Generalized Wiener filtering for positive alpha-stable random variables Filtrage de Wiener généralisé pour des variables aléatoires positives alpha-stables

This report provides a mathematical proof of a result which is a generalization of Wiener filtering to Positive α-stable (PαS) distributions, a particular subclass of the α-stable distributions family whose support is [0;+∞[. PαS distributions are useful to model nonnegative data and since they are heavy-tailed, they present a natural robustness to outliers. In applications such as nonnegative source separation, it is paramount to have a way of estimating the isolated components that constitute a mixture. To address this issue, we derive an estimator of the sources which is given by the conditional expectation of the sources knowing the mixture. It extends the validity of the generalized Wiener filtering to PαS distributions. This allows us to extract the underlying PαS sources from their mixture.