APPARATUS AND METHOD FOR TRANSMITTING AND/OR RECEIVING DATA OVER A FIBER-OPTICAL CHANNEL EMPLOYING PERTURBATION-BASED FIBER NONLINEARITY COMPENSATION IN A PERIODIC FREQUENCY DOMAIN

Abstract

An apparatus for determining an interference in a transmission medium during a transmission of a data input signal according to an embodiment has a transform module configured to transform the data input signal from a time domain to a frequency domain comprising a plurality of frequency channels to obtain a frequency-domain data signal comprising a plurality of spectral coefficients, wherein each spectral coefficient is assigned to one of the frequency channels, an analysis module configured to determine the interference by determining one or more spectral interference coefficients, wherein each spectral interference coefficient is assigned to one frequency channel. The analysis module is configured to determine each spectral interference coefficient depending on the spectral coefficients, and depending on a transfer function, wherein the transfer function is configured to receive two or more argument values, wherein each of the argument values indicates one frequency channel, and wherein the transfer function is configured to return a return value depending on the argument values.

Claims

1. An apparatus for determining an interference in a transmission medium during a transmission of a data input signal, comprising: a transform module configured to transform the data input signal from a time domain to a frequency domain comprising a plurality of frequency channels to acquire a frequency-domain data signal comprising a plurality of spectral coefficients, wherein each spectral coefficient of the plurality of spectral coefficients is assigned to one of the plurality of frequency channels, and an analysis module configured to determine the interference by determining one or more spectral interference coefficients, wherein each of the one or more spectral interference coefficients is assigned to one of the plurality of frequency channels, wherein the analysis module configured to determine each of the one or more spectral interference coefficients depending on the plurality of spectral coefficients and depending on a transfer function, wherein the transfer function is configured to receive two or more argument values, wherein each of the two or more argument values indicates one of the plurality of frequency channels, and wherein the transfer function is configured to return a return value depending on the two or more argument values.

2. The apparatus according to claim 1, wherein the transmission medium is a fiber-optical channel.

3. The apparatus according to claim 1, wherein the apparatus further comprises a signal modification module being configured to modify the frequency-domain data signal using the one or more spectral interference coefficients to acquire a modified data signal, wherein the apparatus further comprises an inverse transform module configured to transform the modified data signal from the frequency domain to the time domain to acquire a corrected time-domain data signal.

4. The apparatus according to claim 3, wherein the signal modification module is configured to combine each one of the one or more spectral interference coefficients, or a value derived from said one of the one or more spectral interference coefficients, and one of the plurality of spectral coefficients to acquire the modified data signal.

5. The apparatus according to claim 3, wherein the transform module is configured to transform the data input signal from the time domain to the frequency domain by transforming a plurality of overlapping blocks of the data input signal from the time domain to the frequency domain to acquire a plurality of blocks of the frequency-domain data signal, and wherein the inverse transform module configured to transform the modified data signal from the frequency domain to the time domain by transforming a plurality of blocks from the frequency domain to the time domain and by overlapping said plurality of blocks being represented in the time domain to acquire the corrected time-domain data signal.

6. The apparatus according to claim 1, wherein the apparatus further comprises an inverse transform module configured to transform the one or more spectral interference coefficients from the frequency domain to the time domain, and wherein the apparatus further comprises a signal modification module being configured to modify the data input signal being represented in the time domain using the one or more spectral interference coefficients being represented in the time domain to acquire a corrected time-domain data signal.

7. The apparatus according to claim 6, wherein the signal modification module is configured to combine each one of the one or more spectral interference coefficients being represented in the time domain, or a value derived from said one of the one or more spectral interference coefficients, and a time domain sample of a plurality of time domain samples of the data input signal being represented in the time domain to acquire the corrected time-domain data signal.

8. The apparatus according to claim 6, wherein the transform module is configured to transform the data input signal from the time domain to the frequency domain by transforming a plurality of overlapping blocks of the data input signal from the time domain to the frequency domain to acquire a plurality of blocks of the frequency-domain data signal, and wherein the inverse transform module is configured to transform a plurality of interference coefficients blocks from the frequency domain to the time domain, said plurality of blocks comprising the one or more spectral interference coefficients, and wherein the signal modification module is configured to modify the overlapping blocks of the data input signal, being represented in the time domain, using the plurality of interference coefficients blocks to acquire a plurality of corrected blocks, wherein the signal modification module is configured to overlap the plurality of corrected blocks to acquire the corrected time-domain data signal.

9. The apparatus according to claim 3, wherein the apparatus further comprises a transmitter module configured to transmit the corrected time-domain data signal over the transmission medium.

10. The apparatus according to claim 3, wherein the apparatus further comprises a receiver module configured to receive the data input signal being transmitted over the transmission medium.

11. The apparatus according to claim 1, wherein the analysis module is configured to determine an estimation of a perturbated signal depending on the data input signal using the one or more spectral interference coefficients.

12. The apparatus according to claim 11, wherein the analysis module is configured to determine the estimation of the perturbated signal by adding each one of the one or more spectral interference coefficients with one of the plurality of spectral coefficients.

13. The apparatus according to claim 1, wherein each of the two or more argument values is a channel index being an index which indicates one of the plurality of frequency channels, or wherein each of the two or more argument values is a frequency which indicates one of the plurality of frequency channels, wherein said one of the plurality of frequency channels comprises said frequency.

14. The apparatus according to claim 1, wherein the analysis module is configured to determine each spectral interference coefficient of the one or more spectral interference coefficients by determining a plurality of addends, wherein the analysis module is configured to determine each of the plurality of addends as a product of three or more of the spectral coefficients and of the return value of the transfer function, the transfer function comprising three or more channel indices or three or more frequencies as the two or more argument values of the transfer function, which indicate three or more of the plurality of frequency channels to which said three or more of the spectral coefficients are assigned.

15. The apparatus according to claim 14, wherein the analysis module is configured to determine the interference by applying a regular perturbation approach.

16. The apparatus according to claim 14, wherein the analysis module is configured to determine each spectral interference coefficient depending on: $\delta A_{\lambda }^{\mathrm{SCI}}\left[\mu \right]=-j\frac{{\varphi }_{\mathrm{NL},\rho }}{N_{DFT}^2}\times {.Math.}_{{\mu }_1,{\mu }_2}{A}_{\lambda }\left[{\mu }_1\right]A_{\lambda }^H\left[{\mu }_2\right]A_{\lambda }\left[{\mu }_3\right]H_{\rho }\left[{\mu }_1,{\mu }_2,{\mu }_3\right]$ wherein ΔA.sub.λ.sup.SCI[μ] is said spectral interference coefficient, wherein A.sub.λ[μ.sub.1] is a first one of the three or more spectral coefficients, wherein A.sub.λ[μ.sub.2] is a second one of the three or more spectral coefficients, wherein, A.sub.λ[μ.sub.3] is a third one of the three or more spectral coefficients, wherein .sup.H indicates Hermitian, wherein μ.sub.1 is a first index which indicates a first one of the plurality of frequency channels, wherein μ.sub.2 is a second index which indicates a second one of the plurality of frequency channels, wherein μ.sub.3 is a third index which indicates a third one of the plurality of frequency channels, wherein H.sub.ρ[μ.sub.1, μ.sub.2, μ.sub.3] indicates the transfer function, wherein N.sub.DFT.sup.2 indicates a square of a number of the plurality of frequency channels of the frequency domain, wherein ϕ.sub.NL,ρ is a number.

17. The apparatus according to claim 14, wherein the analysis module is configured to determine each spectral interference coefficient depending on: $\Delta A_{\lambda }^{\mathrm{XCI}}\left[\mu \right]=-j\frac{{\varphi }_{\mathrm{NL},\nu }}{N_{\mathrm{DFT}}^2}\times {.Math.}_{{\mu }_1,{\mu }_2}\left(B_{\lambda }\left[{\mu }_1\right]B_{\lambda }^H\left[{\mu }_2\right]+B_{\lambda }^H\left[{\mu }_2\right]B_{\lambda }\left[{\mu }_1\right]I\right)\times A_{\lambda }\left[{\mu }_3\right]H_{\nu }\left[{\mu }_1,{\mu }_2,{\mu }_3\right]$ wherein ΔA.sub.λ.sup.XCI[μ] is said spectral interference coefficient, wherein B.sub.λ[μ.sub.1] is a first one of the three or more spectral coefficients, wherein B.sub.λ[μ.sub.2] is a second one of the three or more spectral coefficients, wherein, A.sub.λ[μ.sub.3] is a third one of the three or more spectral coefficients, wherein I indicates an identity matrix, wherein .sup.H indicates Hermitian, wherein μ.sub.1 is a first index which indicates a first one of the plurality of frequency channels, wherein μ.sub.2 is a second index which indicates a second one of the plurality of frequency channels, wherein μ.sub.3 is a third index which indicates a third one of the plurality of frequency channels, wherein H.sub.v, [μ.sub.1, μ.sub.2, μ.sub.3 ] indicates the transfer function, wherein N.sub.DFT.sup.2 indicates a square of a number of the plurality of frequency channels of the frequency domain, wherein ϕ.sub.NL,v is a number.

18. The apparatus according to claim 14, wherein the analysis module is configured to determine the interference by applying a regular logarithmic perturbation approach.

19. The apparatus according to claim 14, wherein the analysis module is configured to determine each spectral interference coefficient depending on: $\Delta A_{\lambda }^{\mathrm{SCI}}\left[\mu \right]=-j\frac{{\varphi }_{\mathrm{NL},\rho }}{N_{DFT}^2}\times {.Math.}_u{A}_{\lambda }\left[{\mu }_1\right]A_{\lambda }^H\left[{\mu }_2\right]A_{\lambda }\left[{\mu }_3\right]H_{\rho }\left[{\mu }_1,{\mu }_2,{\mu }_3\right]$ wherein ΔA.sub.2.sup.XCI[μ] is said spectral interference coefficient, wherein A.sub.λ[μ.sub.1] is a first one of the three or more spectral coefficients, wherein A.sub.λ[μ.sub.2] is a second one of the three or more spectral coefficients, wherein, A.sub.λ[μ.sub.3] is a third one of the three or more spectral coefficients, wherein .sup.H indicates Hermitian, wherein μ.sub.1 is a first index which indicates a first one of the plurality of frequency channels, wherein μ.sub.2 is a second index which indicates a second one of the plurality of frequency channels, wherein μ.sub.3 is a third index which indicates a third one of the plurality of frequency channels, wherein H.sub.ρ[μ.sub.1,μ.sub.2, μ.sub.3] indicates the transfer function, wherein N.sub.DFT.sup.2 indicates a square of a number of the plurality of frequency channels of the frequency domain, wherein ϕ.sub.NL,v is a number, wherein
={[μ.sub.1, μ.sub.2]|μ.sub.2≠μ.sub.1Λμ.sub.2≠μ.sub.3}.

20. The apparatus according to claim 14, wherein the analysis module is configured to determine each spectral interference coefficient depending on: $\Delta A_{\lambda }^{XCI}\left[\mu \right]=-j\frac{{\varphi }_{\mathrm{NL},\nu }}{N_{DFT}^2}\times {.Math.}_u\left(B_{\lambda }\left[{\mu }_1\right]B_{\lambda }^H\left[{\mu }_2\right]+B_{\lambda }^H\left[{\mu }_2\right]B_{\lambda }\left[{\mu }_1\right]I\right)\times A_{\lambda }\left[{\mu }_3\right]H_{\nu }\left[{\mu }_1,{\mu }_2,{\mu }_3\right]$ wherein ΔA.sub.λ.sup.XCI[μ] is said spectral interference coefficient, wherein B.sub.λ[μ.sub.1] is a first one of the three or more spectral coefficients, wherein B.sub.λ[μ.sub.2] is a second one of the three or more spectral coefficients, wherein, A.sub.λ[μ.sub.3] is a third one of the three or more spectral coefficients, wherein I indicates an identity matrix, wherein .sup.H indicates Hermitian, wherein t.sub.h is a first index which indicates a first one of the plurality of frequency channels, wherein μ.sub.2 is a second index which indicates a second one of the plurality of frequency channels, wherein μ.sub.3 is a third index which indicates a third one of the plurality of frequency channels, wherein H.sub.v, [μ.sub.1, μ.sub.2, μ.sub.3] indicates the transfer function, wherein N.sub.DFT.sup.2 indicates a square of a number of the plurality of frequency channels of the frequency domain, wherein ϕ.sub.NL,v is a number, wherein
={[μ.sub.1, μ.sub.2].sup.T|μ.sub.2≠μ.sub.1}.

21. The apparatus according to claim 1, wherein the transfer function is normalized and nonlinear.

22. The apparatus according to claim 1, wherein the analysis module is configured to employ Volterra based compensation to determine the one or more spectral interference coefficients.

23. The apparatus according to claim 1, wherein the analysis module is configured to determine the one or more spectral interference coefficients by determining one or more transmit and receive sequences over discrete frequencies from a periodic Nyquist interval.

24. The apparatus according to claim 1, wherein the transfer function depends on $H_{\nu }\left(e^{j\omega T}\right)=\frac{1}{T^3}\underset{m\in Z^3}{.Math.}{H}_{\nu }\left(\omega -\frac{2\pi m}{T}\right).$

25. A method for determining an interference in a transmission medium during a transmission of a data input signal, comprising: transforming the data input signal from a time domain to a frequency domain comprising a plurality of frequency channels to acquire a frequency-domain data signal comprising a plurality of spectral coefficients, wherein each spectral coefficient of the plurality of spectral coefficients is assigned to one of the plurality of frequency channels, and determining the interference by determining one or more spectral interference coefficients, wherein each of the one or more spectral interference coefficients is assigned to one of the plurality of frequency channels, wherein determining each of the one or more spectral interference coefficients is conducted depending on the plurality of spectral coefficients and depending on a transfer function, wherein the transfer function is configured to receive two or more argument values, wherein each of the two or more argument values indicates one of the plurality of frequency channels, and wherein the transfer function is configured to return a return value depending on the two or more argument values.

