially Coherent Detection

Direct Hom./Het. Direct Hom./Het. Coherent Detection Require LO?

No Yes No Yes Yes Require Carrier

Synchronization?

No No No No Yes Can compensate CD

and PMD by a linear

filter after

photodetection?

No Yes No Yes Yes Degrees of freedom per

polarization per carrier

1 1

2 Modulation formats

supported ASK, FSK, Binary PolSK DPSK, CPFSK, Non-binary PolSK PSK, QAM, PolSK, ASK,

FSK, etc.

(a)

(t 2()

t 3()t 1()t DD (b)

()t y i DPSK ,()

4. Modulation formats

In this section, we compare the BER performance of different modulation methods for single-carrier transmission corrupted by AWGN. We assume that all channel impairments other than AWGN – including CD, PMD, laser phase noise and nonlinear phase noise – have been compensated using techniques discussed in Section 5. Since ASE and LO shot noise are Gaussian, the performance equations obtained are valid for both long haul and back-to-back systems, when the definition of SNR defined by Table 1 is used. Owing to fiber nonlinearity, it is desirable to use modulation formats that maximize power efficiency. Unless otherwise stated (PolSK being the only exception), the formulae provided assume transmission in one polarization, where noise in the unused polarization has been filtered. This condition is naturally satisfied when a homodyne or heterodyne downconverter is used. For noncoherent detection, differentially coherent detection and hybrid detection, the received optical signal needs to be passed through a polarization controller followed by a linear polarizer.

Since the two polarizations in fiber are orthogonal channels with statistically independent noises, there is no loss in performance by modulating and detecting them separately. The BER formulae provided are thus valid for polarization-multiplexed transmission provided there is no polarization crosstalk. Polarization multiplexing doubles the capacity while maintaining the same receiver sensitivity in SNR per bit. We write the received signal as:

k k k n x y +=, where k x is the transmitted symbol and k n is AWGN. For the remainder of this paper, our notation shall be as follows:

M is the number of signal points in the constellation.

()M b 2log = is the number of bits encoded per symbol.

T T s b = is the equivalent bit period. ??

??????????=22k k s n E x E γ is the SNR per symbol in single-polarization transmission. ????????????=22k k s E E n x γis the SNR per symbol in dual-polarization transmission (e.g.

polarization-multiplexed or PolSK). b s b γγ= is the SNR per bit

The maximum achievable spectral efficiency (bit/s/Hz) of a linear AWGN channel is governed by the Shannon capacity [1]:

()s

b γ+=1log 2max . If N D identical channels are available for transmission, and we utilize them all by dividing the available power equally amongst the channels, the total capacity is ()D s D N N b γ+=1log 2, which is an increasing function of N D . Hence, the best transmission strategy is to use all the dimensions available. For example, suppose a target spectral efficiency of 4 bits per symbol is needed. Polarization-multiplexed 4-QAM, which uses the inphase and quadrature components of both polarizations, has better sensitivity than single-polarization 16-QAM.

ASK with noncoherent detection

Optical M -ary ASK with noncoherent detection has signal points evenly spaced in nonnegative amplitude [25]. The photocurrents for different signal levels thus form a

(15)

(14)

quadratic series. It can be shown that the optimal decision thresholds are approximately the geometric means of the intensities of neighboring symbols. Assuming the use of Gray coding, it can be shown that for large M and γb , the BER is approximated by [5]:

()()()?????????????????≈1212311M M b erfc M M b M P b ASK b γ. DPSK with differentially coherent detection

Assuming the use of Gray coding, the BER for M -ary DPSK employing differentially coherent detection is [26]:

()()[]()()∫∫??++≈πππηχχηγχηγχπM b b DPSK b d d b b b M P 0sin cos 1exp sin cos 11sin 11.

For binary DPSK, the above formula is exact, and can be simplified to [27]: ()()b DPSK b P γ?=exp 212. For quaternary DPSK, we have [27]:

()()()()[]22101exp 21,4βααββα+??=I Q P DPSK b , where ()2112?=b γα and ()2112+=b γβ. ()y x Q ,1 and ()x I 0 are the Marcum Q function and the modified Bessel function of the zeroth order, respectively.

