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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 (ˆx) =
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 = 0x = 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.
 Fall '11
 AmosLapidoth

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