26. A non-transitory digital storage medium having stored thereon a computer program for performing a method for determining an interference in a transmission medium during a transmission of a data input signal, comprising: transforming the data input signal from a time domain to a frequency domain comprising a plurality of frequency channels to acquire a frequency-domain data signal comprising a plurality of spectral coefficients, wherein each spectral coefficient of the plurality of spectral coefficients is assigned to one of the plurality of frequency channels, and determining the interference by determining one or more spectral interference coefficients, wherein each of the one or more spectral interference coefficients is assigned to one of the plurality of frequency channels, wherein determining each of the one or more spectral interference coefficients is conducted depending on the plurality of spectral coefficients and depending on a transfer function, wherein the transfer function is configured to receive two or more argument values, wherein each of the two or more argument values indicates one of the plurality of frequency channels, and wherein the transfer function is configured to return a return value depending on the two or more argument values, when said computer program is run by a computer.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Embodiments of the present invention will be described below in more detail with reference to the appended drawings, in which:

[0019] FIG. 1a illustrates an apparatus for determining an interference in a transmission medium during a transmission of a data input signal according to an embodiment;

[0020] FIG. 1b illustrates another embodiment, wherein the apparatus further comprises a signal modification module and an inverse transform module;

[0021] FIG. 1c illustrates a further embodiment, wherein the apparatus further comprises a signal modification module and an inverse transform module;

[0022] FIG. 1d illustrates another embodiment, wherein the apparatus further comprises a transmitter module configured to transmit the corrected time-domain data signal over the transmission medium;

[0023] FIG. 1e illustrates a further embodiment, wherein the apparatus further comprises a receiver module configured to receive the data input signal being transmitted over the transmission medium;

[0024] FIG. 2 illustrates a generic fiber-optical transmission system model;

[0025] FIG. 3a illustrates a transmitter frontend of a generic fiber-optical transmission system model;

[0026] FIG. 3b illustrates an optical channel of the generic fiber-optical transmission system model;

[0027] FIG. 3c illustrates an receiver frontend of the generic fiber-optical transmission system model and variables associated with the regular perturbation model;

[0028] FIG. 4a illustrates definitions of variables in the time-domain;

[0029] FIG. 4b illustrates definitions of variables in the frequency-domain;

[0030] FIG. 5 illustrates a magnitude in logarithmic scale of a single-span nonlinear transfer function according to an embodiment;

[0031] FIG. 6 illustrates a magnitude in logarithmic scale of a single-span nonlinear transfer function according to another embodiment;

[0032] FIG. 7a illustrates a contour plot for a single-channel, single-span, lossless fiber scenario in the regular time-domain model according to an embodiment;

[0033] FIG. 7b illustrates a contour plot for a single-channel, single-span, lossless fiber scenario in the regular-logarithmic model according to another embodiment;

[0034] FIG. 8a illustrates an energy of the kernel coefficients in time-domain over the symbol rate according to an embodiment;

[0035] FIG. 8b illustrates an energy of the kernel coefficients in a frequency-domain over the symbol rate according to a further embodiment;

[0036] FIG. 9a illustrates a contour plot in the regular model in the frequency domain of the normalized mean-square error in dB for a single-channel, single-span, lossless fiber according to an embodiment;

[0037] FIG. 9b illustrates a contour plot in the regular-logarithmic model in the frequency domain of the normalized mean-square error in dB for a single-channel, single-span, lossless fiber according to an embodiment;

[0038] FIG. 10a illustrates contour plots of the normalized mean-square error σ.sub.e.sup.2 in dB according to an embodiment, wherein the results are obtained from the regular-logarithmic time-domain model over a standard single-mode fiber with end-of-span lumped amplification, and wherein the symbol rate and the optical launch power are varied for single-span transmission having a fixed roll-off factor;

[0039] FIG. 10b illustrates contour plots of the normalized mean-square error σ.sub.e.sup.2 in dB according to an embodiment, wherein the results are obtained from the regular-logarithmic time-domain model over a standard single-mode fiber with end-of-span lumped amplification, and wherein the roll-off factor and number of spans N.sub.sp are varied with fixed symbol rate having fixed launch power;

[0040] FIG. 11a illustrates an energy of the kernel coefficients in the time-domain;

[0041] FIG. 11b illustrates kernel energies Eh for cross-channel interference (XCI) imposed by a single wavelength channel;

[0042] FIG. 12a illustrates a contour plot of the normalized mean-square error in dB in a time domain, for dual-channel, single-span, lossless fiber according to an embodiment; and

[0043] FIG. 12b illustrates a contour plot of the normalized mean-square error in dB in a frequency domain, for dual-channel, single-span, lossless fiber according to another embodiment.

DETAILED DESCRIPTION OF THE INVENTION

[0044] FIG. 1 a illustrates an apparatus for determining an interference in a transmission medium during a transmission of a data input signal according to an embodiment.

[0045] The apparatus comprises a transform module 110 configured to transform the data input signal from a time domain to a frequency domain comprising a plurality of frequency channels to obtain a frequency-domain data signal comprising a plurality of spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ), wherein each spectral coefficient of the plurality of spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ), is assigned to one of the plurality of frequency channels.

[0046] Moreover, the apparatus comprises an analysis module 120 configured to determine the interference by determining one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI[μ]), wherein each of the one or more spectral interference coefficients (e. g., ΔA.sub.2.sup.SCI[μ]) is assigned to one of the plurality of frequency channels.

[0047] The analysis module 120 configured to determine each of the one or more spectral interference coefficients (e. g., ΔA.sub.λ.sup.SCI[μ]) depending on the plurality of spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ), and depending on a transfer function (H.sub.ρ[μ.sub.1, μ.sub.2, μ.sub.3]; H.sub.v(ω.sub.1, ω.sub.2, ω.sub.3)) wherein the transfer function (H.sub.ρ[μ.sub.1, μ.sub.2, μ.sub.3]: (ω.sub.1, ω.sub.2, ω.sub.3)) is configured to receive two or more argument values (μ.sub.1, μ.sub.2, μ.sub.3; ω.sub.1, ω.sub.2, ω.sub.3), wherein each of the two or more argument values (μ.sub.1, μ.sub.2, μ.sub.3; ω.sub.1, ω.sub.2, ω.sub.3) indicates one of the plurality of frequency channels, and wherein the transfer function is configured to return a return value depending on the two or more argument values (μ.sub.1, μ.sub.2, μ.sub.3; ω.sub.1, ω.sub.2, ω.sub.3).

[0048] In an embodiment, the transmission medium may, e.g., be a fiber-optical channel.

[0049] FIG. 1b illustrates another embodiment, wherein the apparatus further comprises a signal modification module 130 being configured to modify the frequency-domain data signal using the one or more spectral interference coefficients to obtain a modified data signal. The apparatus of FIG. 1 b further comprises an inverse transform module 135 configured to transform the modified data signal from the frequency domain to the time domain to obtain a corrected time-domain data signal.

[0050] According to an embodiment, the signal modification module 130 of FIG. 1 b may, e.g., be configured to combine each one of the one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI[μ]), or a value derived from said one of the one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI[μ]), and one of the plurality of spectral coefficients (A.sub.λ[μ], A.sub.λ[.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ) to obtain the modified data signal. In a particular embodiment, the signal modification module 130 of FIG. 1b may, e.g., be configured to combine each one of the one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI[μ]), or a value derived from said one of the one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI[μ]), and one of the plurality of spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ) to obtain the modified data signal by subtracting [0051] each one of the one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI [μ]), or a value derived from said one of the one or more spectral interference coefficients (e. g., ΔA.sub.λ.sup.SCI[μ]), [0052] from one of the plurality of spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3]), . . . );

[0053] or, in another embodiment, to obtain the modified receive signal/sequence by subtracting [0054] each one of the one or more spectral interference coefficients (e. g. , ΔA.sub.λ.sup.SCI[μ]), or a value derived from said one of the one or more spectral interference coefficients (e. g., ΔA.sub.λ.sup.SCI[μ]), [0055] from one of the plurality of the spectral coefficients of the distorted receive sequence Y.sub.λ[μ];

[0056] or, in a futher embodiment, to inverse Discrete Fourier Transform the spectral interference coefficients ΔA.sub.λ[k] to obtain time domain interference coefficients Δa.sub.λ[k], and to subtract the time domain interference coefficients Δa.sub.λ[k] from the (time-domain) receive sequence Y.sub.λ[k];

[0057] or, in a yet further embodiment, to obtain the modified data or receive signal by subtracting [0058] each one of the one or more spectral interference coefficients (e. g. , ΔA.sub.λ.sup.SCI[μ]), or a value derived from said one of the one or more spectral interference coefficients [0059] (e. g., ΔA.sub.λ.sup.SCI[μ]), and by multiplying each one of the one or more spectral phase and polarization coefficients (exp(−jϕ.sub.λI−j{right arrow over (S)}.sub.λ)) [0060] from one of the plurality of spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ) or from one of the plurality of the spectral coefficients of the distorted receive sequence Y.sub.λ[μ].

[0061] In an embodiment, the transform module 110 of FIG. 1 b may, e.g., be configured to transform the data input signal from the time domain to the frequency domain by transforming a plurality of overlapping blocks of the data input signal from the time domain to the frequency domain to obtain a plurality of blocks of the frequency-domain data signal. The inverse transform module 135 may, e.g., be configured to transform the modified data signal from the frequency domain to the time domain by transforming a plurality of blocks from the frequency domain to the time domain and by overlapping said plurality of blocks being represented in the time domain to obtain the corrected time-domain data signal.

[0062] FIG. 1c illustrates a further embodiment, wherein the apparatus further comprises an inverse transform module 135 configured to transform the one or more spectral interference coefficients (e. g., ΔA.sub.λ.sup.SCI[μ]) from the frequency domain to the time domain. The apparatus of FIG. 1c further comprises a signal modification module 130 being configured to modify the data input signal being represented in the time domain using the one or more spectral interference coefficients being represented in the time domain to obtain a corrected time-domain data signal.

[0063] According to an embodiment, the signal modification module 130 of FIG. 1c may, e.g., be configured to combine each one of the one or more spectral interference coefficients being represented in the time domain, or a value derived from said one of the one or more spectral interference coefficients, and a time domain sample of a plurality of time domain samples of the data input signal being represented in the time domain to obtain the corrected time-domain data signal.

[0064] In a particular embodiment, the signal modification module 130 of FIG. 1c may, e.g., be configured to combine each one of the one or more spectral interference coefficients being represented in the time domain, or a value derived from said one of the one or more spectral interference coefficients, and a time domain sample of a plurality of time domain samples of the data input signal being represented in the time domain to obtain the corrected time-domain data signal by subtracting [0065] each one of the one or more spectral interference coefficients being represented in the time domain, or a value derived from said one of the one or more spectral interference coefficients, [0066] from a time domain sample of a plurality of time domain samples of the data input signal being represented in the time domain.

[0067] In an embodiment, the transform module 110 of FIG. 1c may, e.g., be configured to transform the data input signal from the time domain to the frequency domain by transforming a plurality of overlapping blocks of the data input signal from the time domain to the frequency domain to obtain a plurality of blocks of the frequency-domain data signal. The inverse transform module 135 may, e.g., be configured to transform a plurality of interference coefficients blocks from the frequency domain to the time domain, said plurality of blocks comprising the one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI[μ]). The signal modification module 130 may, e.g., be configured to modify the overlapping blocks of the data input signal, being represented in the time domain, using the plurality of interference coefficients blocks to obtain a plurality of corrected blocks, wherein the signal modification module 130 is configured to overlap the plurality of corrected blocks to obtain the corrected time-domain data signal.

[0068] FIG. 1d illustrates another embodiment, wherein the apparatus further comprises a transmitter module 140 configured to transmit the corrected time-domain data signal over the transmission medium.

[0069] FIG. 1e illustrates a further embodiment, wherein the apparatus further comprises a receiver module 105 configured to receive the data input signal being transmitted over the transmission medium.

[0070] In an embodiment, the analysis module 120 may, e.g., be configured to determine an estimation of a perturbated signal depending on the data input signal using the one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI[μ]).

[0071] According to an embodiment, the analysis module 120 may, e.g., be configured to determine the estimation of the perturbated signal by adding each one of the one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI[μ]) with one of the plurality of spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[.sub.3], . . .).

[0072] In an embodiment each of the two or more argument values may, e.g., be a channel index (μ.sub.1, μ.sub.2, μ.sub.3) being an index which indicates one of the plurality of frequency channels.

[0073] Or, in another embodiment, each of the two or more argument values is a frequency (ω.sub.1, ω.sub.2, ω.sub.3) which indicates one of the plurality of frequency channels, wherein said one of the plurality of frequency channels comprises said frequency.

[0074] In an embodiment, the analysis module 120 may, e.g., be configured to determine each spectral interference coefficient (e.g. ΔA.sub.λ.sup.SCI[μ]) of the one or more spectral interference coefficients (e. g., ΔA.sub.λ.sup.SCI[μ]) by determining a plurality of addends. The analysis module 120 may, e.g., be configured to determine each of the plurality of addends as a product of three or more of the spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ) and of the return value of the transfer function, the transfer function having three or more channel indices or three or more frequencies as the two or more argument values of the transfer function, which indicate three or more of the plurality of frequency channels to which said three or more of the spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ) are assigned.

[0075] In an embodiment, the analysis module 120 may, e.g., be configured to determine each spectral interference coefficient (e. g., ΔA.sub.λ.sup.SCI[μ]) of the one or more spectral interference coefficients (e.g., ΔA.sub.λ.sup.SCI[μ]) by determining a plurality of addends, wherein the analysis module 120 may, e.g., be configured to determine each of the plurality of addends as a product of three or more of the spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ) and of the return value of the transfer function, the transfer function having three or more channel indices or three or more frequencies as the two or more argument values of the transfer function, which indicate three or more of the plurality of frequency channels to which said three or more of the spectral coefficients (A.sub.λ[μ], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.1], A.sub.λ[μ.sub.2], A.sub.λ[μ.sub.3], . . . ) are assigned.

