Polarization-multiplexed nonlinear inverse synthesis with standard and reduced-complexity NFT processing: erratum

Stella Civelli; Sergei K. Turitsyn; Marco Secondini; Jaroslaw E. Prilepsky

doi:10.1364/OE.27.003617

Abstract

We correct a formula for the numerical nonlinear Fourier transform in [1]. The conclusions of our work are unchanged.

In Section 3.1 of [1] the single-step transfer matrix $U^{(n)}$ , Eq. (20) and its derivative Eq. (25) are not correct. The transfer matrix—obtained via eigenvalue decomposition—should be

U_{m, l}^{(n)} = {\begin{array}{l} c_{0} - j λ s_{0} & m = l = 1, \\ q_{l - 1}^{(n)} s_{0} & m = 1 and l \geq 2, \\ - σ q_{m - 1}^{(n) *} s_{0} & m \geq 2 and l = 1, \\ r_{m - 1, l - 1} [c_{0} + j λ s_{0} - e^{j λ δ}] & m = 2 and l \geq 3 or l = 2 and m \geq 3, \\ r_{m - 1, l - 1} [c_{0} + j λ s_{0}] + e^{j λ δ} (1 - r_{m - 1, m - 1}) & m = l = 2 or m, l \geq 3, \end{array}

where

r_{m l} = q_{m}^{(n) *} q_{l}^{(n)} / \sum_{k = 1}^{M} | q_{k}^{(n)} |^{2}

. The derivative

{U^{'}}^{(n)}

follows from (1). Equation (20) in [1], though not exact, is a first order approximation of (1). The use of Eq. (20) in [1] rather than (1) yields different performance at higher powers and oversampling factors, see Figs. 1(a) and 1(b) for M = 2. However, since the oversampling factor in [1] is

N_{D} = N_{I} = 4

, both matrices can be used without any variation. We verified that the performance metrics showed in [1] in Figs. 3(a) and 3(b) are unaffected when the new matrix (1) is used for the signal processing.

Fig. 1 NMSE—Eq. (37) of [1]—versus oversampling factor N_D, with N_I = 16 and N_s = 8.

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References

1. S. Civelli, S. Turitsyn, M. Secondini, and J. Prilepsky, “Polarization-multiplexed nonlinear inverse synthesis with standard and reduced-complexity NFT processing,” Opt. Express 26, 17360–17377 (2018). [CrossRef]

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Fig. 1

Fig. 1 NMSE—Eq. (37) of [1]—versus oversampling factor N_D, with N_I = 16 and N_s = 8.

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Equations (1)

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U_{m, l}^{(n)} = {\begin{array}{l} c_{0} - j λ s_{0} & m = l = 1, \\ q_{l - 1}^{(n)} s_{0} & m = 1 and l \geq 2, \\ - σ q_{m - 1}^{(n) *} s_{0} & m \geq 2 and l = 1, \\ r_{m - 1, l - 1} [c_{0} + j λ s_{0} - e^{j λ δ}] & m = 2 and l \geq 3 or l = 2 and m \geq 3, \\ r_{m - 1, l - 1} [c_{0} + j λ s_{0}] + e^{j λ δ} (1 - r_{m - 1, m - 1}) & m = l = 2 or m, l \geq 3, \end{array}

Abstract

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Equations (1)

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