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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 13 — Jun. 20, 2011
  • pp: 12781–12788
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Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise

Edouard Grellier and Alberto Bononi  »View Author Affiliations


Optics Express, Vol. 19, Issue 13, pp. 12781-12788 (2011)
http://dx.doi.org/10.1364/OE.19.012781


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Abstract

By assuming the nonlinear noise as a signal-independent circular Gaussian noise, a typical case in non-dispersion managed links with coherent multilevel modulation formats, we provide several analytical properties of a new quality parameter – playing the role of the signal to noise ratio (SNR) at the sampling gate in the coherent receiver – which carry over to the Q-factor versus power (or “bell”) curves. We show that the maximum Q is reached at an optimal power, the nonlinear threshold, at which the amplified spontaneous emission (ASE) noise power is twice the nonlinear noise power, and the SNR penalty with respect to linear propagation is 10 Log ( 3 2 ) 1.76 dB , , although the Q-penalty is somewhat larger and increases at lower Q-factors, as we verify for the polarization-division multiplexing quadrature phase shift keying (PDM-QPSK) format. As we vary the ASE power, the maxima of the SNR vs. power curves are shown to slide along a straight-line with slope ≃−2 dB/dB. A similar behavior is followed by the Q-factor maxima, although for PDM-QPSK the local slope is around −2.7 dB/dB for Q-values of practical interest.

© 2011 OSA

1. Introduction

It has recently been shown that, in non-dispersion managed (NDM) systems with coherent reception of multilevel signal formats, both with single polarization and with polarization division multiplexing (PDM), both for single channel and for multichannel propagation, the nonlinear noise [1

1. In the following, we will call noise both the nonlinear perturbations coming from the same channel, which should more properly be called distortions, and cross-channel nonlinear perturbations.

] at the sampling gate can indeed be treated as a signal-independent noise with circular Gaussian statistics [2

2. 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. ECOC’10, paper P4.07.

4

4. G. Bosco, V. Curri, A. Carena, P. Poggiolini, and F. Forghieri, “On the performance of Nyquist-WDM terabit superchannels based on PM-BPSK, PM-QPSK, PM-8QAM or PM-16QAM subcarriers,” J. Lightwave Technol. 29, 53–61 (2011). [CrossRef]

]. While such a Gaussian approximation had already been proposed for other dispersion-managed systems but with limited accuracy [5

5. H. Louchet, A. Hodzic, K. Petermann, A. Robinson, and R. Epworth, “Analytical model for the design of multi-span DWDM transmission systems,” IEEE Photon. Technol. Lett. 17, 247–249 (2005). [CrossRef]

], or to simplify analysis in the study of nonlinear channel capacity [6

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

, 7

7. E. Narimanov and P. P. Mitra, “The channel capacity of a fiber optics communication system: perturbation theory,” J. Lightwave Technol. 20, 530–537 (2002). [CrossRef]

], the novelty is that in NDM the Gaussian approximation becomes excellent [2

2. 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. ECOC’10, paper P4.07.

, 3

3. P. Ramanatanis and Y. Frignac, “Pattern-dependent nonlinear impairments on QPSK signals in dispersion-managed optical transmission systems,” Proc. ECOC’10, paper Mo.1.C.4.

].

Assuming that the Gaussian nonlinear noise power scales as the cube of the signal power, we analytically derive the main properties of the quality parameter, namely:
  1. its asymptotic low- and high-power behavior: we prove that S vs. power increases with a slope of ≃1 dB/dB in the low-power region of operation, and a slope of ≃−2 dB/dB in the high-power region;
  2. we find an expression of the power that maximizes S, and thus Q, the so-called nonlinear threshold (NLT), as well as the S value at NLT. We prove that at NLT the nonlinear noise power is half the linear noise power, and that the SNR penalty with respect to the linear case is 10Log(3/2) ≃1.76 dB. Such a value of 1.76 dB has indeed been observed from simulations [4

    4. G. Bosco, V. Curri, A. Carena, P. Poggiolini, and F. Forghieri, “On the performance of Nyquist-WDM terabit superchannels based on PM-BPSK, PM-QPSK, PM-8QAM or PM-16QAM subcarriers,” J. Lightwave Technol. 29, 53–61 (2011). [CrossRef]

    ].
  3. we prove that, as the power of amplified spontaneous emission (ASE) noise is varied, the maxima of the S-vs-power curve move along a straight line of slope ≃−2 dB/dB shifted by 5Log(3) ≃ 2.38 dB towards lower powers with respect to the high-power asymptote.

Similar laws are then shown to extend to the Q-factor. While this paper was under review, we became aware of very similar work presented by Bosco et al. [8

8. G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance prediction for WDM PM-QPSK transmission over uncompensated links,” in Proc. OFC’11, paper OThO7.

