Unformatted text preview: e minimization of I (X ; X ) can be restricted to conditional laws of the form
PX X (ˆ = 0x = 0) = 1
x
ˆ PX X (ˆ = 1x = 0) = 0
x
ˆ PX X (ˆ = 0x = 1) = p
x
ˆ PX X (ˆ = 1x = 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.
 Fall '11
 AmosLapidoth

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