[0076] According to an embodiment, the analysis module 120 may, e.g., be configured to determine each spectral interference coefficient (e. g., ΔA.sub.λ.sup.SCI[μ]) according to:

[00001]$\Delta A_{\lambda }^{SCI}\left[\mu \right]=-j\frac{{\varphi }_{\mathrm{NL},\rho }}{N_{DFT}^2}\times {.Math.}_{{\mu }_1,{\mu }_2}{A}_{\lambda }\left[{\mu }_1\right]A_{\lambda }^H\left[{\mu }_2\right]A_{\lambda }\left[{\mu }_3\right]H_{\rho }\left[{\mu }_1,{\mu }_2,{\mu }_3\right]$

[0077] wherein ΔA.sub.λ.sup.SCI[μ] is said spectral interference coefficient, wherein A.sub.λ[μ.sub.1] is a first one of the three or more spectral coefficients, wherein A.sub.λ[μ.sub.2] is a second one of the three or more spectral coefficients, wherein , A.sub.λ[μ.sub.3] is a third one of the three or more spectral coefficients, wherein μ.sub.1 is a first index which indicates a first one of the plurality of frequency channels, wherein μ.sub.2 is a second index which indicates a second one of the plurality of frequency channels, wherein μ.sub.3 is a third index which indicates a third one of the plurality of frequency channels, wherein H.sub.ρ[μ.sub.1, μ.sub.2,μ.sub.3] indicates the transfer function, wherein N.sub.DFT.sup.2 indicates a square of a number of the plurality of frequency channels of the frequency domain, wherein ϕNL,ρ is a number.

[0078] In an embodiment, the transfer function may, e.g., be normalized and nonlinear.

[0079] According to an embodiment, the analysis module 120 is configured to determine the interference by applying a regular perturbation approach (e.g., Algorithm 1).

[0080] In an embodiment, the analysis module 120 is configured to determine the interference by applying a regular logarithmic perturbation approach (e.g., Algorithm 2).

[0081] In an embodiment, the frequency domain may, e.g., be a regular-logarithmic frequency domain.

[0082] According to an embodiment, the transfer function may, e.g., depend on

[00002]$H_{\nu }\left(e^{j\omega T}\right)=\frac{1}{T^3}\underset{m\in {\mathbb{Z}}^3}{.Math.}{H}_{\nu }\left(\omega -\frac{2\pi m}{T}\right).$

[0083] In the following, embodiments of the present invention are described in more detail.

[0084] At first, the notation and the overall system model is introduced.

[0085] The notation and basic definitions are now described.

[0086] Sets are denoted with calligraphic letters, e.g., is the set of data symbols, i.e., the symbol alphabet or signal constellation. A set of numbers or finite fields are typeset in blackboard bold typeface, e.g., the set of real numbers is . Bold letters, such as x, indicate vectors. If not stated otherwise, a vector x=[x.sub.1, x.sub.2, . . . , x.sub.n].sup.T of dimension n is a column vector, and the set of indices to the elements of the vector is

[00003]$ℐ\stackrel{\mathrm{def}}{=}\left\{1,\mathrm{.Math.},n\right\}.$

Non-bold italic letters, like x, are scalar variables, whereas non-bold Roman letters refer to constants, e.g., the imaginary number is j with j.sup.2=1. (.Math.).sup.T denotes transposition and (.Math.).sup.H is the Hermitian transposition.

[0087] A real (bandpass) signal is typically described using the equivalent complex baseband (ECB) representation, i.e., we consider the complex envelope x(t) ∈ with inphase (real) and quadrature (imaginary) component. The n -dimensional Fourier transform of a continuous-time signal x(t)=x(t.sub.1, t.sub.2, . . . , t.sub.n) depending on the n-dimensional time vector t=[t.sub.1, t.sub.2, . . . , t.sub.n].sup.T ∈R.sup.n (in seconds) is denoted by X(ω)={x(t)}, and defined as [14, Ch. 4]

[00004]$\begin{array}{cc}X\left(\omega \right)=\left\{x\left(t\right)\right\}\stackrel{\mathrm{def}}{=}{\int }_{{ℝ}^n}x\left(t\right)e^{-j\omega .Math.t}{d}^{n}t& \left(1\right)\\ x\left(t\right)={-1}^{}\left\{X\left(\omega \right)\right\}=\frac{1}{{\left(2\pi \right)}^n}{\int }_{{ℝ}^n}X\left(\omega \right)e^{j\omega .Math.t}{d}^n\omega .& \left(2\right)\end{array}$

[0088] Here, X(ω) is a continuous function of angular frequencies ω=[ω.sub.1, ω.sub.2, . . . , ω.sub.n].sup.T ∈ R.sup.n with ω=2πf and frequency f ∈R (in Hertz). In the exponential we made use of the dot product of vectors in R.sup.n given by ω.Math.t=ω.sub.1t.sub.1+ω.sub.2t.sub.2+ . . . +ω.sub.nt.sub.n. The integral is an n-fold multiple integral over .sup.n and with integration boundaries at −∞ and ∞ in each dimension. We use the expression d.sup.nt as shorthand for dt.sub.1 dt.sub.2 . . . dt.sub.n. For the one-dimensional case with n=1 the variable subscript is dropped. We may also write the correspondence as x(t) ∘-.circle-solid.X(ω) for short.

[0089] The n-dimensional discrete-time Fourier transform (DTFT) of a discrete-time sequence <x[k]> with k=[k.sub.1, k.sub.2, . . . , k.sub.n].sup.T ∈.sup.n with spacing T between symbols is periodic with 1/T in frequency domain and denoted as X(e.sup.jωT)={circumflex over (F)}{x|k|}, and defined as

[00005]$\begin{array}{cc}X\left(e^{j\omega T}\right)=\stackrel{}{^}\left\{x\left[k\right]\right\}\stackrel{\mathrm{def}}{=}\underset{k\in {\mathbb{Z}}^n}{.Math.}x\left[k\right]e^{-j\omega .Math.\mathrm{kT}}& \left(3\right)\\ x\left[k\right)={\stackrel{}{^}}^{-1}\left\{X\left(e^{j\omega T}\right)\right\}={\left(\frac{T}{2\pi }\right)}^n{\int }_{{\mathrm{��}}^n}X\left(e^{j\omega T}\right)e^{j\omega .Math.\mathrm{kT}}{d}^n\omega .& \left(4\right)\end{array}$

[0090] The set of frequencies in the Nyquist interval is

[00006]$\mathrm{��}\stackrel{\mathrm{def}}{=}\left\{\omega \in ℝ|-{\omega }_{\mathrm{Nyq}}\le \omega <{\omega }_{\mathrm{Nyq}}\right\}$

with the Nyquist (angular) frequency

[00007]${\omega }_{Nyq}\stackrel{\mathrm{def}}{=}2\pi /\left(2T\right).$

[0091] If a whole (finite-length) sequence is treated, this is indicated by the square bracket notation, i.e., <x[k]>

[0092] The notation Σ.sub.k∈z.sub.n is short for Σ.sub.k.sub.1.sup.∞=−∞ Σ.sub.k.sub.2.sup.∞=−∞. . . Σ.sub.k.sub.n.sup.∞=−∞.

[0093] Embodiments employ the so-called engineering notation of the Fourier transform with a negative sign in the complex exponential (in the forward, i.e., time-to-frequency, direction) is used. This has immediate consequences for the solution of the electro-magnetic wave equation (cf. Helmholtz equation), and therefore also for the NLSE. In the optical community, there exists no fixed convention with respect to the sign notation, e.g., some of the texts are written with the physicists' (e.g., [15, Eq. (2.2.8)] or [10]) and others with the engineering (e.g., [16], [17, Eq. (A.4)]) notation in mind. Consequently, the derivations shown here may differ marginally from some of the original sources.

[0094] Continuous-time signals are associated with meaningful physical units, e.g., the electrical field has typically units of volts per meter (V/m). The NLSE and the Manakov equation derived thereof are carried out in Jones space over a quantity u(t)=[u.sub.x(t), u.sub.y(t)].sup.T ∈.sup.2 called the optical field envelope. The optical field envelope has the same orientation as the associated electrical field but is renormalized s.t. u.sup.Hu equals the instantaneous power given in watts (W). Here, signals are instead generally treated as dimensionless entities as this considerably simplifies the notation when we move between the various signal domains (see, e.g., discussion in [18, P. 11] or [19, P. 230]). To this end, the nonlinearity coefficient y commonly given in W.sup.−1m.sup.−1 is also renormalized to have units of m.sup.−1, cf. II-B2.

[0095] To distinguish a two-dimensional complex vector u=[u.sub.x, u.sub.y].sup.T in Jones space from its associated three-dimensional real-valued vector in Stokes space, we use decorated bold letters {right arrow over (u)}=[u.sub.1, u.sub.2, u.sub.3].sup.T ∈.sup.3. The (permuted) set of Pauli matrices is given by [20]

[00008]$\begin{array}{cc}{\sigma }_1\stackrel{\mathrm{def}}{=}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]{\sigma }_2\stackrel{\mathrm{def}}{=}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]{\sigma }_3\stackrel{\mathrm{def}}{=}\left[\begin{array}{cc}0& -j\\ j& 0\end{array}\right],& \left(5\right)\end{array}$

[0096] and the Pauli vector is

[00009]$\stackrel{->}{\sigma }\stackrel{\mathrm{def}}{=}{\left[{\sigma }_1,{\sigma }_2,{\sigma }_3\right]}^T$

[0097] where each vector component is a 2×2 Pauli matrix. The relation between Jones and Stokes space can then be established by the concise (symbolic) expression {right arrow over (u)}=u.sup.H{right arrow over (σ)}u to denote the elementwise operation u.sub.i=u.sup.Hσ.sub.ou for all Stokes vector components i=1, 2, 3. The Stokes vector {right arrow over (u)} can also be expanded using the dot product with the Pauli vector to obtain the complex-valued 2×2 matrix with

[00010]$\begin{array}{cc}\stackrel{->}{u}.Math.\stackrel{->}{\sigma }=u_1{\sigma }_1+u_2{\sigma }_2+u_3{\sigma }_3={\left[\begin{array}{cc}{u}_x{u}_x^{*}-u_y{u}_y^{*}& 2u_x{u}_y^{*}\\ 2u_x^{*}{u}_y& u_y{u}_y^{*}-u_x{u}_x^{*}\end{array}\right]}_{\text{?}},\text{?}\text{indicates text missing or illegible when filed}& \left(6\right)\end{array}$

[0098] which will later be used to describe the instantaneous polarization rotation around the Stokes vector {right arrow over (u)} using the Jones formalism. We may also use the equality [20, Eq. (3.9)]

uu.sup.H=½(u.sup.Hu I+{right arrow over (u)}.Math.{right arrow over (σ)})   (7)

[0099] with the identity matrix I and ∥u∥.sup.2=u.sup.Hu=u.sub.xu.sub.x*+u.sub.yu.sub.y*.

[0100] In the following, a system model according to embodiments is considered.

[0101] Some embodiments provide a point-to-point coherent optical transmission over two planes of polarization in a single-mode fiber. This results in a complex-valued 2×2 multiple-input/multiple-output (MIMO) transmission which is typically used for multiplexing. One of the major constraints of today's fiber-optical transmission systems is the bandwidth of electronic devices which is orders of magnitude smaller than the available bandwidth of optical fibers. It is hence routine to use wavelength-division multiplexing (WDM), where a number of so-called wavelength channels are transmitted simultaneously through the same fiber. Each wavelength signal is modulated on an individual laser operated at different wavelengths such that the spectral support of neighboring signals is not overlapping.

[0102] FIG. 2 illustrates a generic fiber-optical transmission system model. In particular, FIG. 2 shows the block diagram of a coherent optical transmission system exemplifying the digital, analog, and optical domains of a single wavelength channel. Within the bandwidth of a wavelength channel, we can consider the optical end-to-end 2×2 MIMO channel as frequency-flat if we neglect the effects of bandlimiting devices (e.g., switching elements in a routed network). The nonlinear property of the fiber-optical transmission medium is the source of interference within and between different wavelength channels. In the following, we will call the channel under consideration the probe channel, while a co-propagating wavelength channel is called interferer. This allows us to discriminate between self-channel interference (SCI) and cross-channel interference (XCI). In FIG. 2 the probe channel in the optical domain is denoted by a subscript P, whereas interferers are labeled by the channel index v with v∈{1, . . . , N.sub.ch|v≠ρ}. The various domains and its entities are discussed in the following.

[0103] FIG. 3a illustrates a transmitter frontend of a generic fiber-optical transmission system model.

[0104] FIG. 3b illustrates an optical channel of the generic fiber-optical transmission system model.

[0105] FIG. 3c illustrates an receiver frontend of the generic fiber-optical transmission system model and variables associated with the regular perturbation model.

[0106] In the following the transmitter frontend of FIG. 3a is described. The transmission system is fed with equiprobable source bits of the probe (and interferer) channel. The binary source generates uniform i.i.d. information bits q|K|∈F.sub.2 at each discrete-time index K ∈Z. F.sub.2 denotes the Galois field of size two and Z is the set of integers. The binary sequence <q(K> is partitioned into binary tuples of length R.sub.m, s.t. q[k]=[q.sub.,[k], . . . q.sub.R.sub.m[k]] ∈{0, 1}.sup.R.sup.m, where k ∈Z is the discrete-time index of the data symbols. Here, R.sub.m is called the rate of the modulation and will be equivalent to the number of bits per transmitted data symbol, if we assume that the size of the symbol set is a power of two. Each R.sub.m-tuple is associated with one of the possible data symbols α=|a.sub.x, a.sub.y|.sup.T∈ C .sup.2 , i.e., with one of the constellation points. We say that the binary R.sup.m-tuples are mapped to the data symbols a ∈ A by a bijective mapping rule : qa.

[0107] The size of the data symbol set is M=|=2.sup.R.sup.m and we can write the alphabet as

[00011]$\mathrm{��}\stackrel{\mathrm{def}}{=}\left\{a_1,\mathrm{.Math.},a_M\right\}.Math.{ℂ}^2.$

[0108] The symbol set has zero mean if not stated otherwise, that is E{a}=0, and we deliberately normalize the variance of the symbol set to

[00012]${\sigma }_a^2\stackrel{\mathrm{def}}{=}E\left\{{.Math.a.Math.}^2\right\}=1$

[0109] (the expectation is denoted by E{.Math.} and the Euclidean vector norm is ∥.Math.∥. For reasons of readability we denote the data symbols of the interfering channels by b.sub.v[k].Math..