], who however did not explore the nonlinear relationship between the S parameter and the Q-factor.

2. Signal detection model

In such an additive Gaussian noise channel, we shall extend the conventional electrical signal to noise ratio (SNR) at the decision gate by including both linear noise and nonlinear noise, and propose the following new quality parameter:
S=PNA+aNLP3.
(2)

3. Analytical properties of the new quality parameter

We now wish to derive the properties of S versus P at a fixed transmission distance, which are summarized in Fig. 1.

Fig. 1 New quality parameter S versus power P, for two values of ASE power NA differing by 3 dB (solid lines). Dashed lines indicate the left and right asymptotes [Eqs. (3), (4)]. Breakpoints are marked with squares. Maxima are marked with circles, and their vertical and horizontal distance from the linear left asymptote is 1.76 dB. As NA is changed, the maxima slide along the shown dash-dotted line with slope −2 dB/dB.

From Eq. (2) we first notice that there are two asymptotic regimes. At low power, when NAaNLP 3, the asymptotic behavior is SPNASL which is the linear SNR. At large powers when NAaNLP 3, the asymptotic behavior is SRPaNLP3. The break-point power discriminating these two regimes is PB=(NAaNL)13. At break-point, ASE power equals nonlinear noise power. The left and right asymptotes in dB become:
SL,dBPdBNA,dBifPPB
(3)
SR,dB2PdBaNL,dBifPPB
(4)
i.e., the left asymptote has slope 1 dB/dB, while the right asymptote has slope −2 dB/dB. Figure 1 shows a sketch of the new quality parameter versus P for two values of ASE power NA, where the two asymptotes are indicated with dashed lines that meet at the breakpoint, marked with a square.

By factoring out NA in the denominator, Eq. (2) can be rearranged as
S=SL1+aNLP3NA
(5)
and therefore the SNR penalty in linear units is
SP=1+aNLP3NA
(6)
which, in dB units, expresses the (vertical/horizontal) distance of the solid S curves in Fig. 1 from the dashed linear asymptote SL.

Fact 1: power at maximum S

It is customary to define the nonlinear threshold (NLT) as the power PNLT that maximizes the bell curve. Such a power is found when dSdP=0. Since dSdP=(NA+aNLP3)P3aNLP2(NA+aNLP3)2, it is seen that the numerator vanishes when
NA=2(aNLP3)
(7)
i.e., at optimal power ASE noise variance is twice the nonlinear noise variance. The NLT power is
PNLT=(NA2aNL)13
(8)
i.e., NLT is 10Log21/3 ≃ 1 dB below the break-point. The maximum S value is reached at NLT:
SNLT=PNLT32NA=(33aNL(NA2)2)13.
(9)

Fact 2: locus of maxima when varying NA

For any link, from Eq. (8) we see that at each doubling of ASE power NA, the NLT PNLT increases by 1dB, and from Eq. (9) the maximum value SNLT decreases by 2 dB. This is exemplified in Fig. 1. Another interesting property shown in Fig. 1 is that the maxima, as we vary NA, slide along a straight-line (dash-dotted magenta line) parallel to the right asymptote, shifted to lower powers by 10Log(3)22.38dB. The proof is simple: since at NLTNA=2aNLPNLT3, then from Eq. (2) we get SNLT=PNLT3aNLPNLT3, which is the right asymptote SR lowered vertically by 10Log(3)≃4.7 dB, i.e., horizontally by 4.72 dB since the slope of SR is −2 dB/dB.

4. Simulation checks

Fig. 2 Q-factor versus SNR for a 28 Gbaud PDM-QPSK signal and DSP-based coherent receiver. Symbols: Monte-Carlo simulations. Solid line: parablic fit Eq. (11).

At NLT we have dQdP=dQdSdSdP=0, i.e., the maximum Q-factor is also reached at NLT. Maximization of the Q-factor can therefore be performed by maximizing the quality parameter S.

To verify this statement, and validate the theory on the parameter S here presented, we performed numerical simulations of nonlinear system performance using the split step Fourier method, with power-adaptive step size of 1/1000 the nonlinear length. We considered the transmission of seven WDM channels modulated at 112Gb/s PDM-QPSK and with 50GHz channel spacing. Channels were modulated with different pseudo-random quaternary sequences (one for each polarisation) of 16384 symbols. The supporting pulses were non-return to zero. The NDM line consisted of 12 uncompensated 100 km spans of single mode fiber (SMF) with dispersion −17 ps/nm/km, attenuation 0.22 dB/km, nonlinear coefficient 1.32 W−1km−1, and zero dispersion slope and zero polarization mode dispersion. Noise was loaded at the receiver, thus neglecting nonlinear singal-noise interactions, which are known to be negligible in NDM SMF lines at 100 Gb/s [9

9. A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal-noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol. 16, 73–85 (2010). [CrossRef]

, 10

10. A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” Proc. ECOC’10, paper Th.10.E.1.

]. The BER of the DSP coherent receiver was estimated from Monte-Carlo error counting stopped after 400 counts. After obtaining the estimated BER from simulations, the Q-factor was derived by inversion of Eq. (10).