Polarization-Shift Keying (PolSK) PolSK is the special case in this section where the transmitted signal naturally occupies both polarizations. Thus, polarization multiplexing cannot be employed to double system capacity. The BER for binary PolSK is [16]:

()()b PolSK b P γ?=exp 212. For higher-order PolSK, the BER is well-approximated by [28], ()()()??????????????????+?≈∫?10tan tan cos 11011θθθθθπθdt t f t n F b M P PolSK b , where

()()()()t t b t F b cos 1cos 1exp 211+???

=γθ, and ()()()()()t b t b t t f b b cos 11cos 1exp 2

sin ++??=γγθ. n , 0θ and 1θ are related to the number of nearest neighbors and the shape of the decision boundaries on the Poincaré sphere. Table 5 shows their values for 4-PolSK and 8-PolSK. Square 4-PolSK denotes the constellation where the signal points lie at the vertices of a square

(16)

(17) (18)

(19) (20)

enclosed by the Poincaré square. In tetrahedral 4-PolSK and cubic 8-PolSK, the signal points lie at the vertices of a tetrahedron and a cube enclosed by the Poincaré square, respectively.

Table 5. Parameters for computing the BER in polarization-shift keying (PolSK).

PSK with coherent detection

Assuming the use of Gray coding, the BER for M -ary PSK employing coherent detection is given approximately by [29]:

()???????

???????≈M b erfc b M P b PSK b πγsin 1. For the special cases of BPSK and QPSK, we have the exact expressions: ()()()b PSK b PSK b erfc P P γ2142==. QAM with coherent detection

Assuming the use of Gray coding, the BER for a square M -QAM constellation with coherent detection is approximated by [29]:

()()???

???????????????≈12312M b erfc M M

b M P b QAM b γ. The BER performance of 8-QAM with the signals points arranged as shown in Fig. 8 is [20]: ()???????

?+≈33316118b QAM b erfc P γ. Fig. 8. 8-QAM constellation.

Using Eq. (16) to (27), we compute the SNR per bit required for each modulation format discussed to achieve a target BER of 10?3, which is a typical threshold for receivers employing forward error-correction coding (FEC). The results are shown in Table 6. In Fig. 9, we plot spectral efficiency vs SNR per bit with polarization multiplexing to obtain fair comparison with PolSK (we also show results for 12-PolSK and 20-PolSK [28]). Since polarization-multiplexed ASK, DPSK and PSK all have two DOF (one per polarization), as is the case with PolSK, they all have similar slopes at high spectral efficiency. Because QAM

(24) (26) (27) (25)

0002

10b b b 001

010

100110011

111

uses all four available DOF for encoding information, it has better SNR efficiency than the other formats, and exhibits a steeper slope at high spectral efficiency.

Table 6. SNR per bit (in dB) required to achieve BER=10?3. Single-Polarization Transmission

Bits per Symbol Constellation Size M ASK with Direct Detection DPSK with Interferometric Detection PSK with Coherent Detection QAM with Coherent

Detection PolSK 1 2 9.8 7.9 6.8 6.8 7.9 2 4 15.0 9.9 6.8 6.8 8.0 3 8 20.0 13.1 10.0 9.0 9.4 4 16 25.0 17.4 14.3 10.5

Fig. 9. Spectral efficiency vs. SNR per bit required for different modulation formats at a target

BER of 10?3. We assume polarization multiplexing for all schemes except PolSK. Also shown

is the Shannon limit (15), which corresponds to zero BER.

5. Channel impairments and compensation techniques in single-carrier systems

In this section, we review the major channel impairments in fiber-optic transmission. We present the traditional methods of combating these, and show how compensation can be done electronically with coherent detection in single-carrier systems. Impairment compensation in multi-carrier systems is discussed in Section 6.

5.1 Linear impairments

5.1.1 Chromatic dispersion

CD is caused by a combination of waveguide and material dispersion [22]. In the frequency domain, CD can be represented by a scalar multiplication:

()()()I H ???????+??=33226121s fiber s fiber L L j CD e ωωβωωβω, where fiber L is the length of the fiber, β2 is the dispersion parameter, β3 is the dispersion slope, and s ω is the signal carrier frequency. Uncompensated CD leads to pulse broadening, causing intersymbol interference (ISI). Long-haul systems use DCF to compensate CD optically [22]. However, inexact matching between the β2 and β3 of transmission fiber and DCF dictates the need for terminal dispersion compensation at high bit rates, typically 40

(28)

SNR per bit (dB)

Gbit/s or higher [30]. In reconfigurable networks, data can be routed dynamically through different fibers, so the residual dispersion can be time-varying. This necessitates tunable dispersion compensators.

