Generalization of Mrs. Gerber’s lemma

Mrs. Gerber’s Lemma (MGL) hinges on the convexity of $H(p*H^{-1}(u))$, where $H(u)$ is the binary entropy function. In this work, we prove that $H(p*f(u))$ is convex in $u$ for every $p \in [0, 1]$ provided $H(f(u))$ is convex in $u$, where $f(u) \colon (a, b) \to [0, \frac{1}{2}]$. Moreover, our result subsumes MGL and simplifies the original proof. We show that the generalized MGL can be applied in binary broadcast channel to simplify some discussion.

Keywords

Mrs. Gerber’s lemma, binary channel

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Published 31 October 2014