Cramér–von Mises Criterion Based Parametric Mapping for Channel Model Substitution

Channel model substitution (CMS) has been shown to be a powerful toolkit for maintaining the equilibrium between the accuracy and mathematical tractability of channel models for complex analysis and optimization. To enable statistically robust parametric mapping for CMS applications, we generalize and study the Cramér-von Mises (C-V) criterion based parametric mapping in this paper. The C-V criterion seeks the optimal parametric mapping relations by minimizing the integrated squared error (ISE) between the cumulative distribution functions (CDFs) of the reference channel model and its distributional substitute. We analyze the corresponding parametric mapping problem based on the C-V criterion and rigorously prove several key analytical attributes. By taking the CMS techniques for the lognormal shadowed channel model as examples, we verify the generic analysis and illustrate the advantages of the C-V criterion based CMS parametric mapping over those enabled by classical moment matching (MM), Kullback-Leibler (K-L), and Kolmogorov-Smirnov (K-S) criteria. The proven lemmas and attributes for the C-V criterion based parametric mapping in this paper serve as a solid mathematical foundation for applying the C-V criterion to other more sophisticated CMS applications.