Lower Bound on the Capacity of the Continuous-Space SSFM Model of Optical Fiber
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The capacity of a discrete-time model of optical fiber described by the split-step Fourier method (SSFM) as a function of the average input power 𝒫 and the number of segments in distance K is considered. It is shown that if K≥𝒫^2/3 and 𝒫→∞, the capacity of the resulting continuous-space lossless model is lower bounded by 1/2log_2(1+SNR) - 1/2+ o(1), where o(1) tends to zero with the signal-to-noise ratio SNR. As K→∞, the intra-channel signal-noise interactions average out to zero due to the law of large numbers and the SSFM model tends to a diagonal phase noise model. It follows that, in contrast to the discrete-space model where there is only one signal degree-of-freedom (DoF) at high powers, the number of DoFs in the continuous-space model is at least half of the input dimension n. Intensity-modulation and direct detection achieves this rate. The pre-log in the lower bound when K=𝒫^1/δ is generally characterized in terms of δ. We consider the SSFM model where the dispersion matrix does not depend on K, e.g., when the step size in distance is fixed. It is shown that the capacity of this model when K≥𝒫^3 and 𝒫→∞ is 1/2nlog_2(1+SNR)+ o(1). Thus, there is only one DoF in this model. Finally, it is shown that if the nonlinearity parameter γ→∞, the capacity of the continuous-space model is 1/2log_2(1+SNR)+ o(1) for any SNR.
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