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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 26 — Dec. 12, 2011
  • pp: B440–B451
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Analytical results on channel capacity in uncompensated optical links with coherent detection

G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri  »View Author Affiliations


Optics Express, Vol. 19, Issue 26, pp. B440-B451 (2011)
http://dx.doi.org/10.1364/OE.19.00B440


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Abstract

Based on a recently introduced model of non-linear propagation, we propose analytical formulas for the capacity limit of polarization-multiplexed ultra-dense WDM uncompensated coherent optical systems at the Nyquist limit, assuming both lumped and ideally distributed amplification. According to these formulas, capacity fundamentally depends on the transmitted power spectral density and on the total optical WDM bandwidth, whereas it does not depend on symbol-rate. Also, capacity approximately decreases by 2 [bit/s/Hz] for every doubling of link length. We show examples of capacity calculations for specific ultra-long-haul links with different polarization-multiplexed (PM) constellations, i.e. ideal PM-Gaussian, PM-QPSK (quadrature-phase shift keying) and PM-QAM (quadrature amplitude modulation). We show that the launch power maximizing capacity is independent of link length and modulation format. We also discuss the usable range of PM-QAM systems and validate analysis with simulations.

© 2011 OSA

1. Introduction

Thanks to the revolutionary breakthrough in the field of optical fiber transmission introduced by coherent detection, the last few years have been characterized by an escalation of new records in terms of transmission distance, spectral efficiency (SE) and/or total capacity. This has been made possible by the use of high-order constellations and/or advanced spectral shaping techniques [1

1. A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Spectrally efficient long-haul WDM transmission using 224-Gb/s polarization-multiplexed 16-QAM,” J. Lightw. Technol. 29, 373–377 (2011). [CrossRef]

6

6. E. Torrengo, R. Cigliutti, G. Bosco, G. Gavioli, A. Alaimo, A. Carena, V. Curri, F. Forghieri, S. Piciaccia, M. Belmonte, A. Brinciotti, A. La Porta, S. Abrate, and P. Poggiolini, “Transoceanic PM-QPSK Terabit superchannel transmission experiments at baud-rate subcarrier spacing,” Proc. European Conference on Optical Communication (2010), paper We.7.C.2. [CrossRef]

], together with distributed amplification and new generation fibers. How far this process can be pushed depends on the actual ultimate performance limits of optical fiber transmission.

Using this model and assuming the optimal Gaussian constellation [21

21. S. Benedetto and E. Biglieri, Principles of digital transmission: with wireless applications (New York: Kluwer, 1999).

], we first write a closed-form expression of the maximum optical channel capacity at the Nyquist limit for systems employing either distributed or EDFA amplification. We then assess the capacity of realistic hard and soft-decision PM-QAM formats, showing a few examples of application of the capacity formulas. We also assess the capacity of hard-decision PM-QAM formats at larger than symbol-rate frequency spacings and with non-rectangular spectra, resorting to the integral formula for the impact of non-linearity provided in [14

14. P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “A simple and accurate model for non-linear propagation effects in uncompensated coherent transmission links,” Proc. International Conference on Transparent Optical Networks (2011), paper We.B1.3

, 15

15. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” submitted to IEEE J. Lightwave Technol.

]. In this case, we validate the capacity results through a comparison with numerical simulations.

This paper is structured as follows: in Sect. 2 we recall relevant results from [13

13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011). [CrossRef]

15

15. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” submitted to IEEE J. Lightwave Technol.

], and extend them as needed. In Sect. 3 we derive the capacity limits for the case of lumped and ideally distributed amplification and point out certain key properties of such expressions. In Sect. 4 we discuss how to calculate the capacity of specific transmission formats, such as polarization-multiplexed (PM) quadrature phase-shift-keying (PM-QPSK) and PM quadrature-amplitude-modulation (PM-QAM). In Sect. 5 we provide a few examples, discussing key aspects of capacity optimization and identify the optimal usability distance ranges for various practical formats, highlighting BER-capacity relationships and ideal FEC requirements. Finally, we draw some conclusions.

2. Non-linear interference modeling

Based on this assumption, the system bit error rate (BER) then depends on a generalized OSNR, defined so as to include NLI noise as follows:
OSNRNL=PTx,chPASE+PNLI
(1)
where PTx,ch is the average per-channel power and PASE is the dual-polarization ASE noise power within the OSNR noise bandwidth Bn. The NLI noise power PNLI can be written as:
PNLI=GNLIBn
(2)
where GNLI is the PSD of NLI. The above formula assumes that GNLI is locally ‘white’. This is not true in general, but it is very well verified at the Nyquist limit, especially over the center channel of the WDM comb.

