A vector generalization of costa's entropy-power inequality with applications

Authors:
Ruoheng Liu;Tie Liu;H. Vincent Poor;Shlomo Shamai
Affiliations:
Department of Electrical Engineering, Princeton University, Princeton, NJ;Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX;Department of Electrical Engineering, Princeton University, Princeton, NJ;Department of Electrical Engineering, Technion-Israel Institute of Technology, Technion City, Haifa, Israel
Venue:
IEEE Transactions on Information Theory
Year:
2010

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Abstract

This paper considers an entropy-power inequality (EPI) of Costa and presents a natural vector generalization with a real positive semidefinite matrix parameter. The new inequality is proved using a perturbation approach via a fundamental relationship between the derivative of mutual information and the minimum mean-square error (MMSE) estimate in linear vector Gaussian channels. As an application, a new extremal entropy inequality is derived fromthe generalized Costa EPI and then used to establish the secrecy capacity regions of the degraded vector Gaussian broadcast channel with layered confidential messages.