# Hence nding the rate distortion function consists of

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Unformatted text preview: ributed uniformly in {1, 2, . . . , m} and the distortion measure is 0 if x = x ˆ 1 if x = x. ˆ d(x, x) = ˆ ˆ ˆ First, we note that the expected distortion E d(X, X ) equals the probability that X diﬀers from X: ˆ E d(X, X ) = p(x) x p(ˆ|x)d(x, x) x ˆ x ˆ ˆ p(x) Pr X = x X = x = x ˆ = Pr X = X . Hence, ﬁnding the rate distortion function consists of minimizing the mutual information under the ˆ constraint Pr X = X ≤ D . To do so, we introduce the help variable E which indicates whether ˆ X equals X or not, i.e., ˆ 0 if X = X E= ˆ 1 if X = X. We further note that for 1, x = 1 ˆ 0, else PX (ˆ) = ˆx − ˆ the expected distortion equals mm 1 . For this choice of distribution we have I (X ; X ) = 0, and since − the rate distortion function R(D ) is nonincreasing in D it follows that R(D ) is zero for D ≥ mm 1 . m−1 Thus, in the following we only consider the case D &lt; m . c Amos Lapidoth, 2012 3 ˆ We have (considering Pr[E = 0] = Pr X = X ≤ D ) ˆ ˆ I (X ; X ) = H (X ) − H (X |X ) ˆ ˆ = H (X ) − H (X |X, E ) − H (E |X ) ˆ ˆ ≥ log m − Pr[E = 0] H (X |X, E = 0) − Pr[E = 1] H (X |X, E = 1) −H (E ) =0 ≤log(m−1) = log m − Pr[E = 0] log(m − 1) − Hb ( Pr[E = 0]) ˆ = Pr[X =X ]...
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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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