But the expected distortion relates to the

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Unformatted text preview: e minimization of I (X ; X ) can be restricted to conditional laws of the form PX |X (ˆ = 0|x = 0) = 1 x ˆ PX |X (ˆ = 1|x = 0) = 0 x ˆ PX |X (ˆ = 0|x = 1) = p x ˆ PX |X (ˆ = 1|x = 1) = 1 − p, x ˆ ˆ with p ∈ [0, 1] satisfying E d(X, X ) ≤ D . But the expected distortion relates to the probability p like ˆ ˆ E d(X, X ) = Pr[X = 1] E d(X, X ) X = 1 p =. 2 c Amos Lapidoth, 2012 2 ˆ Hence, E d(X, X ) ≤ D is achieved if and only if p ∈ [0, 2D ]. On the other hand, the mutual ˆ information I (X ; X ) depends on p as ˆ ˆ I (X ; X ) = H (X ) − H (X |X ) ˆ ˆ = 1 − Pr X = 0 H (X |X = 0) 1 1+p 1 = 1 − (1 + p)Hb 2 . ˆ Since I (X ; X ) is monotonically decreasing in p (for p ∈ [0, 1]), its minimum (under the given constraints) is achieved by p = 2D . The rate distortion function is thus R(D ) = 1 − 1 + D Hb 2 1 1 + 2D , D ∈ 0, 1 . 2 Rate Distortion for Uniform Source with Hamming Distortion Problem 4 We aim for R(D ) = min ˆ PX |X :E[d(X,X )]≤D ˆ ˆ I (X ; X ), where X is dist...
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This note was uploaded on 05/18/2013 for the course EE Informatio taught by Professor Amoslapidoth during the Fall '11 term at Swiss Federal Institute of Technology Zurich.

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