Fig. 1. (a) Power distribution
along a fiber span in forward propagation, (b) constant step size
distribution backpropagation, and (c) logarithmic step size distribution
backpropagation
In a constant step size (CSS) distribution, the step size is uniform for
all steps across the span of the virtual fiber. For such schemes, it is
required to choose a very small step size, typically less than 100m, to
control the generation of fictitious four-wave mixing (FWM) artifacts
(24). Alternatively, by evaluating the smallest step, which allows a
10% accuracy on all in-band FWM artifacts, a constant step size of up
to 400m can be chosen (26). The selection of such a small constant step
size significantly compromises the computational speed of the SSFM
calculation. On the other hand, a large constant step size generates
fictitious FWM tones during SSFM calculation, degrading the accuracy of
nonlinear phase shift compensation.
Logarithmic Step size SSFM
Variable step sizes where the step size increases and power decreases
are usually preferred to the constant step size. The logarithmic step
size (LSS) distribution is an emerging non-uniform distribution
technique proposed to accurately estimate the nonlinear distortion in
fewer steps and suppress the generation of fictitious FWM tones. This
method aims to keep the average nonlinear phase shift after each step
constant (27). For a fiber span of length L , attenuation
coefficient α , the nth step size is given by (24).
33\* MERGEFORMAT ()
where N is the number of steps per span for SSFM calculation.
In (28), the slope coefficient of the conventional logarithmic
distribution has been chosen as 1 to reduce the relative global error. A
modified logarithmic step size (MLSS) has also been introduced. The
slope coefficient is modified as an optimized attenuation adjusting
factor to control the difference in step sizes of adjacent steps for
optimal performance (29). A generalized logarithmic step size
distribution scheme has been proposed to consider high symbol rates and
the number of steps by optimizing the base for the step size calculation
and the nonlinear coefficient scale factor (27).
Principle of Binary Logarithmic Step size
For a small number of steps, the conventional logarithmic step-size
method, which computes sizes using the natural logarithm (ln), results
in a significant variation of the signal waveform over one step and does
not produce optimal SSFM calculation. We present a binary logarithmic
step size (BLSS) for implementing the DBP, where the step size selection
of the conventional LSS is optimized using the binary logarithm
(log2). Also, the optimized factor introduced by (29)
for adjusting the attenuation coefficient is considered. The proposed
BLSS algorithm for SSFM step size selection is given by 4:
44\* MERGEFORMAT
()
where k is the attenuation adjusting factor.
The modification using the binary logarithm better approximates the
nonlinear distortions for compensation with higher accuracy than the
natural logarithmic step size technique. The proposed technique assumes
that the optimised target function is highly multimodal, with a large
number of local optima in terms of step size. This can make it difficult
for traditional optimization methods to converge to the global optimum.
To address this, the technique utilizes a binary logarithmic step size
distribution. The binary aspect allows the algorithm to explore the
complex search space efficiently by allowing for coarse and fine-grained
searches. On the other hand, the logarithmic scale enables the algorithm
to move swiftly across the search space, which helps to overcome the
issue of getting stuck in local optima.
This approach of controlling the variation of the signal waveform over
one step using a binary logarithmic step size is particularly useful in
digital backpropagation. Here, when a small number of steps per span is
used, and the attenuation factor k is optimized, it helps to
ensure that the difference between one step and the next is not too
substantial for nonlinear compensation. The modification produces
optimal step size distribution while maintaining a low computational
overhead.
Fig. 2 compares the variation of adjacent step sizes for different
compensation techniques. We use the optimized adjusting factor k= 0.4 determined by . For an 80 km fiber span, the step sizes for 10
steps using different selection techniques are given in Table 1.