5.1.2 Polarization-mode dispersion

Fig. 10. First-order polarization-mode dispersion.

Polarization-mode dispersion (PMD) is caused by random birefringence in the fiber. In first-order PMD, a fiber possesses a “fast axis” along in polarization and a “slow axis” in the orthogonal polarization (Fig. 10). These states of polarization are known as the principal states of polarization (PSPs), and can be any vector in Stokes space in general. First-order PMD can be written as [31]:

()211DR R H ?=ωPMD , where ()22,DGD DGD j j e e diag τωτω?=D is a diagonal matrix with DGD τ being the differential group delay between the two PSPs, and 1R and 2R are unitary matrices that rotate the reference polarizations to the fiber’s PSPs, which are elliptical in general. When a signal is launched in any polarization state other than a PSP, the receiver will detect two pulses at each reference polarization. Ignoring CD and other effects, the impulse response measured by a polarization-insensitive direct-detection receiver is ()()()()21222DGD DGD t a t a t h τδτδ+??+??=, where a 2 is the proportion of transmitted energy falling in the slow PSP. In this simple two-path model, we see that PMD can lead to frequency-selective fading [32]. In contrast to CD, which is relatively static, PMD (both the PSPs and the DGD) can fluctuate on time scales on the order of a millisecond [33]. Thus PMD compensators thus need to be rapidly adjustable. The statistical properties of PMD have been studied in [34?36], and it has been shown that DGD τ has a Maxwellian distribution, whose mean value DGD τ grows as the square-root of fiber length. In SMF, DGD τ is typically of order km /ps 1.0. PMD is a significant impact on systems at bit rates of 40 Gbit/s and higher, because DGD τ can be a significant fraction of the symbol period. Uncompensated PMD can result in system outage [37].

One method of combating PMD is to use a tunable PC at the transmitter to ensure the input signal is launched into a PSP [38]. Receiver-based compensators for first-order PMD use a continuously tunable PC followed by a variable retarder, which inverts the DGD of the fiber [38,39]. By cascading multiple first-order PMD compensators, one can retrace the PMD vector of the transmission fiber. Such a device can compensate higher-order PMD [40]. Optical PMD compensators have been constructed using nonlinear chirped fiber Bragg gratings [41], planar lightwave circuits (PLC) [42] and polarization-maintaining fibers (PMF) twisted mechanically [40]. Compensation of DGD τ as large as 1.7 symbols was demonstrated by Noé et al [40]. A major limitation of optical PMD compensation is that device performance

(29)

depends on the degree of tunability, and increasing the number degrees of freedom can require costly hardware. However, optical PMD compensators are transparent to the data rate and modulation format of the transmitted signal, and have been successfully employed for very high-data-rate systems, where digital compensation is currently impossible.

Electronic PMD equalization has gained considerable recent interest. Buchali and Bülow studied the use of a feedforward equalizer (FFE) with decision feedback equalizer (DFE) to combat PMD systems using direct detection of OOK [39]. As with electronic CD compensation in direct detection of OOK [43], the loss of phase during detection prevents quasi-exact compensation of PMD.

5.1.3 Other linear impairments

In addition to CD and PMD, a fiber optic link can also have polarization-dependent loss (PDL) due to anisotropy of network components such as couplers, isolators, filters, multiplexers, and amplifiers [44]. In DWDM transmission, arrayed waveguide gratings (AWG), interleavers and reconfigurable add-drop (de)multiplexers (ROADMs) cause attenuation at the band edges of a channel. When a signal has to pass through cascaded elements, bandwidth narrowing can be problematic. This is a major challenge in 40 Gbit/s RZ-DPSK at 50 GHz channel spacing [45]. Bandwidth narrowing can be equalized by tunable optical equalizers, but such devices are costly, introduce loss, and are difficult to make adaptive.

