# But the expected distortion relates to the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online