In [13

13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011). [CrossRef]

] an approximated but very accurate expression of PNLI at the Nyquist limit was provided for the case of lumped (EDFA) amplification:
PNLIEDFA(23)3Nsγ2LeffPTx,ch3ln(π2|β2|LeffNch2Rs2)π|β2|Rs3Bn
(3)
where Ns is the number of spans, γ is the fiber nonlinearity coefficient, Nch is the number of WDM channels and β2 is fiber dispersion. Leff is the effective length, defined as: Leff = [1 – exp(−2αLs)]/(2α), with α the fiber loss coefficient and Ls the span length.

As for PASE in Eq. (1), the standard formulas:
PASEEDFA=NsF(e2αLs1)hνBn
(5)
and
PASEDA=4αLtothνKTBn
(6)
can be used, where F is the EDFA amplifiers noise figure, h is the Plank’s constant, ν is the center frequency of the WDM comb and KT ≥ 1 is a constant which is approximately equal to 1.13 for realistic Raman amplification [12

12. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

].

Regarding the EDFA amplification scenario, the accuracy of performance prediction based on Eqs. (1)(3) at the Nyquist limit was extensively checked for PM quadrature phase-shift keying (PM-QPSK) in [13

13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011). [CrossRef]

]. A more comprehensive test of the NLI model, encompassing PM-BPSK (binary phase-shift keying), PM-QPSK, PM-8QAM and PM-16QAM (quadrature-amplitude modulation with 8 and 16 symbols, respectively), was carried out in [14

14. P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “A simple and accurate model for non-linear propagation effects in uncompensated coherent transmission links,” Proc. International Conference on Transparent Optical Networks (2011), paper We.B1.3

], [15

15. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” submitted to IEEE J. Lightwave Technol.

], addressing also frequency spacings larger than Rs and three different fiber types. All these tests were performed using simulations based on direct error counting at the receiver. A very good agreement was found throughout. In particular, the model proved accurate for all the tested formats, confirming, as its derivation appears to imply [14

14. P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “A simple and accurate model for non-linear propagation effects in uncompensated coherent transmission links,” Proc. International Conference on Transparent Optical Networks (2011), paper We.B1.3

], [15

15. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” submitted to IEEE J. Lightwave Technol.

], that it should be valid with any coherent format of any cardinality. If so, it can then be used together with the ideally Gaussian constellation which provides the maximum achievable capacity [12

12. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

], [25

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

].

Note that a first experimental model validation, based on PM-QPSK, was recently presented in [20

20. E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri,“Experimental validation of an analytical model for nonlinear propagation in uncompensated pptical links,” Proc. European Conference on Optical Communication (2011), paper We.7.B.2.

]. The experiment too showed good agreement with the model, on three different fiber types.

3. Optical channel capacity with Gaussian constellation at the Nyquist limit

Resorting to Shannon’s formula [25

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

] for the unconstrained additive white-Gaussian-noise (AWGN) channel capacity C = log2(1 + SNR) [bit/s/Hz], where SNR is the ratio between the average signal power and the noise variance at the Rx decision stage, it is then possible to derive a similar formula for the polarization-multiplexed (PM) optical channel:
C=2RsΔflog2(1+SNR)[bit/symbol]
(7)
with:
SNR=BnRsOSNRNL.
(8)
The relationship between SNR and OSNRNL assumes matched electrical filtering. Note that since typical Rx adaptive equalizers tend towards matched filtering, this condition is also a realistic one.

Expressing OSNRNL according to Eqs. (1)(6), we obtained the following expressions of the non-linear optical channel capacity for both the EDFA and DA case:
CEDFA=2log2(1+GTxNs(a+bGTx3))
(9)
CDA=2log2(1+GTxLtot(c+dGTx3))
(10)
where the signal PSD GTx = PTx,ch/Rs is a sort of “average energy” of the WDM comb and a (or c) and b (or d) are related, respectively, to the linear and nonlinear noise contributions through the following relationships:
a=(e2αLs1)Fhν,b=(23)3γ2Leffln(π2|β2|LeffBWDM2)π|β2|c=4αhνKT,d=(23)3γ2ln(π2|β2|LtotBWDM2)π|β2|
(11)
where BWDM = Nch · Rs is the total WDM optical bandwidth.