5.1.4 Compensation of linear impairments and computational complexity

Since a dual-polarization downconverter linearly recovers the full electric field, CD and PMD can be compensated quasi-exactly in the electronic domain after photodetection. One approach is to use a tunable analog filter. However, as in the case of optical compensators, it is difficult to implement the desired transfer function exactly, and analog filters are also difficult to make adaptive. In addition, parasitic effects like signal reflections can lead to signal degradation.

With improvements in DSP technology, digital equalization of CD and PMD is becoming feasible. When the outputs of a dual-polarization downconverter are sampled above the Nyquist rate, the digitized signal contains a full characterization of the received E -field, allowing compensation of linear distortions by a linear filter. CD compensation using a digital infinite impulse response (IIR) filter was studied by [46]. Although an IIR filter allows fewer taps, it is more difficult to analyze, and may require greater receiver complexity because of the need to implement time-reversal filters. In this paper, we concentrate on CD and PMD compensation using a finite impulse response (FIR) filter.

Fig. 11. Digital equalization for a dually polarized linear channel.

Linear equalization using an FIR filter for dually polarized coherent systems was studied in [47,48]. The canonical system model is shown in Fig. 11. The analog outputs of a dual-polarization downconverter are passed through anti-aliasing filters with impulse responses ()t p and then sampled synchronously at a rate of s KT M T =1, where K M is a rational oversampling ratio. We assume that the sampling clock has been synchronized using well-known techniques [49]. In theory, the use of a matched filter in conjunction with symbol rate

~~k x ,1k

实业有限公司简介

公司简介怎么写一，首先介绍一下你们公司的名字，什么性质的，什么时间成立的，有多少员工，主要产品是什么及它的应用范围，自成立以来取得什么样的成绩（包括销售方面、技术创新方面、人员培训方面等等），在未来几年内的发展规划等等。注意千万不要写成泛文，要写得有理有据，还要突出重点，具有很强的吸引力。二， 1，公司背景如何成立，建立人，公司的历史，内容发展 2，公司服务内容或者产品介绍 3，公司员工和公司的结构介绍 4，公司的顾客群或者范围介绍 5，公司近期内的重大发展介绍 6，公司的年度报表简介不过最重要的，建议你弄明白看这个公司简介的对象是谁，知己知彼才能百战百胜。不知道对象是谁就提笔写东西，是很不明智的。我给你的是比较常用的方式，不适用于所有对象群体。三， 1.要体现出你的公司的独特个性。 2.不能学习别人的，才会有自己的个性。上海富蔗化工有限公司座落于中国经济、金融、贸易中心--上海。公司位于上海西北部-上海桃浦化工工业园区。是集化学科研，开发，生产，销售，服务为一体

的综合型企业。本公司在奉贤，南汇，松江等工业园区及江浙一带均有生产和联营企业基地。我公司拥有先进的生产设备，完善的产品检测手段和质量保证体系。我公司的员工具有较强的责任感。经过多年的发展，现已形成有机中间体、医药中间体、化工溶剂和化工助剂四大类上百个品种。公司业务涉及医药、农药、染料、涂料、制革等各行业。公司年销售额在数千万人民币左右。我公司与世界各大化工、医药原料供应厂商有着密切的联系，并与其保持着良好的合作关系，能够稳定、成立以来，发挥自身优势，健全销售网络，注重企业形象，开创了上海首家在网络上以小包装零售业带动大批发销售的多元化经营模式，并且不断增加原料品种的配套措施，赢得广大客户的一致好评。企业产品市场辽阔，远销日本、美国、欧洲、印度，东南亚等地。我们富蔗人本着“求实、高效、创新”的团队精神，参与到激烈的市场竞争中来，以一流的产品质量，优惠的产品价格，令人满意的销售服务，赢得您的支持与信赖。我们愿同四海知音、各界同仁携手共同发展。企业使命： ----立足化工行业，开拓进取。诚信服务，以质量和信用为本。规范管理，引进先进、科学方法。加强合作，建立良好关系。开发新品，发展实业，优化产业结构。创立品牌，报效国家，为经济发展、环境保护和社会进步做出贡献。富蔗承诺，为您负责。 [正派] 经营是富蔗的一贯原则[创新] 进取是富蔗的生存基础 [团队] 协作是富蔗的致胜法宝深圳市国冶星光电子有限公司是目前国内少数大型综合性LED光电产品生产性高科技企业之一，公司集科研、开发、生产、销售、服务为一体，专业生产发光二极管、数码、点阵、（室内、室外、半户外）模块、背光源、贴片（SMD）LED等全系列光电产品，产量大、品质高，产品广泛应用于家电、手机、公共场所、交通、广告、银行等，为大型家电、手机厂的配套供应商。公司拥有先进的专业生产设备、世界领先的全自动SMD生产线及一系列品质保证测试设备（自动固晶机、自动焊线机、模造机、自动切割机、自动分光机、自动装带机、冷热冲击机、可程式恒温恒湿机、定点型恒温恒湿机、环境测试机、电脑分析显微镜等），拥有高素质的管理队伍，高水平的技术、开发人员及一大批训练有素的熟练工人。生产原材料均采用当今国外知名企业产品，现有市场已覆盖全国各地，海外业务拓展已具规模，

