A vector generalization of Costa entropy-power inequality and 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:
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 1
Year:
2009

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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. This 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 from the generalized Costa EPI and then used to establish the secrecy capacity regions of the degraded vector Gaussian broadcast channel with layered confidential messages.