  We rederive from first principles and generalize the theoretical framework of
the nonlinear Gaussian noise model to the case of coherent optical systems with
multiple fiber types per span and ideal Nyquist spectra. We focus on the
accurate numerical evaluation of the integral for the nonlinear noise variance
for hybrid fiber spans. This task consists in addressing four computational
aspects: (i) Adopting a novel transformation of variables (other than using
hyperbolic coordinates) that changes the integrand to a more appropriate form
for numerical quadrature; (ii) Evaluating analytically the integral at its
lower limit, where the integrand presents a singularity; (iii) Dividing the
interval of integration into subintervals of size pi and approximating the
integral in each subinterval by using various algorithms; and (iv) Deriving an
upper bound for the relative error when the interval of integration is
truncated in order to accelerate computation. We apply the proposed model to
coherent optical communications systems with hybrid fiber spans composed of
quasi-singlemode fiber and single-mode fiber segments. The accuracy of the
final analytical relationship for the nonlinear noise variance in long-haul
coherent optical communications systems with hybrid fiber spans is checked
using the split-step Fourier method and Monte Carlo simulation. It is shown to
be adequate to within 0.1 dBQ for the determination of the optimal fiber
segment lengths per span that maximize system performance.