小学奥数之容斥原理

五．容斥原理问题 1．有100种赤贫.其中含钙的有68种,含铁的有43种,那么,同时含钙和铁的食品种类的最大值和最小值分别是( ) A 43,25 B 32,25 C32,15 D 43,11 解：根据容斥原理最小值68+43-100＝11 最大值就是含铁的有43种 2．在多元智能大赛的决赛中只有三道题.已知:(1)某校25名学生参加竞赛,每个学生至少解出一道题;(2)在所有没有解出第一题的学生中,解出第二题的人数是解出第三题的人数的2倍:(3)只解出第一题的学生比余下的学生中解出第一题的人数多1人;(4)只解出一道题的学生中,有一半没有解出第一题,那么只解出第二题的学生人数是( ) A，5 B，6 C，7 D，8 解：根据“每个人至少答出三题中的一道题”可知答题情况分为7类：只答第1题，只答第2题，只答第3题，只答第1、2题，只答第1、3题，只答2、3题，答1、2、3题。分别设各类的人数为a1、a2、a3、a12、a13、a23、a123 由（1）知：a1+a2+a3+a12+a13+a23+a123＝25…① 由（2）知：a2+a23＝（a3+ a23）×2……② 由（3）知：a12+a13+a123＝a1－1……③ 由（4）知：a1＝a2+a3……④ 再由②得a23＝a2－a3×2……⑤ 再由③④得a12+a13+a123＝a2+a3－1⑥ 然后将④⑤⑥代入①中，整理得到 a2×4+a3＝26 由于a2、a3均表示人数，可以求出它们的整数解：当a2＝6、5、4、3、2、1时，a3＝2、6、10、14、18、22 又根据a23＝a2－a3×2……⑤可知：a2>a3 因此，符合条件的只有a2＝6，a3＝2。然后可以推出a1＝8，a12+a13+a123＝7，a23＝2，总人数＝8+6+2+7+2＝25，检验所有条件均符。故只解出第二题的学生人数a2＝6人。 3．一次考试共有5道试题。做对第1、2、3、、4、5题的分别占参加考试人数的95%、80%、79%、74%、85%。如果做对三道或三道以上为合格，那么这次考试的合格率至少是多少？答案：及格率至少为71％。假设一共有100人考试 100-95＝5 100-80＝20 100-79＝21 100-74＝26 100-85＝15 5+20+21+26+15＝87（表示5题中有1题做错的最多人数）