Eqs.(9)(11) provide a closed-form capacity limit for the dual-polarization non-linear optical channel with UT, that is a sort of “optical non-linear Shannon-limit”. They also emphasize the fact that capacity does not depend on the symbol-rate, as long as the signal PSD and WDM bandwidth remain constant. Capacity decreases as BWDM is increased, though weakly due to the logarithm. This behavior is characteristic not only of the PM-Gaussian constellation which yields the ultimate capacity limits, but it can be shown to hold for any PM coherent format with UT. For instance, for PM-QPSK at the Nyquist limit a similar performance invariance vs. the symbol-rate was found in [13

13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011). [CrossRef]

], though in terms of BER rather than capacity. A simulative validation was provided too.

In Fig. 1 we plot the capacity limit in bit/symbol (equivalent, at the Nyquist limit, to bit/s/Hz) vs. the launch power per channel. We assume ideal DA with KT = 1 (Fig. 1a) or EDFA amplification with F=5 dB and Lspan=100 km (Fig. 1b). 125 channels at 32 GBaud each are considered, covering a total optical bandwidth BWDM of 4 THz (approximately the C band). Each curve refers to a different system length, from 500 to 8000 km for DA and from 100 to 1600 km for EDFA amplification. The fiber is standard single-mode (SSMF) with same parameters as in [12

12. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

]: γ = 1.27 1/W/km, α = 0.22 dB/km, β2 = −21.7 ps2/km.

Fig. 1 Capacity limit vs. launch power per channel at different system lengths with ideal distributed-amplification (a) and EDFA amplification with F=5 dB, Ls=100 km (b). Assumptions: UT and PM-Gaussian constellation, 125 channels at 32 GBaud, channel spacing equal to symbol-rate, resulting in a total optical bandwidth of 4 THz. Dashed lines: Shannon limit - Eq. (7). Solid lines: non-linear capacity limit - Eq. (9),(10).

Figure 1 appears to be similar to Fig. 3 in [12

12. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

]. Our values are slightly less than twice those in [12

12. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

]. The doubling stems from the fact that we assume dual-polarization, whereas [12

12. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

] assumes single. Also, Eqs. (9)(10) include cross-polarization interference, which could in part justify values slightly lower than double, together with the fact that [12

12. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

] used (single-channel) backward propagation non-linear compensation in the receiver simulation.

Fig. 3 Maximum capacity vs. transmission distance, over SSMF with span length 100 km and EDFA noise Fig. 5 dB. Solid curves: soft decision. Dashed curved: hard decision. WDM transmission over the whole C-band at the Nyquist limit (total optical bandwidth BWDM=4 THz, symbol-rate spacing, rectangular spectra).

4. Optical channel capacity for realistic constellations

To obtain capacity estimates for generic PM coherent formats in UT links, we resorted to the standard formulas of capacity over AWGN [21

21. S. Benedetto and E. Biglieri, Principles of digital transmission: with wireless applications (New York: Kluwer, 1999).

], [25

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

], specific of each format, adapting them to the polarization-multiplexed optical channel. Assuming that all symbols have the same a priori probability, the channel capacity for hard-decision is given by:
Chard=2RsΔf1MaX,bYPY|X(b|a)log2PY|X(b|a)PY(b)
(15)
where M is the number of constellation points, X = {x1,..., xM} is the set of possible transmitted symbols,Y = {y1,..., yM} is set of the output symbols after hard-decision, PY|X (b|a) is the probability of receiving the symbol b when the symbol a has been transmitted, PY (b) is the probability of receiving the symbol b.

Similarly, assuming soft-decision, we get:
Csoft=2RsΔf1MaXpY|X(y|a)log2pY|X(y|a)pY(y)
(16)
where y is the soft value at the output of the channel, pY|X (y|a) is the probability density function of the random variable y, pY (y), conditioned to the transmission of the symbol a.

Assuming transmission over an AWGN channel and using standard information theory results [21

21. S. Benedetto and E. Biglieri, Principles of digital transmission: with wireless applications (New York: Kluwer, 1999).

