We derive analytic formulas for the improvement in effective optical signal-to-noise ratio brought by a digital nonlinear compensator for dispersion uncompensated links. By assuming Gaussian distributed nonlinear noise, we are able to take both nonlinear signal-to-signal and nonlinear signal-to-noise interactions into account. In the limit of weak nonlinear signal-to-noise interactions, we derive an upper boundary of the OSNR improvement. This upper boundary only depends on fiber parameters as well as on the total bandwidth of the considered wavelength-division multiplexing (WDM) signal and the bandwidth available for back propagation. We discuss the dependency of the upper boundary on different fiber types and also the OSNR improvement in practical system conditions. Furthermore, the analytical formulas are validated by numerical simulations.
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