Fig. 3 Q 2 (left) and SNR S (right) vs. channel power P for an SMF NDM 12x100 km link and 7 channels with 112Gb/s PDM-QPSK modulation on a 50 GHz grid, for NA = [−10, −9.2, −8.4] dBm. Symbols: simulations. Solid lines: Analytical best fit. Left and right asymptotes and locus of maxima are also shown for reference.

Finally, fitted analytical S values were converted to fitted Q values using Eq. (11), as shown in solid lines in Fig. 3(left).

Using the fitted NA and aNL values, we plotted in dashed black lines in Fig. 3(right) both the linear asymptotes SL,dB (shown only for the top and bottom data sets) and the nonlinear asymptote SR,dB. We also plotted the locus of maxima of coordinates given by Eqs. (8), (9) as a dash-dotted magenta line, which is a straight line with slope −2 dB/dB. The same asymptotes and locus of maxima, after warping through Eq. (11), were plotted in Fig. 3(left). In this case the locus of maxima has a parabolic shape, and if linearized around the shown 3 maxima (red circles, white filled) it corresponds to a line with slope ∼−2.7 dB/dB. The linear asymptotes allow an appreciation of the SNR penalty at NLT, which is confirmed to be very close to the theoretical SPdB = 1.76 dB. Also the Q-penalty at NLT can be appreciated as the distance from the linear asymptotes to the top of the bell curve, and its numerical values are plotted as symbols in Fig. 4 for ASE power varied over the range NA = −10 : −8 dBm in steps of 0.4 dB. It is seen that the Q-penalty is around 2 dB at lower Q values at NLT (large NA), and remains above 1.8 dB over the measured range. Using the parabolic fit Eq. (11), it is easy to see that the Q-penalty at NLT has equation
QPdB=SPdB[BA(SPdB+2(SdB+b)]
(12)
where S at NLT is given in Eq. (9). Such a formula is also plotted in Fig. 4 in solid line. However, the parabolic fit Eq. (11) is accurate up to Q-values of about 14 dB, and beyond such value it underestimates the Q-factor. Hence Eq. (12) ceases to hold at very small NA (where the linear Q exceeds 14 dB). The true Q-penalty at NLT will asymptotically decrease to 1.76 dB as NA decreases, i.e., Q-factor at NLT increases.

Fig. 4 Q-penalty at NLT vs. Q-factor at NLT for 28 Gbaud PDM-QPSK signal and DSP coherent receiver. Symbols: Monte-Carlo simulations. Solid line: Eq. (12). aNL = 0.0066 (mW)−2.

5. Conclusions

We have exploited an elementary Gaussian nonlinear model for the received signal field in coherent transmissions, in order to analytically prove the salient features of the bell curves of Q-factor versus transmitted channel power. Such a model holds whenever the line strength is large enough that nonlinear-signal noise interactions (a manifestation of which is nonlinear phase noise) are weak [9

9. A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal-noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol. 16, 73–85 (2010). [CrossRef]

, 10

10. A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” Proc. ECOC’10, paper Th.10.E.1.

], and the received field statistics are circular complex Gaussian. The model establishes that at maximum Q the ASE power is twice that of the nonlinear noise, yielding an SNR penalty of 1.76 dB from back to back, and a slightly larger Q-penalty, which approaches 2dB at smaller NLT Q-values for a 28 Gbaud PDM-QPSK format. As we change the linear noise power, the locus of maxima of SNR versus power slide along a straight-line with slope ≃−2 dB/dB, while the corresponding slope for the Q-factor of a 28 Gbaud PDM-QPSK modulation is around −2.7 dB/dB. While we proposed here the SNR S as a transmission quality parameter, it is worth mentioning that in wireless communications with in-phase/quadrature modulation formats the sum of noise, co-channel and cross-channel interference (i.e., linear plus nonlinear noise in our parlance) is known as the error vector, and the standard deviation of the error vector (called the error vector magnitude, EVM) is often used as a design parameter (see, e.g., [13

13. A. Georgiadis “Gain, phase imbalance, and phase noise effects on error vector magnitude,” IEEE Trans. Veh. Technol. 53, 443–449 (2004). [CrossRef]

]), even when the error vector statistics are not necessarily neither Gaussian nor signal-independent, as they approximately are instead in NDM links.