深圳市大诠实业有限公司简介及产品介绍

深圳市大诠实业有限公司简介深圳市大诠实业有限公司大诠旗下网络变压器厂于2007年6月成立，公司主要生产脉冲变压器、滤波器、磁环线圈等磁性元件。2016年注册属于大诠实业的自主网络滤波器品牌NMS.工厂面积5000多平方米，前期项目总投资三百多万元人民币，目前年产值达6000多万元人民币。公司属于内资企业，总部设于深圳市龙华新区大浪街道，现有员工300人左右。其中一些工厂在云南文山市环城西路中部，下辖13个分厂约7000人。大诠实业东莞工厂位于广东省东莞市高埗镇芦溪村银涌工业区创鸿基工业园四楼。公司目前正处于高速发展时期，是同行客户信得过的企业。大诠优势所有产品均符合全球业界各项质量及环保规定。公司拥有一个经验丰富的管理团队，有一批爱岗敬业，开拓进取，团结一心，坦诚从事之优秀员工，有公正严明的规章制度及相对完善的管控体系。公司经过多年的锤炼，积累了丰富的集生产加工，开发，品质，后勤保障等各种管控经验，有多渠道固定之客户群，客户遍布珠三角，长三角以及欧美，东南亚等发达国家和地区。也有永久持续，品质交期稳定之产品前段供应链。大诠文化愿景：成为全球网络变压器最具领导地位的生产商！使命：为网络通讯行业提供最具竞争力的多元化网络变压器！理念：与员工共成长，与客户求共赢！以质量创品牌，以诚信谋发展！精神：开拓、创新，立足市场求发展；优质、高效，专心服务为用户! 价值观：以人为本，客户至上，诚信经营，大家共赢！工作作风：产前准备严格核对；生产过程严格要求；员工作业严格规范；出厂成品严格把关。服务宗旨：每位客户的满意是环宇人存在价值的最好体现！大诠产能（300K/天） 12PIN（超薄）SMD 20K/天 16PIN （常规）SMD 50K/天 24PIN（小）SMD 60K/天 24PIN（大）SMD 20K/天 48PIN（常规）SMD 10K/天 18PIN（单排常规）DIP 5K/天 36PIN（单排常规）DIP 10K/天 24PIN（双排常规）DIP 10K/天 48PIN（双排常规）DIP 15K/天设备仪器列表序号设备名称数量用途序号设备名称数量用途 1 TCP 1 集高压测试、综合测试、外观检验、包装于一体的设备。 14 切脚机 6 DIP产品切脚设备 2 射频网络分析仪 1 产品特性测试15 显微镜8 产品局部放大，用于分析产品异常。

2015国家公务员考试行测：数学运算-容斥原理和抽屉原理

【导读】国家公务员考试网为您提供：2015国家公务员考试行测：数学运算-容斥原理和抽屉原理，欢迎加入国家公务员考试QQ群：242808680。更多信息请关注安徽人事考试网https://www.360docs.net/doc/8d14556680.html, 【推荐阅读】 2015国家公务员笔试辅导课程【面授+网校】容斥原理和抽屉原理是国家公务员考试行测科目数学运算部分的“常客”，了解此两种原理不仅可以提高做题效率，还可以提高自己的运算能力，扫平所有此类计算题。中公教育专家在此进行详细解读。一、容斥原理在计数时，要保证无一重复，无一遗漏。为了使重叠部分不被重复计算，在不考虑重叠的情况下，把包含于某内容中的所有对象的数目先计算出来，然后再把计数时重复计算的数目排斥出去，使得计算的结果既无遗漏又无重复，这种计数的方法称为容斥原理。 1.容斥原理1——两个集合的容斥原理如果被计数的事物有A、B两类，那么，先把A、B两个集合的元素个数相加，发现既是 A类又是B类的部分重复计算了一次，所以要减去。如图所示：公式：A∪B=A+B-A∩B 总数=两个圆内的-重合部分的【例1】一次期末考试，某班有15人数学得满分，有12人语文得满分，并且有4人语、数都是满分，那么这个班至少有一门得满分的同学有多少人? 数学得满分人数→A，语文得满分人数→B，数学、语文都是满分人数→A∩B，至少有一门得满分人数→A∪B。A∪B=15+12-4=23，共有23人至少有一门得满分。 2.容斥原理2——三个集合的容斥原理如果被计数的事物有A、B、C三类，那么，将A、B、C三个集合的元素个数相加后发现两两重叠的部分重复计算了1次，三个集合公共部分被重复计算了2次。如图所示，灰色部分A∩B-A∩B∩C、B∩C-A∩B∩C、C∩A-A∩B∩C都被重复计算了1 次，黑色部分A∩B∩C被重复计算了2次，因此总数A∪B∪C=A+B+C-(A∩B-A∩B∩C)-(B∩ C-A∩B∩C)-(C∩A-A∩B∩C)-2A∩B∩C=A+B+C-A∩B-B∩C-C∩A+A∩B∩C。即得到：公式：A∪B∪C=A+B+C-A∩B-B∩C-C∩A+A∩B∩C

