D we have i x x h x h x x

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Unformatted text preview: ” X = 1 ˆ = Pr X = “?” . Hence, finding the rate distortion function consists of minimizing the mutual information under ˆ the constraint Pr X = “?” ≤ D . We have ˆ ˆ I (X ; X ) = H (X ) − H (X |X ) ˆ ˆ ˆ ˆ ˆ ˆ = 1 − Pr X = 0 H (X |X = 0) + Pr X = 1 H (X |X = 1) + Pr X = “?” H (X |X = “?”) =0 =0 ≤D ≤1 ≥ 1 − D · 1, where the last inequality is met with equality if PX |X (ˆ|x) = x ˆ 1 − D if x = x ˆ D if x = “?”. ˆ A simple coding scheme would be as follows. The encoder describes only the first (1 − D )n bits of X n . The decoder then pads the remaining D n p ositions with the erasure symbol. Rate Distortion Function with Infinite Distortion Problem 3 First, we note that for the computation of the rate distortion function we only need to consider ˆ D ∈ [0, 1/2], since for rate R = 0 we can achieve the distortion D = 1/2 by having X = 0 with probability 1. Note also that achieving finite distortion requires PX |X (ˆ = 0|x = 1) = 0. Hence, x ˆ ˆ th...
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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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