Acknowledgments

The authors are glad to acknowledge fruitful discussions with S. Bigo, O. Rival, F. Vacondio of Alcatel-Lucent, and P. Serena of Parma University.

References and links

1.

In the following, we will call noise both the nonlinear perturbations coming from the same channel, which should more properly be called distortions, and cross-channel nonlinear perturbations.

2.

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. ECOC’10, paper P4.07.

3.

P. Ramanatanis and Y. Frignac, “Pattern-dependent nonlinear impairments on QPSK signals in dispersion-managed optical transmission systems,” Proc. ECOC’10, paper Mo.1.C.4.

4.

G. Bosco, V. Curri, A. Carena, P. Poggiolini, and F. Forghieri, “On the performance of Nyquist-WDM terabit superchannels based on PM-BPSK, PM-QPSK, PM-8QAM or PM-16QAM subcarriers,” J. Lightwave Technol. 29, 53–61 (2011). [CrossRef]

5.

H. Louchet, A. Hodzic, K. Petermann, A. Robinson, and R. Epworth, “Analytical model for the design of multi-span DWDM transmission systems,” IEEE Photon. Technol. Lett. 17, 247–249 (2005). [CrossRef]

6.

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

7.

E. Narimanov and P. P. Mitra, “The channel capacity of a fiber optics communication system: perturbation theory,” J. Lightwave Technol. 20, 530–537 (2002). [CrossRef]

8.

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance prediction for WDM PM-QPSK transmission over uncompensated links,” in Proc. OFC’11, paper OThO7.

9.

A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal-noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol. 16, 73–85 (2010). [CrossRef]

10.

A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” Proc. ECOC’10, paper Th.10.E.1.

11.

E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009). [CrossRef]

12.

G. Charlet, J. Renaudier, H. Mardoyan, P. Tran, O. Bertran Pardo, F. Verluise, M. Achouche, A. Boutin, F. Blache, J. Dupuy, and S. Bigo, “Transmission of 16.4-bit/s capacity over 2550 km using PDM QPSK modulation format and coherent receiver,” J. Lightwave Technol. 27, 153–157 (2009). [CrossRef]

13.

A. Georgiadis “Gain, phase imbalance, and phase noise effects on error vector magnitude,” IEEE Trans. Veh. Technol. 53, 443–449 (2004). [CrossRef]

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: February 28, 2011
Revised Manuscript: April 22, 2011
Manuscript Accepted: May 6, 2011
Published: June 17, 2011

Citation
Edouard Grellier and Alberto Bononi, "Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise," Opt. Express 19, 12781-12788 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12781


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References

  1. In the following, we will call noise both the nonlinear perturbations coming from the same channel, which should more properly be called distortions, and cross-channel nonlinear perturbations.
  2. 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. ECOC’10, paper P4.07.
  3. P. Ramanatanis and Y. Frignac, “Pattern-dependent nonlinear impairments on QPSK signals in dispersion-managed optical transmission systems,” Proc. ECOC’10 , paper Mo.1.C.4.
  4. G. Bosco, V. Curri, A. Carena, P. Poggiolini, and F. Forghieri, “On the performance of Nyquist-WDM terabit superchannels based on PM-BPSK, PM-QPSK, PM-8QAM or PM-16QAM subcarriers,” J. Lightwave Technol. 29, 53–61 (2011). [CrossRef]
  5. H. Louchet, A. Hodzic, K. Petermann, A. Robinson, and R. Epworth, “Analytical model for the design of multi-span DWDM transmission systems,” IEEE Photon. Technol. Lett. 17, 247–249 (2005). [CrossRef]
  6. J. Tang, “The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission,” J. Lightwave Technol. 19, 1104–1109 (2001). [CrossRef]
  7. E. Narimanov and P. P. Mitra, “The channel capacity of a fiber optics communication system: perturbation theory,” J. Lightwave Technol. 20, 530–537 (2002). [CrossRef]
  8. G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance prediction for WDM PM-QPSK transmission over uncompensated links,” in Proc. OFC’11 , paper OThO7.
  9. A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal-noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol. 16, 73–85 (2010). [CrossRef]
  10. A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” Proc. ECOC’10 , paper Th.10.E.1.
  11. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009). [CrossRef]
  12. G. Charlet, J. Renaudier, H. Mardoyan, P. Tran, O. Bertran Pardo, F. Verluise, M. Achouche, A. Boutin, F. Blache, J. Dupuy, and S. Bigo, “Transmission of 16.4-bit/s capacity over 2550 km using PDM QPSK modulation format and coherent receiver,” J. Lightwave Technol. 27, 153–157 (2009). [CrossRef]
  13. A. Georgiadis “Gain, phase imbalance, and phase noise effects on error vector magnitude,” IEEE Trans. Veh. Technol. 53, 443–449 (2004). [CrossRef]

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