[0110] The discrete-time data symbols a[k] are converted to the continuous-time transmit signal 8(t) by means of pulse-shaping constituting the digital-to-analog (D/A) transition, cf. FIG. 3a. We can express the transmit signal 8(t)=[s.sub.1(t), s.sub.2(t), s.sub.2(t)].sub.T ∈.sup.2 as a function of the data symbols with

[00013]$\begin{array}{cc}s\left(t\right)=T.Math.\underset{k\in \mathbb{Z}}{.Math.}a\left[k\right]h_T\left(t-\mathrm{kT}\right),& \left(8\right)\end{array}$

[0111] where 8(t) is a superposition of a time-shifted (with symbol period T) basic pulses h.sub.T(t) weighted by the data symbols. The pre-factor T is used to preserve a dimensionless signal in the continuous-time domain (cf. [18, P. 11]). We assume that the transmit pulse has √{square root over (Nyquist)} property, i.e., |H.sub.T(ω)|.sup.2 has Nyquist property with the Fourier pair h.sub.T(t) ∘-.circle-solid.H.sub.T(ω). To keep the following derivations tractable, all wavelength channels transmit at the same symbol rate

[00014]$R_s\stackrel{\mathrm{def}}{=}1/T$

[0112] as me prooe cnannei. ne pulse energy E.sub.T of the probe channel is given by [18, Eq. (2.2.22)]

[00015]$\begin{array}{cc}{E}_T={\int }_{-\infty }^{\infty }{.Math.T.Math.h_T\left(t\right).Math.}^2\mathrm{dt}=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{.Math.T.Math.H_T\left(\omega \right).Math.}^{2}d\omega .& \left(9\right)\end{array}$

[0113] The pulse energy E.sub.T has the unit seconds due to the normalization of the signals. Using the symbol energy

[00016]$E_s\stackrel{\mathrm{def}}{=}{\sigma }_a^2{E}_T.$

[0114] the average signal power P calculates to [18, Eq. (4.1.1)]

[00017]$\begin{array}{cc}P\stackrel{\mathrm{def}}{=}\frac{1}{T}{\int }_0^{T}E\left\{{.Math.s\left(t\right).Math.}^2\right\}\mathrm{dt}=\frac{{\sigma }_a^2}{T}{E}_T=\frac{E_s}{T}.& \left(10\right)\end{array}$

[0115] Since, see above, the variance of the data symbols σ.sub.a.sup.2 is fixed to 1, the transmit power P is directly adjusted via the pulse energy E.sub.T. The corresponding quantities related to one of the interferers are indicated by the subscript V.Math..

[0116] In the following, an optical channel according to FIG. 3b is described. The electrical-to-optical (E/O) conversion is performed by an ideal dual-polarization (DP) inphase-quadrature (IQ) converter. The two elements of the transmit signal 8.sub.v(t) correspond to the modulated optical signals in the x- and y-polarization. The optical field envelope u.sub.v(z, t) of each wavelength channel

u.sub.v(0, t)=s.sub.v(t) exp (jΔω.sub.vt),   (11)

[0117] is modulated at its angular carrier frequency ω.sub.v=ω.sub.0+Δω.sub.v at the input of the optical transmission line z=0. Here, ω.sub.0=2πf.sub.0 is the center frequency of the signaling regime of interest. For the probe channel, the carrier frequency ω.sub.σ is to coincide with ω.sub.O such that Δω.sub.σ=0 and u.sub.σ(0, t)=8.sub.σ(t). The transmitter frontend of the probe channel is shown in FIG. 3a.

[0118] The N.sub.ch wavelength signals u.sub.v(0, t) at z=0 are combined by an ideal optical multiplexer to a single WDM signal, cf. FIG. 3b. The optical field envelope before transmission is then

[00018]$\begin{array}{cc}u\left(0,t\right)=\underset{v=1}{\overset{N_{\mathrm{ch}}}{.Math.}}{u}_v\left(0,t\right)=\underset{v=1}{\overset{N_{\mathrm{ch}}}{.Math.}}{s}_v\left(t\right)\exp \left({j\mathrm{\Delta \omega }}_{v}t\right)& \left(12\right)\\ & \\ U\left(0,\omega \right)=\underset{v=1}{\overset{N_{\mathrm{ch}}}{.Math.}}{U}_v\left(0,\omega \right)=\underset{v=A}{\overset{N_{\mathrm{ch}}}{.Math.}}{S}_v\left(\omega -{\mathrm{\Delta \omega }}_v\right),& \left(13\right)\end{array}$

[0119] with the Fourier pairs 8.sub.v(t) S.sub.v(ω) and u(0, t) U (0, ω). Any initial phase and laser phase noise (PN) are neglected to focus only on deterministic distortions. The optical field envelope is the ECB representation of the optical field u.sub.o(z, t) in the passband notation

[00019]$\begin{array}{cc}{u}_o\left(z,t\right)\stackrel{\mathrm{def}}{=}u\left(z,t\right).Math.\exp \left({j\omega }_{0}t-{j\beta }_0\left(z\right)z\right),& \left(14\right)\end{array}$

[0120] which is known as the slowly varying amplitude approximation [15, Eq. (2.4.5)]. For consistency of notation we treat the optical field envelope as a dimensionless entity (in accordance with the electrical signals). The optical field propagates in Z-direction (the dimension Z has units of meter) with the local propagation constant β.sub.0(z)=β(z, ω.sub.0), β(z, ω) is the space and frequency-dependent propagation constant. A Taylor expansion of β(z, ω) is performed around ω.sub.0 with the derivatives of β(z, ω) represented by the coefficients [15, Eq. (2.4.4)]

[00020]$\begin{array}{cc}{\beta }_n\left(z\right)\stackrel{\mathrm{def}}{=}\frac{{\partial }^n\beta \left(z,\omega \right)}{\partial {\omega }^n}{|}_{\omega ={\omega }_0},n\in \mathbb{N}.& \left(15\right)\end{array}$

[0121] Here, we only consider coefficients up to second order, i.e., n ∈{0,1, 2}. We also introduce the path-average4 dispersion length

[00021]$\begin{array}{cc}{L}_D\stackrel{\mathrm{def}}{=}\frac{1}{2\pi .Math.{\beta }_2.Math.R_s^2},& \left(16\right)\end{array}$

[0122] which denotes the distance after which two spectral components spaced B=R.sub.s Hertz apart, experience a differential group delay of T=1/R.sub.s due to chromatic dispersion (CD). We can equivalently define the walk-off length of the probe and one interfering wavelength channel as

[00022]$\begin{array}{cc}{L}_{\mathrm{wo},v}\stackrel{\mathrm{def}}{=}\frac{1}{.Math.{\mathrm{\Delta \omega }}_v{\beta }_2.Math.R_s},& \left(17\right)\end{array}$

[0123] which quantifies the fiber length that must be propagated in order for the v.sup.th wavelength channel to walk off by one symbol from the probe channel.

[0124] 4We discriminate between local (i.e., α(z), β(z), γ(z)) and path-average (i.e., α, β, γ) properties of the transmission link. The latter are implicitly indicated if the z-argument of the local property is omitted, e.g.,

[00023]${\beta }_2\stackrel{\Delta }{=}\frac{1}{L}{\int }_0^L{\beta }_2\left(\zeta \right)d\zeta .$

[0125] Now, signal propagation is considered.

[0126] In the absence of noise, the two dominating effects governing the propagation of the optical signal in the fiber are dispersion—expressed by the z-profile of the fiber dispersion coefficient β.sub.2(z)—and nonlinear signal-signal interactions. Generation of the so-termed local NLI depends jointly on the local fiber nonlinearity coefficient γ(z) and the z-profile of the optical signal power. For ease of the derivation, we assume that all z-dependent variation in γ(z) can be equivalently expressed in a variation of either a local gain g(z) or the local fiber attenuation α(z). We also neglect the time- (and frequency-) dependency of the attenuation, gain, and nonlinearity coefficient.

[0127] The interplay between the optical signal, dispersion, and nonlinear interaction is all combined in the noiseless Manakov equation. It is a coupled set of partial differential equations in time-domain for the optical field envelope u(z, t) in the ECB, and the derivative is taken w.r.t. propagation distance z ∈R and to the retarded time t ∈R. The retarded time is defined as

[00024]$t\stackrel{\mathrm{def}}{=}{t}^{\prime }-z/v_g,$

[0128] where t′ is me pnysical time and v.sub.g is the (path-average) group velocity v.sub.g=1/β.sub.1 of the probe channel [15, Eq. (2.4.8)]. It can be understood as a time frame that moves at the same average velocity as the probe to cancel out any group delay at the reference frequency ω.sub.σ=ω.sub.0. All other frequencies experience a residual group delay relative to the reference frequency due to CD.

[0129] The propagation of u(z, t) in the signaling regime of interest is governed by [17, Eq.(6.26)]

[00025]$\begin{array}{cc}\frac{\partial }{\partial z}u=j\frac{{\beta }_2\left(z\right)}{2}\frac{{\partial }^2}{\partial z^2}u+\frac{g\left(z\right)-\alpha \left(z\right)}{2}u-j\gamma \left(z\right)\frac{8}{9}{.Math.u.Math.}^{2}u.& \left(18\right)\end{array}$

[0130] The space- and time-dependency of u(z, t) is omitted here for compact notation. By allowing the local gain coefficient g(z) to contain Dirac δ-functions one can capture the z-dependence of an amplification scheme, i.e., based on lumped erbium-doped fiber amplifier (EDFA) or Raman amplification. Polarization-dependent effects such as birefringence and polarization mode dispersion (PMD) are neglected limiting the following derivations to the practically relevant case of low-PMD fibers. We also assume that all wavelength channels are co-polarized, i.e., modulated on polarization axes parallel to the ones of the probe channel.

[0131] Now, the dispersion profile is considered.

[0132] The accumulated dispersion is a function that satisfies [21, Eq. (8)]

[00026]$\begin{array}{cc}\frac{dℬ\left(z\right)}{dz}={\beta }_2\left(z\right).& \left(19\right)\end{array}$

[0133] Here, (z) can be used to express a z-dependency in the dispersion profile, i.e., lumped dispersion compensation by inline dispersion compensation or simply a transmission link with distinct fiber properties across multiple spans. We obtain

(z)=∫.sub.0.sup.zβ.sub.2(ζ)dζ+.sub.0,   (20)

[0134] where

[00027]${ℬ}_0\stackrel{\mathrm{def}}{=}ℬ\left(0\right)$

[0135] is the amount of pre-dispersion (in units of squared seconds, typically given in ps.sup.2) at the beginning of the transmission line.

[0136] Now, the power profile is considered.

[0137] To describe the power evolution of u(z, t). we introduce the normalized power profile (z) as a function that satisfies the equation [21, Eq. (7)]

[00028]$\begin{array}{cc}\frac{d\left(z\right)}{dz}=\left(g\left(z\right)-\alpha \left(z\right)\right)\left(z\right),& \left(21\right)\end{array}$

[0138] with boundary condition P(0)=P(1,)=1, i.e., the last optical amplifier resets the signal power to the transmit power.

[0139] The Z -dependence on α(z) allows for varying attenuation coefficients over different spans. In writing (21) we assumed that both the local gain coefficient and attenuation coefficient are frequency-independent. We may also define the logarithmic gain/loss profile as

[00029]$\begin{array}{cc}\left(z\right)\stackrel{\mathrm{def}}{=}\ln \left(\mathrm{��}\left(z\right)\right)={\int }_0^z\left(g\left(\zeta \right)-\alpha \left(\zeta \right)\right)d\zeta .& \left(22\right)\end{array}$

[0140] The last expression in (22) is obtained by solving (21) for P(z)=The boundary conditions on P(z) immediately give the boundary condition (0)=(L)=0.

[0141] We can now define the impulse response and transfer function of the linear channel—that is, when the fiber nonlinearity coefficient is zero, i.e., γ=0 in (18). To that end, we define the optical field envelope u.sub.LIN(z, t) U.sub.LIN(z, ω) that propagates solely according to linear effects with the boundary condition u.sub.LIN(0, t)=u(0, t) at the input of the transmission link. The linear channel transfer function and impulse response is then given by

[00030]$\begin{array}{cc}{H}_C\left(z,\omega \right)\stackrel{\mathrm{def}}{=}\exp \left(\frac{\left(z\right)-j{\omega }^2ℬ\left(z\right)}{2}\right)& \left(23\right)\\ h_C\left(z,t\right)=\frac{1}{\sqrt{2\pi }}\frac{1}{\sqrt{jℬ\left(z\right)}}\exp \left(\frac{\left(z\right)+j{t}^2/ℬ\left(z\right)}{2}\right),& \left(24\right)\end{array}$

[0142] which represents the joint effect of chromatic dispersion and the gain/loss variation along the link. We finally have the linear channel relation in time-domain u.sub.LIN(z) t)=h.sub.C(z, t)* u.sub.LIN(0 ,t) and frequency-domain U.sub.LIN(z, ω)=H.sub.C(z, ω)U.sub.LIN (0, ω). which will be used in the following to derive the first-order perturbation method.

[0143] In the following, a receiver frontend according to FIG. 3c is described. Again, we assume ideal optical-to-electrical (O/E) and analog-to-digital (A/D) conversion. The received continuous-time, optical signal u(L, t) is first matched filtered w.r.t. the linear channel response and transmit pulse and then sampled at the symbol period T, cf. FIG. 3 (c). The receiver frontend hence also compensates for any residual link loss and performs perfect CD compensation. Note, that the analog frontend is usually realized using an oversampled digital representation. E.g., CD compensation is typically performed in the (oversampled) digital domain. Here, we favour to conceptually incorporate it in the analog domain since it significantly simplifies notation in the derivation of the end-to-end channel model. The transfer function of the entire cascade of the receiver frontend is given by

[00031]$\begin{array}{cc}{H}_R\left(\omega \right)=\frac{T}{E_T}{H}_C^{*}\left(L,\omega \right)H_T^{*}\left(\omega \right).& \left(25\right)\end{array}$

[0144] The factor T/E.sub.T re-normalizes the received signal to the variance of the constellation σ.sub.a.sup.2. Since we only consider T-spaced sampling any fractional sampling phase-offset or timing synchronization is already incorporated as suited delay in the receive filter h.sub.R(t). s.t. the transmitted and received sequence of the probe are perfectly aligned in time.

