Abstract
The nonlinear Schrödinger equation can be solved by split-step
methods, where in each step, linear dispersion and nonlinear effects are treated
separately. This paper considers the optimal design of an FIR filter as the
time-domain implementation for the linear part. The objective is to minimize
the integral of the squared error between the FIR frequency response and the
desired dispersion characteristics over the band of interest. This least square
(LS) problem is solved in two approaches: the normal equation approach gives
the explicit solution, whereas the singular value decomposition approach,
which is based on the theory of discrete prolate spheroidal sequences, provides
geometrical insights and reveals that the normal equation could be ill-conditioned.
In addition, the frequency response might exhibit singular behaviors such
as overshoot. We propose two filters that both can mitigate these shortcomings:
the regularized LS filter achieves this by adding a regularization term to
the objective function; the quadratically constrained quadratic programming-based
filter addresses overshooting more efficiently by imposing a maximum magnitude
constraint on the frequency response. Numerical results show that these filters
can suppress the overshoots, control the squared error, reduce the filter
length and lower the computational complexity. For both single channel and
wavelength-division multiplexing channels, the proposed methods generate similar
outputs as the standard split-step Fourier method.
© 2012 IEEE
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