  Coherent dual-polarization (DP) optical transmission systems encode
information on the four available degrees of freedom of an optical field: the
two polarization states, each with two quadrature components. Such systems
naturally operate based on a four-dimensional (4D) signal space. Having a
general analytical model to accurately estimate nonlinear interference (NLI) is
key to analyze such transmission systems as well as to study how different
DP-4D formats are affected by NLI. However, the available models in the
literature are not completely general. They either do not apply to the entire
DP-4D formats or do not consider all the NLI contributions. In this paper, we
develop a model that applies to all DP-4D modulation formats with independent
symbols. Our model takes self-channel interference, cross-channel interference
and multiple-channel interference effects into account. As an application of
our model, we further study the effects of signal-noise interactions in
long-haul transmission via the proposed model. When compared to previous
results in the literature, our model is more accurate at predicting the
contribution of NLI for both low and high dispersion fibers in single- and
multi-channel transmission systems. For the NLI, we report an average gap from
split step Fourier simulation results below 0.15 dB. The simulation results
further show that by considering signal-noise interactions, the proposed model
in long-haul transmission can reduce the transmission reach prediction error by
4%.