[0145] Note, that the time delay L/v.sub.g at ω.sub.0 and any initial phase β.sub.0 has already been canceled from the propagation equation.

[0146] In the following, first-order perturbation is considered.

[0147] A concept of fiber-optical channel models based on the perturbation method is to assume that nonlinear distortions are weak compared to its source, i.e., the linearly propagating signal. Starting from this premise the regular perturbation (RP) approach for the optical end-to-end channel is written as

u(L, t)=u.sub.LIN(L, t) Δu(L, t),   (26)

[0148] where u.sub.LIN(z, t) ∈.sup.2 is the signal propagating according to the linear effects, i.e., according to (23), (24). In this context, the nonlinear distortion Δu(z, t) ∈.sup.2 is termed perturbation, which is generated locally according to nonlinear signal-signal interaction and is then propagated linearly and independently of the signal u.sup.LIN(z, t) to the end of the optical channel at z=L. We assume that the optical perturbation at z=0 is zero, i.e., Δu(0, t)=0. The received signal is then given as the sum of the solution for the linearly propagating signal and the accumulated perturbation representing the accumulated nonlinear effects. An objective here is to develop the input/output relation of the equivalent discrete-time end-to-end channel in the form of

y[k]=a[k]+Δa[k],   (27)

[0149] where the total NLI is absorbed into a single discrete-time perturbative term Δa[k], cf. FIG. 3 (c). To that end, we start with a known RP solution of the optical end-to-end relation and successively incorporate the used components according to FIG. 2 and FIG. 3.

[0150] Now, the optical end-to-end channel is considered.

[0151] The solution to the optical perturbation after transmission at z=L is given in frequency-domain by [4, Eq. (12)], [22, Eq. (2)], [23, Eq. (4)], [24, Eq. (24)-(27)],

[00032]$\begin{array}{cc}\Delta U\left(L,\omega \right)=-j\gamma \frac{8}{9}\frac{L_{\mathrm{eff}}}{{\left(2\pi \right)}^2}{H}_C\left(L,\omega \right)\times {\int }_{{ℝ}^2}\underset{¯}{U}\left(\omega ,v_1,v_2\right)H_{NL}\left(v_1,v_2\right)d^{2}v,& \left(28\right)\end{array}$

[0152] with the normalized nonlinear transfer function H.sub.NL(V.sub.1, v.sub.2) and

[00033]$\underset{\_}{U}\left(\omega ,v_1,v_2\right)\stackrel{\mathrm{def}}{=}U\left(0,\omega +v_2\right)U^H\left(0,\omega +v_1+v_2\right)U\left(0,\omega +v_1\right),$

[0153] i.e., a term that depends on the optical field envelope at the input of the transmission system. Note, that we made use of the common variable substitution

[00034]$\begin{array}{cc}{\omega }_1\stackrel{\mathrm{def}}{=}\omega +v_1& \left(29\right)\\ {\omega }_2\stackrel{\mathrm{def}}{=}\omega +v_1+v_2& \left(30\right)\\ {\omega }_3\stackrel{\mathrm{def}}{=}\omega -{\omega }_1+{\omega }_2=\omega +v_2,& \left(31\right)\end{array}$

[0154] to express the field U in terms of difference frequencies υ.sup.1 and υ.sup.2 relative to ω. FIG. 4a and FIG. 4b summarizes definitions of the time- and frequency variables that are used throughout this text. The integral over .sup.2 in (28) can also be performed w.r.t. ω.sub.1 and ω.sub.2

[0155] FIG. 4a illustrates definitions of variables in the time-domain. FIG. 4b illustrates definitions of variables in the frequency-domain. Both T.sub.1,T.sub.2 and υ.sub.1, υ.sub.2 can take positive and negative values in R.

[0156] Equation (28) shows that the first-order RP method can be understood as a FWM process with un-depleted pumps where three wavelengths affect a fourth. Equivalently, one can think of the joint annihilation and creation of two two-photon pairs (i.e., with four frequencies involved) preserving both energy (frequency matching) and momentum (phase matching) during the interaction [25, FIG. 7.2.5]. The conjugate field corresponds to the inverse process where photon creation and annihilation is interchanged.

[0157] FIG. 5 illustrates a magnitude in logarithmic scale of a single-span nonlinear transfer function for β.sub.2=−21 ps.sup.2/km, .sub.0=0 ps.sup.2, 10 log.sub.10 e.sup.α=0.2 dB/km and L.sub.sp=100 km over the difference frequencies υ.sub.1 and υ.sub.2 normalized to R.sub.s=64 GBd. The red line denotes H.sub.NL(ξ) which only depends on the scalar ξ=υ.sub.1υ.sub.2. (Part for υ.sub.1>υ.sub.2 not shown).

[0158] The normalized nonlinear transfer function is a measure of the phase matching condition and defined as

[00035]$\begin{array}{cc}\begin{array}{c}{H}_{NL}\left(v_1,v_2\right)\stackrel{\mathrm{def}}{=}\frac{1}{L_{\mathrm{eff}}}{\int }_0^L\exp \left(\left(\zeta \right)+j{v}_1{v}_2ℬ\left(\zeta \right)\right)d\zeta \\ =\frac{1}{L_{\mathrm{eff}}}{\int }_0^L{H_C^{*}\left(\zeta ,\sqrt{v_1{v}_2}\right)}^{2}d\zeta .\end{array}& \left(34\right)\end{array}$

[0159] The pre-factor is the effective length of the whole transmission link defined as

[00036]$\begin{array}{cc}{L}_{\mathrm{eff}}\stackrel{\mathrm{def}}{=}{\int }_0^L\left(\zeta \right)d\zeta ={\int }_0^L\exp \left(\left(\zeta \right)\right)d\zeta ,& \left(35\right)\end{array}$

[0160] and acts as a normalization constant s.t. H.sub.NL(0, 0)=1.

[0161] The phase mismatch Δβ. i.e., the difference in the (path-average) propagation constant due to dispersion, is defined as [15, Eq. (6.3.19)]

[00037]$\begin{array}{cc}\begin{array}{c}\mathrm{\Delta \beta }\stackrel{\mathrm{def}}{=}\beta \left(\omega \right)-\beta \left({\omega }_1\right)+\beta \left({\omega }_2\right)-\beta \left({\omega }_3\right)\\ =\frac{{\beta }_2}{2}\left({\omega }^2-{\omega }_1^2+{\omega }_2^2-{\left(\omega -{\omega }_1+{\omega }_2\right)}^2\right)\\ ={\beta }_2\left({\omega }_1-\omega \right)\left({\omega }_2-{\omega }_1\right)={\beta }_2{v}_1{v}_2,\end{array}& \left(36\right)\end{array}$

[0162] where the propagation constants at the four frequencies are developed in a second-order Taylor series according to (15). E.g., for transmission systems without inline dispersion compensation and zero pre-dispersion .sub.0=0, we have (z)=β.sub.2z and the phase mismatch Δβ can be found in the argument of the exponential in (34) with v.sub.1v.sub.2(z)=Δβz.

[0163] In the context of the equivalent approach following the regular VSTF [3], [4], [24], the nonlinear transfer function H.sub.NL(v.sub.1, v.sub.2) is also referred to as 3rd-order Volterra kernel. Closed form analytical solutions to (34) can be obtained for single-span or homogeneous multi-span systems [24], [26]. It is noteworthy, that H.sub.NL(v.sub.1, v.sub.2) contains all information about the transmission link characterized by the dispersion profile (including CD pre-compensation .sub.0, cf. (20)) and the gain/loss profile.

[0164] FIG. 5 shows the magnitude of H.sub.NL(v.sub.1, v.sub.2) exemplifying a single-span standard single-mode fiber (SSMF) link. Note, that H.sub.NL(v.sub.1, v.sub.2) depends in fact on the product

[00038]$\xi \stackrel{\mathrm{def}}{=}{v}_1{v}_2$

[0165] and is hence a hyperbolic function in two dimensions [27, Sec. VIII] (cf. the contour in FIG. 5). The bold red line drawn into the diagonal cross section in FIG. 5 is the corresponding nonlinear transfer function H.sub.NL(ξ) which only depends on the scalar variable

[00039]$\xi \stackrel{\mathrm{def}}{=}{v}_1{v}_2.$

[0166] FIG. 6 illustrates a magnitude in logarithmic scale of a single-span nonlinear transfer function for β.sub.2=−21 ps.sup.2/km, .sub.0=0 ps.sup.2, 10 log.sub.10 e.sup.α=0.2 dB/km and L.sub.sp8=100 km over ξ=v.sub.1v.sub.2. The normalization by (2πR.sub.s).sup.2 relates H.sub.NL(ξ) to the probe's spectral width. The width of |H.sub.NL(ξ/R.sub.s.sup.2)|.sup.2 is then proportional to 1/.sub.T,σ=L.sub.D/L.sub.eff ∝ R.sub.s.sup.−2, i.e., doubling R.sub.s reduces the spectral width by a factor of 4.

[0167] FIG. 6 shows the H.sub.NL(ξ) over the normalized variable ξ/2πR.sub.s).sup.2 to relate the nonlinear transfer function to the spectral width of the probe channel. The spectral width of |H.sub.NL(ξ/(2πR.sub.s).sup.2)|.sup.2 is proportional to the inverse dimensionless map strength

[00040]$1/{T,\rho }_{}\stackrel{\mathrm{def}}{=}{L}_D/L_{\mathrm{eff}}$

[0168] closely related to the nonlinear diffusion bandwidth defined in [22]. Conversely, the map strength .sub.T,σ quantifies the number of nonlinearly interacting pulses in time over the effective length L.sub.eff within the probe channel [28]. It is therefore a direct measure of intra-channel (i.e., SCI) nonlinear effects [29]. The relevant quantity for inter-channel (i.e., XCI) effects is given by

[00041]$S_{T,v}\stackrel{\mathrm{def}}{=}{L}_{\mathrm{eff}}/L_{\mathrm{wo},v}$

[0169] (with v≠σ) where the temporal walk-off between wavelength channels is the relevant length scale. In [23] it was shown that H.sub.NL(ξ) is related to the power-weighted dispersion distribution (PWDD) by a (one-dimensional) Fourier transformation (w.r.t. the scalar variable ξ) and has a time-domain counterpart which is discussed in the next paragraph.

[0170] In the following, the electrical end-to-end channel is considered.

[0171] To derive the discrete-time end-to-end channel model the filter cascade of the linear receiver frontend is subsequently applied to ΔU(L, ω). The perturbation ΔS (ω) (i.e., the perturbation in the electrical domain following our terminology, cf. FIG. 3c) is obtained by

ΔS(ω)=H.sub.C*(L, ω)ΔU(L, ω),   (39)

[0172] which cancels out the leading term H.sub.C(L, ω) in (28) since |H.sub.C(L, ω)|=1. The result is shown in (32) at the bottom of this page. Remarkably, there exists an equivalent time-domain representation Δs(t) ΔS(ω) shown in (33) where the Fourier relation is derived in Appendix A. The time-domain perturbation Δs(t) has the same form as its frequency-domain counterpart, i.e., the integrand is constituted by the respective time-domain representation of the optical signal and the double integral is performed over the time variables T.sub.1 and T.sub.2 (cf. FIGS. 4 (a) and [23], [30]).

[0173] The frequency matching with

[00042]${\omega }_3\stackrel{\mathrm{def}}{=}\omega -{\omega }_1+{\omega }_2$

[0174] is translated to a temporal matching 8

[00043]$t_3\stackrel{\mathrm{def}}{=}t-t_1+t_2$

[0175] (cf. [31 ]), i.e., the selection rules of FWM apply both in time and frequency. The temporal matching is not to be confused with the phase matching condition in (34), (36).

[0176] Remarkably, the time-domain kernel h.sub.NL(T.sub.1, T.sub.2) is related to h.sub.NL(υ.sub.1, υ.sub.2) by an inverse two-dimensional (2D) Fourier transform (cf. [30, Appx.] and [28, Eq. (6)]) which can be written as

[00044]$\begin{array}{cc}\begin{array}{c}{h}_{NL}\left({\tau }_1,{\tau }_2\right)=h_{NL}\left(\tau \right)={-1}^{}\left\{H_{NL}\left(v\right)\right\}\\ =\frac{1}{L_{\mathrm{eff}}}{\int }_0^L\frac{1}{2\pi .Math.ℬ\left(\zeta \right).Math.}\exp \left(\left(\zeta \right)-j\frac{{\tau }_1{\tau }_2}{ℬ\left(\zeta \right)}\right)d\zeta \\ \stackrel{ℬ\left(z\right)\le 0}{=}\frac{j}{L_{\mathrm{eff}}}{\int }_0^L{h_C^{*}\left(\zeta ,\sqrt{{\tau }_1{\tau }_2}\right)}^{2}d\zeta ,\end{array}& \left(40\right)\end{array}$

[0177] with the tuples T=[T.sub.1, T.sub.2].sup.T and υ=[υ.sub.1υ.sub.2].sup.T. The time-domain kernel maintains its hyperbolic form as it is a function of the product T.sub.1T.sub.2. Also note the duality to (34), where in both representations the nonlinear transfer function can be understood as the path-average (cf. [32]) over an expression related to the linear channel response h.sub.C(z,t) H.sub.C(z,ω). Note, that in (40) the condition on β(z)≤0 (which is typically fulfilled in the anomalous dispersion regime with β.sub.2<0) is used to obtain the simple result without cumbersome differentiation of the term |β(z)|.

[0178] The next step is to resolve the perturbation Δs (t) ΔS(ω) into contributions originating from SCI, XCI or multichannel interference (MCI). We notice from FIG. 6 that, given R.sub.s is sufficiently large, |H.sub.NL(ξ)|.sup.2 vanishes if ϵ»(2πR.sub.s), i.e., if the phase matching condition is not properly met. Conversely, if the spectral width of |N.sub.NL(ξ/R.sub.s.sup.2)|.sup.2 (or equivalently the inverse map strength 1/δ.sub.T,ρ) is small enough, the integrand in (32), (33) can be factored into a SCI and XCI term, i.e., mixing terms that originate either from within the probe channel (both υ.sub.1<2πR.sub.s and υ.sub.2<2πR.sub.s) or from within the probe channel and a single interfering wavelength channel (either υ.sub.1<2πR.sub.s, or υ.sub.2<2πR.sub.s). Mixing terms originating from MCI are only relevant for small R.sub.s. We hence neglect any FWM terms involving more than two wavelength channels.