], it is possible to evaluate analytically all transition probabilities in Eqs. (15)(16) in terms of SNR at the receiver. In a PM optical channel, the values of SNR can be derived, through Eq. (8), from the generalized OSNR of Eq. (1). When the channel spacing Δf is equal to the symbol-rate Rs, the amount of PNLI, required to estimate the OSNRNL, is evaluated using Eqs. (3)(4). If Δf > Rs, PNLI can be obtained by resorting to the integral formula provided in [13

13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011). [CrossRef]

15

15. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” submitted to IEEE J. Lightwave Technol.

], solving it by numerical integration.

5. Examples of application of the capacity formulas

5.1. Terrestrial link with EDFA amplification - Nyquist limit

Fig. 2 Capacity vs. launch power spectral density after 1000 km (a–c) and 5000 km (b–d) over SSMF with span length 100 km. WDM transmission over the whole C-band (BWDM=4 THz) at the Nyquist limit (rectangular spectra with spacing Δf = Rs). Markers: curve maxima.

Notice the somewhat counter-intuitive result, which is found for any fiber length or system parameters, that higher-cardinality constellations potentially deliver a higher net capacity than lower-cardinality ones (though needing more complex FECs to do so). The same result is found in Shannon’s theory for the AWGN channel and the reason why the non-linear UT optical channel behaves similarly is that NLI noise can be approximated as AWGN as well.

Note however that the capacity maximum is reached at very different BER values among formats. It is in fact their high pre-FEC BER values that make PM-16/64QAM capacity substantially lower than ideal. The values of pre-FEC BER corresponding to the maximum capacity points (diamond markers in Fig. 2) are shown in Table 1. Notice that pre-FEC values are obviously hard-decision-based, and therefore they are the same, whether the subsequent FEC is either hard or soft. Using soft-decision FECs, higher values of capacity can be achieved with respect to hard-decision FECs, essentially because soft-FECs need a lower redundancy than hard-FECs to obtain virtually error-free, but the advantage is quite moderate: as an example, at 1000 km the maximum capacity increases from 7.6 to 7.8 bits/symbol for PM-16QAM and from 8.2 to 9 bits/symbol for PM-64QAM.

Table 1. Raw (pre-FEC) BER values corresponding to the maximum capacity points (diamond markers in Fig. 2).

table-icon
View This Table

Note also that the ratio between the lost (hard or soft) capacity and the maximum capacity of a format corresponds to the minimum required ideal (hard or soft) FEC overhead necessary to obtain an arbitrarily low BER [21

21. S. Benedetto and E. Biglieri, Principles of digital transmission: with wireless applications (New York: Kluwer, 1999).

]. When using soft decision, the loss in capacity is lower, meaning that a lower FEC overhead is required than in the hard decision case. Practical FECs of course need higher overheads, although state-of-the-art FECs come rather close to the minimum required overhead.

5.2. Terrestrial link with EDFA amplification - 32 Gbaud with 50 GHz spacing

Figure 4 addresses a more realistic case of transmission of 32-Gbaud channels with 50 GHz spacing, where the number of channels was set equal to 11 in order to be able to validate the results through numerical simulations (dots in figure). Capacity was predicted using Eq. (7), together with Eq. (4) in [13

13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011). [CrossRef]

]. The latter provides PNLI when the spacing is different from the symbol-rate, and requires numerical integration. Note that the plateau values of channel capacity in Fig. 4 are lower than ideal due to the loss of spectral efficiency induced by the channel spacing being larger than the symbol-rate (Rsf = 32/50 = 0.64). The simulation dots were obtained by estimating the capacity from BER values [12

12. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

], [21

21. S. Benedetto and E. Biglieri, Principles of digital transmission: with wireless applications (New York: Kluwer, 1999).

] and show a good agreement with the predicted capacity. This confirms the accuracy of the NLI model [13

13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011). [CrossRef]

15

15. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” submitted to IEEE J. Lightwave Technol.

] and, as a consequence, of capacity estimation based on it.

Fig. 4 Maximum hard-decision capacity vs. transmission distance for 11 channels at 32 Gbaud and 50 GHz spacing over SSMF with span length 100 km and EDFA noise figure 5 dB. Solid curves: analytical prediction. Dots: simulations.

Note that, at any given distance the capacity values of Fig. 3 are always larger than those of Fig. 4, which shows that from the viewpoint of maximizing capacity, tighter spectral packing should be pursued: the greater impact of non-linearity is not enough to offset the capacity gain.

