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A binary analog to the entropy-power inequality

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2 Author(s)
S. Shamai ; AT&T Bell Lab., Murray Hill, NJ, USA ; A. D. Wyner

Let {Xn}, {Yn} be independent stationary binary random sequences with entropy H( X), H(Y), respectively. Let h(ζ)=-ζlogζ-(1-ζ)log(1-ζ), 0⩽ζ⩽1/2, be the binary entropy function and let σ(X)=h-1 (H(X)), σ(Y)=h-1 (H(Y)). Let zn=XnYn , where ⊕ denotes modulo-2 addition. The following analog of the entropy-power inequality provides a lower bound on H(Z ), the entropy of {Zn}: σ(Z)⩾σ(X)*σ(Y), where σ(Z)=h-1 (H(Z)), and α*β=α(1-β)+β(1-α). When {Y n} are independent identically distributed, this reduces to Mrs. Gerber's Lemma from A.D. Wyner and J. Ziv (1973)

Published in:

IEEE Transactions on Information Theory  (Volume:36 ,  Issue: 6 )