In this paper we propose a generalization of the usual deBruijn identity that links the Shannon differential entropy (or the Kullback-Leibler divergence) and the Fisher information (or the Fisher divergence) of the output of a Gaussian channel. The generalization makes use of φ-entropies on the one hand, and of φ-divergences (of the Csiza`r class) on the other hand, as generalizations of the Shannon entropy and of the Kullback-Leibler divergence respectively. The generalized deBruijn identities induce the definition of generalized Fisher informations and generalized Fisher divergences; some of such generalizations exist in the literature. Moreover, we provide results that go beyond the Gaussian channel: we are then able to characterize a noisy channel using general measures of mutual information, both for Gaussian and non-Gaussian channels.

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