[0179] The optical field envelope u(0, t) U(0, ω) in (32), (33) is now expanded according to (12), (13). By definition we have Δω.sub.ρ=0 and we can expand the triple product of U(0, ω) in (32) as

[00045]$\begin{array}{cc}{\mathrm{UU}}^{H}U=\underset{\underset{\mathrm{SCI}}{︸}}{U_{\rho }{U}_{\rho }^H{U}_{\rho }}+\underset{v\ne \rho }{.Math.}\underset{\underset{\mathrm{XCI}}{︸}}{\left(U_v{U}_v^H{U}_{\rho }+U_{\rho }{U}_v^H{U}_v\right)}& \left(41\right)\end{array}$

[0180] where the frequency-dependency of U(0, ω) is omitted for short notation. The XCI term has two contributionsthe first results from an interaction where ω.sub.3 and ω.sub.2 are from the v.sup.th interfering wavelength channel and ω and ω.sub.1 are within the probe's support (υ.sub.2.fwdarw.Δω.sub.v in FIG. 4 (b)). The second involves an interaction where ω.sub.2 and ω.sub.1 are from the interfering wavelength channel and ω and ω.sub.3 are from the probe channel (υ.sub.1.fwdarw.Δω.sub.v).

[0181] We can exploit the symmetry of the nonlinear transfer function H.sub.NL(υ.sub.1, υ.sub.2)=H.sub.NL(υ.sub.2, υ.sub.1) to simplify the XCI expression in (41). Since U.sub.v.sup.HU.sub.v is a scalar, we have U.sub.ρU.sub.v.sup.HU.sub.v=U.sub.v.sup.HU.sub.vU.sub.ρ. The 2×2 identity matrix I is used to factor the XCI expression in a v- and ρ-dependent term. We obtain with the definition of the electrical signal of each wavelength channel (cf. (12), (13)) after rearranging some terms

[00046]$\begin{array}{cc}U\left(0,{\omega }_3\right)U^H\left(0,{\omega }_2\right)U\left(0,{\omega }_1\right)H_{NL}\left({\omega }_2-{\omega }_3,{\omega }_2-{\omega }_1\right)=S_{\rho }\left({\omega }_1\right)S_{\rho }^H\left({\omega }_2\right)S_{\rho }\left({\omega }_3\right)H_{NL}\left({\omega }_2-{\omega }_1,{\omega }_2-{\omega }_3\right)+\underset{v\ne \rho }{.Math.}\left(S_v\left({\omega }_1\right)S_v^H\left({\omega }_2\right)+S_v^H\left({\omega }_2\right)S_v\left({\omega }_1\right)I\right)S_{\rho }\left({\omega }_3\right)\times H_{NL}\left(\underset{\underset{v_2}{︸}}{{\omega }_2-{\omega }_1},\underset{\underset{v_1}{︸}}{{\omega }_2-{\omega }_3}-\Delta w_v\right),& \left(42\right)\end{array}$

[0182] which now corresponds to the case that ω.sub.3 always lays in the support of the probe 10. The signals of the interfering wavelength channels are now represented in their respective ECB and the relative frequency offset Δω.sub.v is accounted for in the modified nonlinear transfer function H.sub.NL.

[0183] At this point, considering (32) and (42), we formulated the relation between the perturbation at the probe ΔS(ω) after chromatic dispersion compensation and the transmit spectra S.sub.v(ω) of the probe and the interferers in their respective baseband. The remaining operation in the receiver cascade is to perform matched filtering w.r.t. the transmit pulse and then to perform T-spaced sampling. An alternative formulation with ω1 in the support of the probe is obtained by exchanging the subscripts of ω1 and ω3 in frequency-domain and t1 and t3 in time-domain.

[0184] Now, the discrete-time end-to-end channel is considered.

[0185] We recap that the periodic spectrum X(e.sup.jωT) of the sampled signal

[00047]$x\left[k\right]\stackrel{\mathrm{def}}{=}x\left(\mathrm{kT}\right)$

[0186] is related to the aliased spectrum of the continuous-time signal x(t) over the Nyquist interval by

[00048]$\begin{array}{cc}X\left(e^{j\omega T}\right)\stackrel{\mathrm{def}}{=}\mathrm{ALIAS}\left\{X\left(\omega \right)\right\}=\frac{1}{T}\underset{m\in \mathbb{Z}}{.Math.}X\left(\omega -\frac{2\pi m}{T}\right).& \left(43\right)\end{array}$

[0187] The matched filter H.sub.T*(ω) and the aliasing operator are used to translate (32), (33) to the equivalent discrete-time form in (37), (38) exemplarily for the SCI contribution Δa.sup.SCI. The total perturbation inflicted on the probe channel is Δa[k]=Δa.sup.SCI[k]+Δa.sup.SCI[k]. In (37), (38) we use the 1/T-periodic spectrum A(e.sup.jωT) which is related to the discrete-time sequence a[k] by a DTFT A(e.sup.jωT)={a[k]}. The channel-dependent nonlinear length is

[00049]$L_{\mathrm{NL},\nu }\stackrel{\mathrm{def}}{=}1/\left(\gamma P_{\nu }\right)$

and P.sub.v is the optical launch power of the v.sup.th wavelength channel. The normalized nonlinear end-to-end transfer function H.sub.v(ω)=H.sub.v(ω.sub.1, ω.sub.2, ω.sub.3) characterizes the nonlinear cross-talk from the v.sup.th wavelength channel to the probe channel. In particular, H.sub.ρ(ω) describes SCI and H.sub.v(ω) with v≠ρ describes XCI. It is defined as

[00050]$\begin{array}{cc}{H}_v\left(\omega \right)\stackrel{\mathrm{def}}{=}T.Math.H_{T,v}\left({\omega }_1\right)T.Math.H_{T,v}^{*}\left({\omega }_2\right)/P_v\times T.Math.H_{T,\rho }\left({\omega }_3\right)T.Math.H_{T,\rho }^{*}\left({\omega }_1-{\omega }_2+{\omega }_3\right)/E_{T^{\prime }}\times H_{NL}\left({\omega }_2-{\omega }_1,{\omega }_2-{\omega }_3-\Delta {\omega }_v\right),& \left(44\right)\end{array}$

[0188] and its periodic continuation, i.e., the aliased discrete-time equivalent is given by

[00051]$\begin{array}{cc}{H}_v\left(e^{j\omega T}\right)=\frac{1}{T^3}\underset{m\in {\mathbb{Z}}^3}{.Math.}{H}_{\nu }\left(\omega -\frac{2\pi m}{T}\right),& \left(45\right)\end{array}$

[0189] where the three-fold aliasing is done along each frequency dimension with ω=[ω.sub.1, ω.sub.2, ω.sub.3].sup.T and m=[m.sub.1, m.sub.2, m.sub.3].sup.T. The normalization in (44) is done s.t. H.sub.ρ(e.sup.j0T)=1 and dimensionless. Note, that by definition the optical launch power P.sub.v of the v.sup.th wavelength channel is related to the pulse energy of H.sub.T,v(ω) in (9), (10).

[0190] The nonlinear end-to-end transfer function in (44) depends on the characteristics of the transmission link, comprised by H.sub.NL(.Math., .Math.), the characteristics of the pulse-shapes of the probe and interfering wavelength channel (assuming matched filtering w.r.t. the channel and the probe's transmit pulse) and the frequency offset Δω.sub.v between probe and interferer.

[0191] It is remarkable that the integration in (37) is over the twofold tuple [ω.sub.1, ω.sub.2].sup.T while the time-domain summation in (38) is over three independent variables k=[k.sub.1, k.sub.2, k.sub.3].sup.T ∈.sup.3. This is a consequence of the time-frequency relation between convolution and element-wise multiplication. The temporal matching used for the optical field in (33) is now canceled in (38) due to the convolution with the matched filter h.sub.T*(−T), i.e., k.sub.3 does not depend on k.sub.1 and k.sub.2 unlike

[00052]$t_3\stackrel{\mathrm{def}}{=}t-t_1+t_2.$

[0192] Note, that the frequency variable ω.sub.3 in (37) still complies with the frequency matching ω.sub.3=ω−ω.sub.1+ω.sub.2 but may be outside the Nyquist interval . Due to the 1/T-periodicity of the spectrum A(e.sup.jωT) any frequency component outside , is effectively folded back into the Nyquist interval by addition of integer multiples of ω.sup.Nyq (denoted by the FOLD{.Math.} operation in (37)).

[0193] The XCI complement to (37) reads

[00053]$\begin{array}{cc}\Delta A^{\mathrm{XCI}}\left(e^{j\omega T}\right)=-j\underset{\nu \ne p}{.Math.}\frac{8}{9}\frac{L_{\mathrm{eff}}}{L_{NL,\nu }}\frac{T^2}{{\left(2\pi \right)}^2}{\int }_{{\mathrm{��}}^2}\times \left(B_{\nu }\left(e^{j{\omega }_{1}T}\right)B_{\nu }^H\left(e^{j{\omega }_{2}T}\right)+B_{\nu }^H\left(e^{j{\omega }_{2}T}\right)B_{\nu }\left(e^{j{\omega }_{1}T}\right)I\right)\times A\left(e^{j{\omega }_{3}T}\right)H_{\nu }\left(e^{j\omega T}\right)d^2\omega .& \left(46\right)\end{array}$

[0194] The time-domain description of the T-spaced channel model in (38) is equivalent to the pulse-collision picture (cf. [13, Eq. (3-4)] and [33, Eq. (3-4)]) and the XCI result is repeated here for completeness

[00054]$\begin{array}{cc}\Delta a^{\mathrm{XCI}}\left[k\right]=-j\underset{v\ne \rho }{.Math.}\frac{8}{9}\frac{L_{\mathrm{eff}}}{L_{NL,\nu }}\underset{\kappa \in {\mathbb{Z}}^3}{.Math.}\left(b_{\nu }\left[k+{\kappa }_1\right]b_{\nu }^H\left[k+{\kappa }_2\right]+b_{\nu }^H.Math.k+{\kappa }_2.Math.b_{\nu }.Math.k+{\kappa }_1.Math.I\right)a.Math.h+{\kappa }_3.Math.h_{\nu }.Math.\kappa .Math..& \left(47\right)\end{array}$

[0195] The time-domain and aliased frequency-domain kernel are related by a three-dimensional (3D) DTFT according to

h.sub.v[k]=.sup.−1{H.sub.v(e.sup.jωT)}.   (48)

[0196] The kernel h.sub.v[k]=h.sub.v[k.sub.1, k.sub.2, k.sub.3] is equivalent to the kernel derived via an integration over time and space in [10, Eq. (61), (62)] and used in [13].

[0197] Now, the relation to the GN-model and to system design rules is explained.

[0198] Parseval's theorem applied to (48) yields

[00055]$\begin{array}{cc}{E}_{h,\nu }\stackrel{\mathrm{def}}{=}\underset{\kappa \in {\mathbb{Z}}^3}{.Math.}{.Math.h_{\nu }\left[\kappa \right].Math.}^2={\left(\frac{T}{2\pi }\right)}^3{\int }_{{\mathrm{��}}^3}{.Math.H_{\nu }\left(e^{j\omega T}\right).Math.}^2{d}^3\omega ,& \left(49\right)\end{array}$

[0199] where the right-hand side can be interpreted as an alternative formulation of the (frequency-domain) Gaussian noise (GN)-model [27] in 1/T-periodic continuous-frequency domain. In (49) the common pre-factor

[00056]${\left(\frac{8}{9}\frac{L_{\mathrm{eff}}}{L_{\mathrm{NL},\nu }}\right)}^2$

[0200] is omitted here and the energy in time- and frequency domain is calculated over the whole support of the probe and interfering wavelength channel, whereas [27, Eq. (1)] is evaluated only at a single frequency ω. Beyond that, to include all SCI and XCI contributions one needs to sum over all v—the GN-model in its standard form also includes MCI. This is the dual representation to the original work where the optical signal is constructed as a continuous-time signal with period T.sub.0 and discrete frequency components (c.f. the Karhunen-Loève formula in [26], [34]). In other words, the discretization in one domain and the periodicity in the other is exchanged in (49) compared to the GN-model. In this view, the result obtained by the GN-model corresponds to the kernel energy E.sub.h,v of the corresponding end-to-end channel.

[0201] At the same time, the (system relevant) variance of the perturbation

[00057]${\sigma }_{\Delta a}^2\stackrel{\mathrm{def}}{=}E\left\{{.Math.\Delta a.Math.}^2\right\}$

[0202] depends as well on the properties of the modulation format A which in turn is a problem addressed by the extended Gaussian noise (EGN)-model [34], cf. also the discussion in [5, Sec. F and Appx.]. Note, that the derivation of (49) does not require any assumptions on the signal (albeit its pulse-shape)—in particular no Gaussian assumption.

[0203] We can identify three relevant system parameters that characterize the nonlinear response: the map strength .sub.T,ρ=L.sub.eff/L.sub.D (or equivalently the v-dependent .sub.T,v=L.sub.eff/L.sub.wo,v) which is a measure of the temporal extent, i.e., the memory of the nonlinear interaction. Secondly, the (V-dependent) nonlinear phase shift

[00058]${\varphi }_{\mathrm{NL},v}\stackrel{\mathrm{def}}{=}\frac{8}{9}\frac{L_{\mathrm{eff}}}{L_{\underset{\_}{N}L,\nu }}$

[0204] that depends via L.sub.NL,v linearly on the launch power P.sub.v and essentially acts as a scaling factor to the nonlinear distortion Δα[k]. And at last, the total kernel energy E.sub.h,v which charactarizes the strength of the nonlinear interaction—independent of the launch power.

[0205] Now, applications to fiber nonlinearity compensation according to embodiments is described.

