Existence and uniqueness of GEXIT curves via the Wasserstein metric

In the analysis of iterative coding systems it is often necessary to compare two densities and to measure how close they are. Sometimes it is convenient to compare their entropy or their Battacharyya parameter. But sometimes a more powerful measure is required. The Wasserstein metric is a convenient choice. We derive some basic properties of the Wasserstein metric which are important in the context of iterative coding. In particular, we will see how the Wasserstein metric compares to some other natural measures (such as the difference of entropies or Battacharyya parameters) and how the Wasserstein metric behaves under “natural” operations, like variable - or check-node convolution or under convex combinations. As an “application” we show how to prove the existence of the belief propagation Generalized EXIT curve for a non-trivial portion of the parameters.