雷达原理复习

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壹玖公司简介河南壹玖实业有限公司成立于2014年7月的，壹玖实业是一家融实体企业经营与商业培训为一体的民营企业，由董事长袁国顺先生创立的“免费商业模式”培训为切入点的，企业家智慧及资源密集体。壹玖实业是河南唯一一家以企业发展落地为基础，从商业模式学习切入、实操落地的新经济生态大系统，致力于中小企业未来根本核心竞争力的研究，通过帮助企业商业模式升级带动企业稳定，飞速发展，在行业里脱颖而出。壹玖实业以河南为中心向全国乃至世界辐射，截止2016年底已在全国发展100家事业部，力争三年内发展1000家，目前已拥有中小企业家会员1万多人。挽救濒临倒闭民营中小企业120家以上，盘活10亿以上烂尾楼项目15个。间接帮助社会多创造10万个以上就业机会。河南壹玖实业有限公司目前正致力于帮助会员企业打造更具竞争力的免费商业模式，以振兴民族经济为己任，用免费模式，让奇迹发生，助推中华民族伟大复兴的中国梦早日实现。公司自成立至2016年年底，已拥有高素质员工500多人，参股、控股多家公司及项目，初步形成了以中原为核心，辐射全国的战略布局。直接影响到的企业家会员超过1万名，间接影响到的企业超过8万余家。壹玖实业以袁国顺老师为核心的企业家领导团队，用自身经营企业20多年的经验帮助会员企业。壹玖系统可以彻底帮助您解决现在面临的企业问题，为您的企业装上更具竞争力的武器，更轻松的经营企业。壹玖资本，等您加入！

袁国顺，河南壹玖实业集团董事长，一个实实在在践行免费模式的企业家！ l38O年25月6日9622 分電联年月日分，找袁国顺老师。“企业的一切问题都是老板的问题，老板行，一切都行!”“企业家不是努力不够，而是思维不够。” 袁国顺先生搏击商海二十年，运用哲学、逻辑学和心理学，从自己和他人成功或失败中剖析深层次的原理，形成了独特新颖而且具有现实指导意义的新的商业思维模式。“一花独放不是春，百花盛开春满园”。为了帮助哪些和他一样在商海中打拼的企业家少走弯路，袁国顺先生走上讲台，帮助多家企业创造了行业神话和奇迹。壹玖资本自助式大平台主办的“聚世界资本、创商海经济”广东峰会上，国内顶级讲师、河南壹玖实业有限公司董事长袁国顺出场便语出惊人。他指出，在竞争激烈的当下，免费模式可以帮助企业实现腾飞。“中国企业家普遍认为免费就是无常付出、倒贴，其实这只是表象。根据企业产品特性，设定一个免费的时间段做铺垫，调试时间的浓度来让客户形成惯性消费行为，最终变成你的企业的忠实客户。” 纵观整个社会经济发展，50年前，开工厂能盈利，30年前做贸易很赚钱，现在互联网如此发达、产品库存极其庞大的情况下，很多企业家必须遵循当下互联网经济的规律——平台分享、合作共赢。如果企业家还停留在“机制、约定、管理员工、经营方法”的层面，企业是很难发展的，这也是为什么传统企业最近几年举步维艰的原因，当然这不是企业家的过错，也不是社会的过错，而是整个经济的发展到了新的周期。袁国顺在接受采访时表示：请广大企业家不要认为免费就是打劫、白送，而是根据你的企业产品特性，设定一个免费的时间段做铺垫，调试时间的浓度来让客户形成惯性消费行为，最终变成你的企业的忠实客户。来自全国各地近600个企业家参加峰会连续三天的“顶尖免费商业模式”学习。其中不乏广东商业巨头，如广药集团高管、喜喜传媒、国龙集团高管等。在为期三天的峰会中，还举办了壹玖资本广东分公司的启动仪式。广东壹玖资本负责人汪先付在接受采访时表示，“我们实战峰会灌输的是一种商业模式：弱化员工能力，强化模式的价值。”而公司的互助经济大平台不是简单的企业家培训、关系的抱团，而是默契的利益共同体。平台里的企业家必须有共同的理念和价值观，“免费模式”只是系统里的一环，它要实现的并不是真正的免费，而是通过免费来延伸利润链条。

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崇和实业宣传册文案市场部 2014年7月17日走近崇和

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