[0206] The derived channel models also finds applications for fiber nonlinearity compensation, where implementation complexity is of particular interest. An experimental demonstration of intra-channel fiber nonlinearity compensation based on the time-domain model in (38) has been presented in [35]. In terms of computational efficiency a frequency-domain implementation can be superior to the time-domain implementation, in particular, for cases where the number of nonlinear interacting pulses is large.

[0207] This is typically the case for large map strengths .sub.T,≯, large relative frequency offsets Δω.sub.v, i.e., large .sub.T,v, and pulse shapes h.sub.T(t) that extend over multiple symbol durations, e.g., a root-raised cosine (RRC) shape with small roll-off factor ρ. Then, the number of coefficients of the time-domain kernel h.sub.v[k] exceeding a relevant energy level grows very rapidly leading to a large number of multiplications and summations. The frequency-domain picture comprises only a double integral instead of a triple sum and can be efficiently implemented using standard signal processing techniques.

TABLE-US-00001 Algorithm 1: REG-PERT-FD for the SCI contribution  1 a.sub.λ[k] = overlapSaveSplit(  a[k]  , N.sub.DFT, K)  2 k, μ, μ.sub.1, μ.sub.2 ∈ {0, 1, . . . , N.sub.DFT − 1}  3 [00059]$H_{\rho }\left[{\mu }_1,{\mu }_2,{\mu }_3\right]=H_{\rho }\left[\mu \right]=H_{\rho }\left(e^{j\frac{2\pi }{N_{\mathrm{DFT}}}\mu }\right)$  4 forall λ do  5  A.sub.λ[μ] = DFT{a.sub.λ[k]}  6  forall μ do  7   μ.sub.3 = mod.sub.N.sub.DFT(μ − μ.sub.1 + μ.sub.2)  8   [00060]$\Delta A_{\lambda }^{\mathrm{SCI}}\left[\mu \right]=-j\frac{{\varphi }_{\mathrm{NL},\rho }}{N_{\mathrm{DFT}}^2}\times {.Math.}_{{\mu }_1,{\mu }_2}{A}_{\lambda }\left[{\mu }_1\right]A_{\lambda }^H\left[{\mu }_2\right]A_{\lambda }\left[{\mu }_3\right]H_{\rho }\left[{\mu }_1,{\mu }_2,{\mu }_3\right]$  9   Y.sub.λ.sup.PERT[μ] = A.sub.λ[μ] + ΔA.sub.λ.sup.SCI[μ] 10  end 11  y.sub.λ.sup.PERT[k] = DFT.sup.−1{Y.sub.λ.sup.PERT[μ]} 12 end 13 y.sup.PERT[k]   = overlapSaveAppend(y.sub.λ.sup.PERT[k], N.sub.DFT, K)

[0208] Exemplarily for the SCI contribution, Algorithm 1 realizes the regular perturbation (REG-PERT) procedure in 1/T-periodic discrete frequency-domain (FD) corresponding to the continuous-frequency relation in (38). Here, the overlap-save algorithm is used to split the sequence α[k] into overlapping blocks α.sub.λ[k]A.sub.λ[μ] of size N.sub.DFT enumerated by the subindex λ ∈ [36]. The block size is equal to the size of the discrete Fourier transform (DFT) and the overlap between successive blocks is K. The one-dimensional DFT is performed on each vector component of α.sub.λ[k] and the correspondence always relates the whole blocks of length NDFT.

[0209] The aliased frequency-domain kernel is discretized to obtain the coefficients

[00061]$\begin{array}{cc}{H}_{\rho }\left[{\mu }_1,{\mu }_2,{\mu }_3\right]=H_{\rho }\left[\mu \right]\stackrel{\mathrm{def}}{=}{H}_{\rho }\left(e^{j\frac{2\pi }{N_{DFT}}\mu }\right)& \left(50\right)\end{array}$

[0210] where NDFT is the number of discrete-frequency samples. The discrete-frequency indices μ.sub.1 and μ.sub.2 are elements of the set {0, 1, . . . , N.sub.DFT−1} whereas μ.sub.3 must be (modulo) reduced to the same number set due to the 1/T-periodicity of ω.sub.3 in (37). The number of coefficients can be decreased by pruning, similar to techniques already applied to VSTF models [37]. However, note that in contrast to VSTF models the proposed algorithm operates on the 1/T-periodic spectrum of blocks of transmit symbols a.sub.λ[k] and the filter coefficients are taken from the aliased frequency-domain kernel. Line 8 of the algorithm effectively realizes equation (37) where the (double) sum is performed over all μ.sub.1 and μ.sub.2* After frequency-domain processing the blocks of perturbed receive symbols Y.sub.λ.sup.PERT[μ]y.sub.λ.sup.PERT[k] are transformed back to time domain where the N.sub.DFT-K desired output symbols of each block are appended to obtain the perturbed sequence y.sup.PERT[k]. Algorithm 1 can be generalized to XCI analogously to (46).

[0211] According to an embodiment, Algorithm 1 for XCI reads as follows:

TABLE-US-00002 Algorithm 1: REG-PERT-FD for the XCI contribution of the v.sup.th wavelength channel  1 a.sub.λ[k] = overlapSaveSplit(  a[k]  , N.sub.DFT, K)  2 b.sub.λ[k] = overlapSaveSplit(  b.sub.v[k]  , N.sub.DFT, K)  3 k, μ, μ.sub.1, μ.sub.2 ∈ {0, 1, . . . , N.sub.DFT − 1}  4 [00062]$H_v\left[{\mu }_1,{\mu }_2,{\mu }_3\right]=H_v\left[\mu \right]=H_v\left(e^{j\frac{2\pi }{N_{\mathrm{DFT}}}\mu }\right)$  5 forall λ do  6  A.sub.λ[μ] = DFT{a.sub.λ[k]}  7  B.sub.λ[μ] = DFT{b.sub.λ[k]}  8  forall μ do  9   μ.sub.3 = mod.sub.N.sub.DFT(μ − μ.sub.1 + μ.sub.2) 10   [00063]$\Delta A_{\lambda }^{\mathrm{XCI}}\left[\mu \right]=-j\frac{{\varphi }_{\mathrm{NL},v}}{N_{\mathrm{DFT}}^2}\times {.Math.}_{{\mu }_1,{\mu }_2}\left(B_{\lambda }\left[{\mu }_1\right]B_{\lambda }^H\left[{\mu }_2\right]+B_{\lambda }^H\left[{\mu }_2\right]B_{\lambda }\left[{\mu }_1\right]I\right)\times A_{\lambda }\left[{\mu }_3\right]H_v\left[{\mu }_1,{\mu }_2,{\mu }_3\right]$ 11   Y.sub.λ.sup.PERT[μ] = A.sub.λ[μ] + ΔA.sub.λ.sup.XCI[μ] 12  end 13  y.sub.λ.sup.PERT[k] = DFT.sup.−1{Y.sub.λ.sup.PERT[μ]} 14 end 15 y.sup.PERT[k]   = overlapSaveAppend(y.sub.λ.sup.PERT[k], N.sub.DFT, K)

[0212] The time- and frequency-domain picture of the regular perturbation approach are equivalent due to the DTFT in (37), (38) which interrelates both representations. Algorithm 1 represents a practical realization in discrete-frequency which produces the same (numerical) results as the discrete-time model as long as N.sub.DFT and K are chosen sufficiently large for a given system scenario. To that end, below, the regular discrete-time and -frequency model and the reference channel model implemented via the SSFM are compared. Then, the regular model is extended to a combined regular-logarithmic model where a subset of the perturbations are considered as multiplicative, i.e., perturbations that cause a rotation in phase or in the state of polarization (SOP).

[0213] Now, a regular-logarithmic model in the discrete-time domain is provided.

[0214] It was already noted in [38] that the regular VSTF approach (or the equivalent RP method) in (26) reveals an energy-divergence problem if the optical launch power P is too high—or more precisely if the nonlinear phase shift ϕ.sub.NL is too large.. Using a first-order RP approach, a pure phase rotation is approximated by exp(jϕ)≈1+jϕ. While multiplication with exp(jϕ) is an energy conserving transformation (i.e., the norm is invariant under phase rotation), the RP approximation is obviously not energy conserving. In the context of optical transmission, already a trivial (time-constant) average phase rotation due nonlinear interaction is not well modeled by the RP method.

[0215] This inconsistency was first addressed in the early 2000s [4], [39] and years later revived in the context of intra-channel fiber nonlinearity mitigation. E.g. in [40], [41] it turned out that a certain subset of symbol combinations in the time-domain RP model deterministically creates a perturbation oriented into the -j-direction from the transmit symbol a[k]. Similarly, in the pulse-collision picture [11]-[13] a subset of degenerate cross-channel pulse collisions were properly associated to distortions exhibiting a multiplicative nature. In the same series of contributions, these subsets of degenerate, in the sense that not all four interacting pulses are distinct, distortions were first termed two- and three-pulse collisions, i.e., symbol combinations k ∈.sup.3 in (47) with k.sub.3=0 in our terminology. While the pulse collision picture covers only cross-channel effects, we will extent the analysis also to intra-channel effects.

[0216] In this context, we review some properties of the kernel coefficients relevant for inter-channel (v≠ρ) two- and three-pulse collisions [13]

h.sub.v[k.sub.1, k.sub.2, 0]∈. if k.sub.1=k.sub.2   (51)

h.sub.v[k.sub.1, k.sub.2, 0]=h.sub.v*[k.sub.2, k.sub.1, 0]∈if k.sub.1≠k.sub.2,   (52)

[0217] where two-pulse collisions with k.sub.1=k.sub.2 in (51) are doubly degenerate and the kernel is real-valued. The transmit pulse-shape h.sub.T (t) is assumed to be a real-valued (root) raised-cosine.

[0218] In case of three-pulse collisions, the kernel is generally complex-valued but due to its symmetry property in (52) and the double sum over all (nonzero) pairs of [k.sub.1, k.sub.2].sup.T in (47) the overall effect is still multiplicative.

[0219] Additionally, for intra-channel contributions (v=ρ) we find the following symmetry properties of the kernel

h.sub.ρ[k.sub.1, k.sub.2, k.sub.3]=h.sub.ρ[k.sub.3, k.sub.2, k.sub.1]  (53)

h.sub.ρ[k.sub.1, k.sub.2, k.sub.3]=h.sub.ρ[-k.sub.1, -k.sub.2, -k.sub.3],   (54)

[0220] and we identify a second degenerate case with k.sub.1=0 as source for multiplicative distortions, cf. the symmetric form of (38) w.r.t. k.sub.1 and k.sub.3.

[0221] In the following, the original RP solution is modified such that perturbations originating from certain degenerate mixing products are associated with a multiplicative perturbation. Similar to [13], [41], [42], we extend the previous RP model to a combined regular-logarithmic model. It takes the general form of

y[k]=exp (jΦ[k]+j{right arrow over (s)}[k].Math.{right arrow over (σ)}) (a[k]+Δa[k]).   (55)

[0222] In addition to the regular, additive perturbation Δa[k] we now also consider a phase rotation by exp(jΦ[k]) and a rotation in the state of polarization by exp(j{right arrow over (s)}[k].Math.{right arrow over (σ)}). Here, exp(.Math.) denotes the matrix exponential. All perturbative terms combine both SCI and XCI effects, i.e., the additive perturbation Δa[k] ∈.sup.2 is the sum of SCI and XCI contributions. The time-dependent phase rotation is given by exp(jΦ[k]) with the diagonal matrix Φ[k] ∈.sup.2×2 defined as

[00064]$\begin{array}{cc}\Phi \left[k\right]\stackrel{\mathrm{def}}{=}{\varphi }^{\mathrm{SCI}}\left[k\right]I+{\varphi }^{\mathrm{XCI}}\left[k\right]I,& \left(56\right)\end{array}$

[0223] i.e., we find a common phase term for both polarizations originating from intra- and inter-channel effects. The combined effect of intra- and inter-channel cross-polarization modulation (XPoIM) is expressed by the Pauli matrix expansion {right arrow over (s)}[k].Math.{right arrow over (σ)} ∈.sup.2×2 using (6), with the notation adopted from [20] and [43]. The expansion defines a unitary rotation in Jones space of the perturbed vector a[k]+Δa[k] around the time-dependent Stokes vector {right arrow over (s)}[k] and is explained in more detail in the following.

[0224] 1) SCI Contribution: TAT o discuss the SCI contribution we first introduce the following symbol sets

[00065]$\begin{array}{cc}{SCI}^{}\stackrel{\mathrm{def}}{=}\{{\left[{\kappa }_1,{\kappa }_2,{\kappa }_3\right]}^T\in {\mathbb{Z}}^3.Math.{.Math.h_{\rho }\left[\kappa \right]/h_{\rho }\left[0\right].Math.}^2>{\Gamma }^{\mathrm{SCI}}\}& \left(57\right)\\ {\varphi }_{\oplus }^{}\stackrel{\mathrm{def}}{=}{SCI}^{}.Math.{\kappa }_1=0⩓{\kappa }_2\ne 0⩓{\kappa }_3\ne 0\}& \left(58\right)\\ {\varphi }_{\ominus }^{}\stackrel{\mathrm{def}}{=}\{{SCI}^{}.Math.{\kappa }_3=0⩓{\kappa }_2\ne 0⩓{\kappa }_1\ne 0\}& \left(59\right)\\ {\varphi }_{SCI}^{}\stackrel{\mathrm{def}}{=}{\varphi }_{\oplus }^{}.Math.{\varphi }_{\ominus }^{}.Math.\left\{\kappa =0\right\}& \left(60\right)\\ {\Delta }_{SCI}^{}\stackrel{\mathrm{def}}{=}{SCI}^{}\setminus {\varphi }_{SCI}^{},& \left(61\right)\end{array}$

[0225] where (57) defines the base set including all possible symbol combinations that exceed a certain energy level Γ.sup.SCI normalized to the energy of the center tap at k=0. In (58), (59) the joint set of degenerate two- and three-pulse collisions for SCI are defined which follow directly from the kernel properties in (51),(52) for k.sub.3=0, and (53),(54) for k.sub.1=0. The set of indices for multiplicative distortions K.sub.ϕ.sup.SCI in (60) also includes the singular case k=0. Then, the additive set is simply the complementary set of K.sub.ϕ.sup.SCI) w.r.t. the base set .sup.SCI.

