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Unformatted text preview: in assuming that minj d(i, j ) = 0 for all i ∈
{1, 2, . . . , m}, it remains to note that for every distortion measure d(·, ·), there exists a corresponding distortion measure d (·, ·) with minj d (i, j ) = 0 for every i ∈ {1, 2, . . . , m} (by choosing
wi = minj d(i, j )). And as we have shown, the rate distortion function R(·) then follows directly
from R (·) by
R(D ) = R (D − w ).
¯
c Amos Lapidoth, 2012 1 Erasure Distortion Problem 2 Note ﬁrst that for rate R = 0 the distortion D = 1 is achievable through the choice PX X (“?”x) = 1.
ˆ
Furthermore, d(1, 0) = ∞ and d(0, 1) = ∞ imply that the rate distortion function is certainly only
ˆ
ˆ
achieved by conditional laws satisfying Pr X = 0 X = 1 = 0 and Pr X = 1 X = 0 = 0, since
otherwise the expected distortion would be inﬁnite. It follows that
ˆ
E d(X, X ) = ˆ
Pr X = x X = x d(x, x)
ˆ
ˆ Pr[X = x]
x 1
=
2 x
ˆ ˆ
ˆ
ˆ
Pr X = 0 X = 0 · 0 + Pr X = “?” X = 0 · 1 + Pr X = 1 X = 1 · 0 ˆ
+ Pr X = “?” X = 1 · 1
= 1
2 ˆ
ˆ
Pr X = “?” X = 0 + Pr X = “?...
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 Fall '11
 AmosLapidoth

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