5.3. Submarine link with EDFA amplification and PSCF fiber - Nyquist limit

Fig. 5 Maximum capacity vs. transmission distance, over PSCF with span length 50 km and EDFA noise figure 5 dB. Solid curves: soft decision. Dashed curved: hard decision. WDM transmission over the whole C-band at the Nyquist limit (total optical bandwidth BWDM=4 THz, symbol-rate spacing, rectangular spectra).

Obviously, decreasing the span length allows to increase the maximum reachable distance for all modulation formats. This plot shows that, potentially, also PM-16QAM could reach ultra-long-haul distances (beyond 8,000 km with 20% hard-FEC overhead) over submarine-like links.

6. Conclusion

We have derived simple analytical capacity-limit formulas, based on a recently proposed nonlinear propagation model for uncompensated transmission, both with lumped and ideal distributed amplification. We have also shown how to evaluate the capacity of any specific transmission format at the Nyquist limit. The found results are consistent with the simulation results in [12

12. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

].

Acknowledgments

This work was supported by CISCO Systems (SRA contract) and by the EURO-FOS project, a Network of Excellence funded by the European Commission through the 7th ICT-Framework Programme. The simulator OptSim was supplied by RSoft Design Group Inc.

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H. Louchet, A. Hodzic, and K. Petermann, “Analytical model for the performance evaluation of DWDM transmission systems,” IEEE Photon. Technol. Lett. 15, 1219–1221 (2003). [CrossRef]

20.

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri,“Experimental validation of an analytical model for nonlinear propagation in uncompensated pptical links,” Proc. European Conference on Optical Communication (2011), paper We.7.B.2.

21.

S. Benedetto and E. Biglieri, Principles of digital transmission: with wireless applications (New York: Kluwer, 1999).

22.

G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Proc. European Conference on Optical Communication (2011), paper We.7.B.3.

23.

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. Tapia Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” Proc. European Conference on Optical Communication (2010), paper P4.07. [CrossRef]

24.

F. Vacondio, C. Simonneau1, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. European Conference on Optical Communication (2011), paper We.7.B.1.

25.

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

26.

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri,“Performance prediction for WDM PM-QPSK transmission over uncompensated links,” Proc. Optical Fiber Communication Conference (2011), paper OThO7.

27.

A. Bononi and E. Grellier, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19, 12781–12788 (2011). [CrossRef] [PubMed]

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems
(060.4080) Fiber optics and optical communications : Modulation

ToC Category:
Transmission Systems and Network Elements

History
Original Manuscript: September 30, 2011
Revised Manuscript: November 16, 2011
Manuscript Accepted: November 16, 2011
Published: November 22, 2011

Virtual Issues
European Conference on Optical Communication 2011 (2011) Optics Express

Citation
G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, "Analytical results on channel capacity in uncompensated optical links with coherent detection," Opt. Express 19, B440-B451 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-26-B440


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References

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  13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23, 742–744 (2011). [CrossRef]
  14. P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “A simple and accurate model for non-linear propagation effects in uncompensated coherent transmission links,” Proc. International Conference on Transparent Optical Networks (2011), paper We.B1.3
  15. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” submitted to IEEE J. Lightwave Technol.
  16. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express15, 15777–15810 (2008). [CrossRef]
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  19. H. Louchet, A. Hodzic, and K. Petermann, “Analytical model for the performance evaluation of DWDM transmission systems,” IEEE Photon. Technol. Lett.15, 1219–1221 (2003). [CrossRef]
  20. E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri,“Experimental validation of an analytical model for nonlinear propagation in uncompensated pptical links,” Proc. European Conference on Optical Communication (2011), paper We.7.B.2.
  21. S. Benedetto and E. Biglieri, Principles of digital transmission: with wireless applications (New York: Kluwer, 1999).
  22. G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Proc. European Conference on Optical Communication (2011), paper We.7.B.3.
  23. A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. Tapia Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” Proc. European Conference on Optical Communication (2010), paper P4.07. [CrossRef]
  24. F. Vacondio, C. Simonneau1, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. European Conference on Optical Communication (2011), paper We.7.B.1.
  25. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J.27, 379–423 (1948).
  26. G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri,“Performance prediction for WDM PM-QPSK transmission over uncompensated links,” Proc. Optical Fiber Communication Conference (2011), paper OThO7.
  27. A. Bononi and E. Grellier, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express19, 12781–12788 (2011). [CrossRef] [PubMed]

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