[0226] We start with the additive perturbation defined above in (38) which now reads

[00066]$\begin{array}{cc}\Delta a^{\mathrm{SCI}}\left[k\right]=-j{\varphi }_{\mathrm{NL},\rho }\underset{{\Delta }_{SCI}^{}}{.Math.}a\left[k+{\kappa }_1\right]a^H\left[k+{\kappa }_2\right]a\left[k+{\kappa }_3\right]h_{\rho }\left[\kappa \right],& \left(62\right)\end{array}$

[0227] where the triple sum is now restricted to the set K.sub.Δ.sup.SCI excluding all combinations which result in a multiplicative distortion, cf. (61).

[0228] To calculate the common phase ϕ.sup.SCI[k] and the intra-channel Stokes rotation vector {right arrow over (s)}.sup.SCI[k] we first analyse the expression a[k+k.sub.1]a.sup.H[k+k.sub.2]a[k+k.sub.3] from the original equation in (38). For the set K.sub.ϕ.sup.⊕ with k.sub.1=0 the triple product factors into the respective transmit symbol a[k] and a scalar value a.sup.H[k+k.sub.2]a[k+k.sub.3]. After multiplication with h.sub.ρ[0, k.sub.2, k.sub.3] and summation of all k ∈K.sub.ϕ.sup.⊕ the perturbation is strictly imaginary-valued (cf. symmetry properties in (53),(54)).

[0229] On the other hand, for K.sub.ϕ.sup.↑ with k.sub.3=0 we have to rearrange the triple product using the matrix expansion from (7) to factor the expression accordingly as16

aa.sup.Ha=½(a.sup.Ha I+(a.sup.H{right arrow over (σ)}a).Math.{right arrow over (σ)}) a.   (63)

[0230] (multiplication with h.sub.ρ[k] and summation over k ∈.sub.ϕ.sup.SCI are implied)

[0231] The first term a.sup.Ha I also contributes to a common phase term, whereas the second term (a.sup.H{right arrow over (σ)}a).Math.{right arrow over (σ)} ∈.sup.2×2 is a traceless and Hermitian matrix exp(j(a.sup.H{right arrow over (σ)}a).Math.{right arrow over (σ)}) is a unitary polarization rotation. Since the Pauli expansion {right arrow over (μ)}.Math.{right arrow over (σ)} in (6) is Hermitian, the expression exp(j {right arrow over (u)}.Math.{right arrow over (σ)}) is unitary.

[0232] The multiplicative perturbation exp(jϕ.sup.SCI[k] with ϕ.sup.SCI[k] ∈ is then given by

[00067]$\begin{array}{cc}\begin{array}{c}{\varphi }^{SCI}\left[k\right]=-{\varphi }_{\mathrm{NL},\rho }\underset{{\varphi }_{\oplus }^{}}{.Math.}{a}^H\left[k+{\kappa }_2\right]a\left[k+{\kappa }_3\right]h_{\rho }\left[\kappa \right]-\\ \frac{1}{2}{\varphi }_{\mathrm{NL},\rho }\underset{{\varphi }_{\ominus }^{}}{.Math.}{a}^H\left[k+{\kappa }_2\right]a\left[k+{\kappa }_1\right]h_{\rho }\left[\kappa \right]-\\ {\varphi }_{\mathrm{NL},\rho }{.Math.a\left[k\right].Math.}^2{h}_{\rho }\left[0\right]\end{array}& \left(64\right)\\ \begin{array}{c}=-\frac{3}{2}{\varphi }_{\mathrm{NL},\rho }\underset{{\varphi }_{\ominus }^{}}{.Math.}{a}^H\left[k+{\kappa }_2\right]a\left[k+{\kappa }_1\right]h_{\rho }\left[\kappa \right]-\\ {\varphi }_{\mathrm{NL},\rho }{.Math.a\left[k\right].Math.}^2{h}_{\rho }\left[0\right].\end{array}& \left(65\right)\end{array}$

[0233] Given a wide-sense stationary transmit sequence a[k], the induced nonlinear phase shift has a time-average value ϕ.sup.SCI, around which the instantaneous phase ϕ.sup.SCI[k] may fluctuate (cf. also [44]).

[0234] The instantaneous rotation of the SOP due to the expression exp(j{right arrow over (s)}.sup.SCI[k].Math.{right arrow over (σ)}) ∈C.sup.2×2 causes intra-channel XPoIM [45]. It is given by

[00068]$\begin{array}{cc}\begin{array}{c}{\stackrel{.fwdarw.}{s}}^{SCI}\left[k\right].Math.\stackrel{.fwdarw.}{\sigma }=-\frac{1}{2}\left({\varphi }_{\mathrm{NL},\rho }\right)\underset{{\varphi }_{\ominus }^{}}{.Math.}(2a\left[k+{\kappa }_1\right]a^H\left[k+{\kappa }_2\right]-\\ a^H\left[k+{\kappa }_2\right]a\left[k+{\kappa }_1\right]I)h_{\rho }\left[\kappa \right],\end{array}& \left(66\right)\end{array}$

[0235] where we made use of the relation in (6). The rotation matrix exp(j{right arrow over (s)}.sup.SCI[k].Math.{right arrow over (σ)}) is unitary and {right arrow over (s)}.sup.SCI[k].Math.{right arrow over (σ)} is Hermitian and traceless. The physical meaning of the transformation described in (66) is as follows: The perturbed transmit vector (a[k]+Δa[k]) in (55) is transformed into the polarization eigenstate {right arrow over (s)}[k] (i.e., into the basis defined by the eigenvectors of {right arrow over (s)}.sup.SCI[k].Math.{right arrow over (σ)}). There, both vector components receive equal but opposite phase shifts and the result is transformed back to the x/y-basis of the transmit vector. In Stokes space, the operation can be understood as a precession of ({right arrow over (a)}[k]+{right arrow over (a)}[k]) around the Stokes vector {right arrow over (s)}.sup.SCI[k] by an angle equal to its length ∥{right arrow over (s)}.sup.SCI[k]. The intra-channel Stokes vector {right arrow over (s)}.sup.SCI[k] depends via the nonlinear kernel h.sub.ρ[k] on the transmit symbols within the memory of the nonlinear interaction S.sub.T,ρ around a[k]. Similar to the nonlinear phase shift—for a wide-sense stationary input sequence—the Stokes vector {right arrow over (s)}.sup.SCI[k] has a time-constant average value around which it fluctuates over time.

[0236] 2) XCI Contribution: The same methodology is now applied to cross-channel effects. The symbol set definitions for XCI follow from the considerations described above.

[00069]$\begin{array}{cc}{\nu }_{XCi}^{}=\{{\left[{\kappa }_1,{\kappa }_2,{\kappa }_3\right]}^T\in {\mathbb{Z}}^3.Math.{.Math.h_{\nu }\left[\kappa \right]/h_{\nu }\left[0\right].Math.}^2>{\Gamma }_{\nu }^{\mathrm{XCI}}\}{\varphi ,\nu }_{XCI}^{}\stackrel{\mathrm{def}}{=}d.Math.f\left\{K_{\nu }^{\mathrm{XCI}}|{\kappa }_3=0⩓{\kappa }_2\ne 0⩓{\kappa }_1\ne 0\right\}& \left(67\right)\\ .Math.\left\{\kappa =0\right\}& \left(68\right)\\ {\Delta ,\nu }_{XCI}^{}\stackrel{\mathrm{def}}{=}{\nu }_{\mathrm{XCI}}^{}\setminus {\varphi ,\nu }_{XCI}^{},& \left(69\right)\end{array}$

[0237] where the subscript v indicates the channel number of the respective interfering channel. For .sub.ϕ,v.sup.XCI, only the degenerate case .sub.3=0 has to be considered due to the kernel properties of h.sub.v[k.sub.1, k.sub.2, 0] in (51),(52). Similar to (63), the expression bb.sup.H+b.sup.Hb I from (47) is rearranged to obtain

[00070]$\begin{array}{cc}\frac{3}{2}-\underset{b^H\mathrm{bI}}{\underbrace{\left[\begin{array}{cc}{b}_x{b}_x^{*}+b_y{b}_y^{*}& 0\\ 0& b_y{b}_y^{*}+b_x{b}_x^{*}\end{array}\right]}}+\frac{1}{2}\underset{2{\mathrm{bb}}^H-b^H\mathrm{bI}=(b^h\stackrel{.fwdarw.}{\sigma }b).Math.\stackrel{.fwdarw.}{\sigma }}{\underbrace{\left[\begin{array}{cc}{b}_x{b}_x^{*}+b_y{b}_y^{*}& 2b_x{b}_y^{*}\\ 2b_y{b}_x^{*}& b_y{b}_y^{*}+b_x{b}_x^{*}\end{array}\right]}},& \left(70\right)\end{array}$

[0238] where the argument and subscript v is omitted for concise notation. The multiplicative cross-channel contribution is again split into a common phase shift in both polarizations and an equal but opposite phase shift in the basis given by the instantaneous Stokes vector of the v.sup.th interferer. We define the total, common phase shift due to cross-channel interference as

[00071]$\begin{array}{cc}{\varphi }^{\mathrm{XCI}}\left[k\right]=-\underset{\nu \ne \rho }{.Math.}\frac{3}{2}{\varphi }_{NL,\nu }\underset{{\varphi ,\nu }_{\mathrm{XCI}}^{}}{.Math.}{b}_{\nu }^H\left[k+{\kappa }_1\right]b_{\nu }\left[k+{\kappa }_2\right]h_{\nu }\left[\kappa \right]& \left(71\right)\end{array}$

[0239] which depends on the instantaneous sum over all interfering channels and the sum of b.sub.v.sup.Hb.sub.v bv over [k.sub.1, k.sub.2].sup.T. The effective, instantaneous cross-channel Stokes vector {right arrow over (s)}[k] is given by

[00072]$\begin{array}{cc}\begin{array}{c}{\stackrel{.fwdarw.}{s}}^{\mathrm{XCI}}\left[k\right].Math.\stackrel{.fwdarw.}{\sigma }=-\underset{\nu \ne p}{.Math.}\frac{1}{2}{\varphi }_{NL,\nu }\underset{{\varphi ,\nu }_{\mathrm{XCI}}^{}}{.Math.}(2b_{\nu }\left[k+{\kappa }_1\right]b_{\nu }^H\left[k+{\kappa }_2\right]-\\ b_{\nu }^H\left[k+{\kappa }_2\right]b_{\nu }\left[k+{\kappa }_1\right]I)h_{\nu }\left[\kappa \right].\end{array}& \left(72\right)\end{array}$

[0240] Note, that the expressions in (71), (72) include both contributions from two- and three pulse collisions (cf. [13, Eq. (10)- (13)]).

[0241] 3) Energy of Coefficients in Discrete-Time Domain: The energy of the kernel coefficients is defined according to Parseval's theorem in (49) for the subsets given in (57-61). We find for the different symbol sets

[00073]$\begin{array}{cc}{E}_h^{\mathrm{SCI}}\stackrel{\mathrm{def}}{=}\underset{{\varphi }_{SCI}^{}}{.Math.}{.Math.h_{\rho }\left[\kappa \right].Math.}^2& \left(73\right)\\ E_{h,\Delta }^{SCI}\stackrel{\mathrm{def}}{=}\underset{{\varphi }_{SCI}^{}}{.Math.}{.Math.h_{\rho }\left[\kappa \right].Math.}^2& \left(74\right)\\ E_{h,\varphi }^{SCI}\stackrel{\mathrm{def}}{=}\underset{{\varphi }_{SCI}^{}}{.Math.}{.Math.h_{\rho }\left[\kappa \right].Math.}^2,& \left(75\right)\end{array}$

[0242] with the clipping factor Γ.sup.SCI in (57) equal to zero. The energy for cross-channel effects is defined accordingly with the sets from (67-69). Since the subsets for additive and multiplicative effects are always disjoint we have E.sub.h=E.sub.h,Δ+E.sub.h,ϕ.

[0243] Now, a regular-logarithmic model in frequency domain is provided.

[0244] Similar to the above, we first review some kernel properties of the aliased frequency-domain kernel coefficients

[00074]$\begin{array}{cc}\begin{array}{cc}{H}_v\left(e^{j\omega T}\right)\in ℝ,& \mathrm{if}{\omega }_2={\omega }_1\iff {\omega }_3=\omega \iff v_2=0,\end{array}& \left(76\right)\\ \begin{array}{cc}{H}_{\rho }\left(e^{j\omega T}\right)\in ℝ,& \mathrm{if}{\omega }_2={\omega }_1\iff {\omega }_3=\omega \iff v_2=0\\ & \bigvee {\omega }_2={\omega }_3\iff {\omega }_1=\omega \iff v_1=0,\end{array}& \left(77\right)\end{array}$

[0245] where the two (doubly) degenerate cases ω.sub.1=ω.sub.2 and ω.sub.3=ω.sub.2 correspond to classical inter- and intra-channel cross-phase modulation (XPM). Accordingly, the frequency domain model is now modified such that these contribution will be associated with multiplicative distortions. However, due to the multiplicative nature, only average values can be incorporated into the frequency-domain model as they are both constant over time and frequency and can be treated as a common pre-factor in both pictures. We will see in the following that this already leads to significantly improved results compared to the regular model. Note that, in contrast to the regular models, the regular-logarithmic model in time and frequency are no longer equivalent.

[0246] The general form of the combined regular-logarithmic model in frequency is given by

Y(e.sup.jωT)=exp (j{right arrow over (Φ)}+J{right arrow over (S)}.Math.{right arrow over (σ)})×(A(e.sup.jωT)+ΔA(e.sup.jωT)),   (78)

[0247] where the phase- and polarization-term take on a frequency-constant value, i.e., independent of e.sup.jωT (and vice-versa independent of k in the time-domain picture). Following the same terminology as before, we introduce the average multiplicative perturbation of the common phase term

[00075]$\begin{array}{cc}\stackrel{\_}{\Phi }\stackrel{\mathrm{def}}{=}{\stackrel{\_}{\varphi }}^{\mathrm{SCI}}I+{\stackrel{\_}{\varphi }}^{\mathrm{XCI}}I,& \left(79\right)\end{array}$